The effect of an external magnetic field on the gas-liquid transition in the Ising spin fluid

4 Pages·0.21 MB·English

by  R. O. Sokolovskii

The effect of an external magnetic field on the gas-liquid transition in the Ising spin
fluid
R.O.Sokolovskii∗
Institute for Condensed Matter Physics, Svientsitskii 1, Lviv 290011, Ukraine.
9 The theoretical phase diagrams of the magnetic (Ising) lattice fluid in an external magnetic
9
field is presented. It is shown that, depending on the strength of the nonmagnetic interaction
9
between particles, various effects of external field on the Ising fluid take place. In particular, at
1
moderatevaluesofthenonmagneticattractionthefieldeffectonthegas-liquidcriticaltemperature

n is nonmonotoneous. A justification of such behavior is given. If short-range correlations are taken
a into account (within a cluster approach), the Curie temperature also depend on the nonmagnetic
J
interaction.

5
PACS numbers 64.70.Fx, 77.80.Bh, 75.50.Mm, 64.60.Kw.
]
h
c Anisotropic liquids are very sensitive matters. Such properties of the quenched site-disordered Ising model
e
are nematic liquid crystals and ferrofluids. Many efforts and the annealed one [7]). Such drawbacks can be over-
m
have been made in order to investigate effects of shape comewith the two-sitecluster approximation(TCA)[8].
- and flexibility of the molecules, of long and short-range We shall show within both the MFA and the TCA
t
a interactions on the properties of anisotropic liquids. Ex- that at different values of the nonmagnetic interaction
t
s ternalfield effects are still worthierof attention, because betweenparticlesthe Isingfluiddemonstratesvariousef-
.
t theapplicationofanexternalfieldallowstochangeprop- fects of external fields. In particular,at moderate values
a
m erties of anisotropic liquids dynamically (in contrast to ofthe nonmagneticattractionthe fieldeffect onthe crit-
the effects of molecules’ shape, etc., which are static). ical temperature is nonmonotoneous.
-
d In ferrofluids an external magnetic field removes the In lattice models of a fluid its particles are allowed to
n magnetic order-disordertransition, nevertheless the first occupy only those spatialpositions which belong to sites
o order transitions between ferromagnetic phases of dif- ofachosenlattice. Theconfigurationalintegralofasim-
c
ferent densities remain. The external field deforms the ple fluid is in such a way substituted by the partition
[
phase diagram of a magnetic fluid shifting coexistence function
1 linesbetweenthesephases. Kawasakistudiedamagnetic
v
4 latticegas[1],whichimplementsoneofthewaystomodel Z =Spexp(−βH), β =1/(kBT),
2 properties of magnetic fluids. On temperature-density 1
H=H −µN =− I n n −µ n ,
0 phasediagramsinRef.[1]onecanseethatthegas-liquid 2 ij i j i
1 binodal of the magnetic fluid significantly lowers after Xij Xi
0 the application of an external magnetic field. Vakarchuk
9 where n , which equals 0 or 1, is a number of particles
with coworkers [2] and Lado et al. [3,4] studied another i
9 at site i. Sp means a summation over all occupation
modelofmagneticfluids—thefluidofhardsphereswith
/
t embeddedHeisenbergspins. Theyconcludedthatinsuch patterns. The total number N of particles is allowed to
a
fluctuate, µ is a chemical potential, which should be de-
m a system the temperature of the gas-liquid critical point
termined from the relation
(the top of binodal) increases after the application of an
-
d external field. The nature of such a discrepancy can be
n various, because the types of models (continuum fluids N = n ; h···i =Z−1Sp(···)exp(−βH). (1)
o i H
[2–4] and the lattice gas [1]) as well as approximations * +
c Xi H
: used differ.
v
Inthis letterresultsofamoredetailedinvestigationof Lattice models, due to particles can not approach closer
i
X the magnetic (Ising) lattice gas are reported. The first than the lattice spacing allows, automatically preserve
r newpointistheinclusionofthenonmagneticinteraction the essential feature of molecular interaction: nonover-
a
between particles. Another one consists in overcoming lappingofparticles. The latticefluidwithnearestneigh-
of limitations applied by the mean field approximation bor interaction is known to demonstrate the gas-liquid
(MFA). It is known that this approximation is good for transition only. Nevertheless, the lattice gas with inter-
very long-range potentials only (and becomes accurate actingfurtherneighborspossessesarealistic(thatmeans,
forafamily oftheinfinite-rangedones,the so-calledKac argon-like) phase diagram with all transitions between
potentials [5,6]). The MFA can not reproduce some es- the gaseous, liquid and solid phases being present [9].
