The FCC-BCC-Fluid triple point for model pair interactions with variable softness

0.17 MB·English

by  Sergey A. Khrapak

FCC-BCC-Fluid triple point for model pair interactions with variable softness
∗
Sergey A. Khrapak and Gregor E. Morfill
Max-Planck-Institut fu