sential features of the systems with the nearest-neighbor We shall consider a magnetic fluid in which the parti-
interaction (such as the percolation phenomena in the cles carry Ising spins S =±1 and there is also an addi-
i
quenched diluted Ising model, differences in magnetic tional exchange interaction between the particles
1
1
H −µN =− JijSiniSjnj −h Sini whereJ0 = jJij =zK andI0 = jIij =zV areinte-
2 gralinteractionstrengths,K andV denote, respectively,
Xij Xi P P
the magnetic coupling and the nonmagnetic attraction
1
− Iijninj −µ ni (2) between nearest neighbor sites. The expression (6) can
2
ij i becomputedexplicitlyintermsofthefieldsϕandψ and
X X
Each site can be in three states: empty (i), occupied by model parameters. The other thermodynamic potentials
the particle with spin up (ii) or down (iii). The trace in canbe found in a straightforwardway. For example, the
the partition function implies a summation over all 3N grand thermodynamic potential Ω of the model satisfies
states, where N is a number of sites. the following Gibbs-Helmholtz equation
An interaction of fluctuations are totally neglected in
∂βΩ
themeanfieldapproximationusedinthepreviousstudies =U −µN. (7)
∂β
ofthemodel[1]. Thiscanpartiallyberecoveredusingthe
ideaof“clusters”. Thepartitionfunctionofafinitegroup The solution of this differential equation, taking into ac-
ofparticlesinanexternalfieldcanbeevaluatedexplicitly. count relations (5), reads
A contribution of the other particles may be expressed
z
in terms of the effective field, and this field has to be βΩ/N =z′lnSpexp(−βH )− lnSpexp(−βH ). (8)
i ij
2
evaluatedselfconsistently. Fromsucha pointofview the
MFA is a one-site cluster approximation, in which each It is possible to build isotherms of the fluid using the
cluster comprises one site. Increasing the size of clusters thermodynamic relation Ω = −PV and expression (8)
one may expect to obtainmore accurateresults. Indeed, andsolvingthesystemofnonlinearselfconsistencyequa-
the results of the two-site cluster approximation turns tions (5). It is possible and convenient to exclude the
to be accurate for the one-dimensional systems [10] and chemicalpotentialµ andthe field ψ from finalequations
on the Cayley tree [7]. Here we shall formulate such an of state:
approximation for the Ising lattice gas with the nearest
z
neighborinteractions. Forthesakeofbrevityweshallnot βPV/N =−ln(1−n)+ ln[(1−n)(1−x)+xr], (9)
2
usetheclusterexpansionformulationwhichhassomead-
sinh2βh˜′
vantages, such as the possibility to calculate corrections tanhβh˜ =p
of a higher order and correlation functions of the model cosh2βh˜′+exp(−2βK)
[8]. Instead we shall rely on the first order approxima- +(1−p)tanhβh˜′, (10)
tion and closely follow the derivation by Vaks and Zein
[7]. Let us introduce the effective-field Hamiltonian of a where
single site
p=1−(1−n)/r, r =0.5+ (n−0.5)2+n(1−n)/x,
Hi−µni ≡Hi =−h˜Sini−µ˜ni, (3) 2exp(−βV −βK)cosh2βph˜′
whereh˜ =h+zϕ,µ˜ =µ+zψ,ϕandψareeffectivefields x= cosh2βh˜′+exp(−2βK) , (11)
substituting for interactions with nearest neighbor sites,
z is a first coordination number of the lattice. In the p is a probability that a randomly chosen nearest neigh-
two-site Hamiltonian the interaction between a pair of bor site to a given particle is occupied. In the limit
thenearestneighborsitesistakenintoaccountexplicitly z → ∞ and K → 0, V → 0 (keeping J0 and V0 con-
H −µn −µn ≡H =−J S n S n −h˜′S n −˜h′S n stant) TCA formulae (9–11) turn into the results of the
ij i j ij ij i i j j i i j j
MFA.
′ ′
−I n n −µ˜n −µ˜ n , (4)
ij i j i j At low temperatures the isotherms of the fluid con-
whereh˜′ =h+z′ϕ,µ˜′ =µ+z′ψ,andz′ =z−1duetoone tain the “liquid” and “gaseous” parts separated by a re-
oftheneighborsisalreadytakenintoaccount. Thefields gionofthenegativecompressibility. Thethermodynamic
havetobefoundfromtheselfconsistencyconditionsthat states, in which the compressibility of the uniform fluid
require an equality of averagevalues calculated with the is negative (∂P < 0), constitute the spinodal region on
∂n
one-site and two-site Hamiltonians. To determine ϕ and the temperature-density phase diagram. In this region
ψ it is sufficient to impose these conditions on the aver- the fluid is thermodynamically unstable and must sep-
age values of spin m = hS n i and of the occupation arate on phases of different densities. The densities of
i i H
number n=hn i , coexisting phases can be determined with the Maxwell
i H
rule of areas applied to the nonmonotoneous sections of
hn i =hn i ; hS n i =hS n i . (5)
i Hi i Hij i i Hi i i Hij the isotherms. Also at sufficiently low temperatures and
This approximationleads to the following expression for h = 0 an isotherm of the model has a break at the den-
the internal energy sityinwhichnonzerosolutionofselfconsistencyequation
1 (10) for the field ϕ appears, and the second order phase
U/N =−2J0hSiniSjnjiHij −hhSiniiHij transition to the ferromagnetic phase occurs.
1
−2I0hninjiHij, (6) 2
t ∞ (a) t (b)
1.2 1.2
0.5 0.01
0.1 ∞ 0.01
1 1 0.5
0.1
0.8 0.5 ∞ 0.01 Y=4 0.8
Y=2 0.5 0.01 Y=4
0.1 ∞ 0.1 Y=2
0.6 0.6
0.4 0.4
Y=0
Y=0
0.2 0.01 0.1 ∞ 0.2
0.01 0.1 ∞
0 n 0 n
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
FIG. 1. The phase diagram of the magnetic (Ising) lattice gas within the mean field (a) and two-site cluster (b) approx-
imations. n = N/N and t = kBT/J0 are reduced density and temperature, respectively. Picture b represents TCA results
for the model with the nearest neighbor interactions on the simple cubic lattice (z = 6). The results of MFA (picture a)
are independent of interaction range and lattice structure. Three families of lines are built for three values of nonmagnetic
interactionv=I0/J0. ThethicklinesarebinodalsandCurielinesatzeroexternalfield. Thethinlineswithattachednumbers
represent thebinodals at nonzero values of the externalfield h/J0.
Figure 1a shows the temperature-density phase dia- odal, stronger fields shift it up.
gram of the model within the mean field approximation
In Fig. 1b one can see that the TCA, besides quan-
at different model parameters. There are three fami-
titative differences, gives some qualitative corrections to
lies of lines for three values of the nonmagnetic inter-
the MFA results. Within the TCA the Curie line be-
action strength v = I0/J0. The bold lines correspond comes slightly concave, and the nonmagnetic attraction
to zero field case (h = 0). The straight line is a line of
between particles increases the Curie temperature. The
Curie points that separates paramagnetic and ferromag-
lattereffectcanbejustifiedbythequalitativearguments.
netic regions. Within the mean field approximation the
Indeed, the nonmagnetic attraction increases the proba-
slope of the Curie line is independent of v. Under bin-
bility that a randomly chosen pair of the nearest neigh-
odals (convex lines resting on the points (0,0) and (1,0)
borsitesisoccupied. Sinceatthissitesparticlesinteract
on the phase diagram) a phase separation takes place:
magnetically, the magnetic interaction becomes more ef-
oneofthe phases(vapor)is rarefied,the other (liquid) is
fective,andtheCurietemperatureincreasesalso. There-
denser. Thenonmagneticattractionbetweenparticles,of
foretheaccountofdensityfluctuationsintheTCAleads
course,favorsthe phaseseparation—the binodalmoves
to the dependence of the Curie temperature on v. In
upwardwith increasing v. At small v the phase diagram
the case of the non-compressible fluid (n = 1) the den-
possesses of the tricritical point: the top of binodal lays
sity fluctuations are absent, and the Curie temperature
onthe Curie line,the liquidis ferromagneticandthe gas
is independent of the nonmagnetic attraction.
isparamagnetic. Thetricriticalpointdisappearsatlarge
TheTCApredictionsconcerningtheeffectoffieldsup-
v, inthis casethe topofbinodaldeviates fromthe Curie
port the MFA results. The variety of field effects may
line in the paramagnetic region.
be explained by the existence of two concurrent ten-
The external field dissolves the ferromagnetic transi- dencies. The first, the external field aligns the spins,
tion and therefore eliminates the Curie line. The gas- whichleadstothemoreeffectiveattractionbetweenpar-
liquid binodals in presence ofthe externalmagnetic field ticles (let us remind that at v =0 particles with parallel
are depicted with thin lines. The attached numbers are spins (S S = 1) attract and those with opposite spins
i j
strengthsof the fieldh/J0. At smallv there is atemper- (SiSj = −1) repulse). This raises the binodal (for ex-
ature interval, where the external field suppresses phase ample, in simple nonmagnetic fluids the binodal goes up
separation—thetopofbinodalshiftsdownwardinagree- when the interaction increases). The second tendency
ment with the results of Kawasaki [1]. Nevertheless, at takes place, if the susceptibility of the rarefied phase is
large nonmagnetic attractions (e.g., v = 4) the reverse larger than that of the coexisting dense phase. In this
field effect takes place. At moderate values of the non- case the magnetization and, consequently, the effective
magnetic interaction (for example, v =2) the field effect attraction between particles grow better in the rarefied
is nonmonotoneous — weak fields lower the top of bin- phase. This decreases the energetical gain of the phase
3
separation. Therefore the second tendency suppresses Still more illuminating confirmation of the “bi-
the gas-liquidseparationinthe fluidandcounteractsthe tendency” explanation one can see in Fig. 2. There is
first tendency. The second tendency is very strong at the phase diagramof special topology,which takes place
h = 0 and v = 0 in the region of the tricritical point, at intermediate v. The model at h = 0 and t = 0.63
wherethevapor(paramagnetic)branchofbinodalalmost undergoestwofirst-orderphasetransitions. At this tem-
coincides with the Curie line (where the susceptibility perature the fluid can be in three phases: paramagnetic
tends to infinity), whereas the branch of the coexistent gas(atn<0.35),paramagneticliquid(0.65<n<0.69),
liquid phase rapidly deviates from the Curie line. As a ferromagnetic liquid (n > 0.83). What we would like to
result, the external field lowers the top of the binodal. emphasizeisthatweakexternalfields(e.g.,h/J0 =0.01)
Thesecondtendencygetsweakanddisappearswhenthe raise the binodal at n = 0.5 and lower it at n = 0.75.
susceptibilities of the coexistent phases levels; they are Such behavior completely fit into the “bi-tendency” ex-
comparable,forexample,ifbothliquidandvaporphases planation: at n = 0.5 and h = 0 both phases are para-
are paramagnetic. The behavior of v = 4 binodal (see magnetic, the second tendency is absent, therefore the
Fig.1)demonstratesthisfeature. Arelationbetweenthe external field favors the phase separation; at n = 0.75
susceptibilitiesresultsfromvariousfactors. Forexample, the second tendency wins at small fields, like in the case
a short-rangeness of the interactions levels the suscepti- v =2 (see Fig. 1).
bilities and weakens the second tendency [11]. It can be One can see that the lattice gas approachmay be suc-
seen from the following observation of the field effect at cessfullyusedfordescriptionofcomplexfluidswhencon-
v =2: in Fig. 1a the top of binodalat h=0.01is higher tinual approacheslead to too complex calculations or do
than that at h = 0.1, whereas in Fig. 1b the reverse sit- notgivesatisfactoryresults. Itisthissituationthattakes
uationtakes place. Since the results ofthe MFA (as well place when one determines the influence of the nonmag-
as those of the TCA in the limit z → ∞) are correct neticattractionontheCurietemperature[11,12]. Inthis
for the long-ranged potentials, whereas for our case the case the account of the short-range correlations within
TCAismuchmoreaccurate,the correctionsprovidedby the cluster approach yields qualitatively new results in
theTCAhavetobeattributedtodifferencesbetweenthe comparison with current continual methods.
systems with the long-range and short-range potentials.
t
0.9
∞
∗
0.8 E-mail: [email protected]
0.5 0.1 [1] T. Kawasaki, Progr. Theor. Phys. 58, 1357 (1977).
0.7
[2] I.O. Vakarchuk, H.V. Ponedilok, and Yu.K. Rudavskii,
Teor. Mat. Fiz. 58, 445 (1984).
0.6 0.05 0.01 [3] F. Lado, E. Lomba, and J. J. Weis, Phys. Rev. E 58,
0.5 0.7 0.1 3478 (1998).
0.68 0.05 [4] F.LadoandE.Lomba,Phys.Rev.Lett.80,3535(1998).
0.4 [5] J. Lebovitz, O.Penrose, J. Math. Phys. 7, 1016 (1966).
0.66
[6] R. Balescu, Equilibrium and nonequilibrium statistical
0.3
0.64 mechanics (Wiley, New-York,1975), Chap. 9.4
0.2 0.62 [7] V.G. Vaks, N.Ye. Zein, Zh. Eksp. Teor. Fiz. 67, 1082
0.01
(1974).
0.6
0.1 [8] R.R.Levitskii,S.I.Sorokov,R.O.Sokolovskii,ActaPhys-
0 0.5 1
ica Polonica A 92, 383 (1997).
0 n [9] C.K. Hall and G. Stell, Phys.Rev. A7,1679 (1973).
0 0.2 0.4 0.6 0.8 1
[10] B.Ja. Balagurov, V.G. Vaks, R.O. Zajtsev, Fiz. Tverd.
FIG. 2. The phase diagram of the model at v = 3 within Tela 16, 2302 (1974).
the TCA. The inset shows the vicinity of the top of h = 0 [11] T.G. Sokolovska, R.O. Sokolovskii, cond-mat/9810244.
binodal in a more detail. [12] B.GrohandS.Dietrich,Phys.Rev.Lett.72,2422(1994)
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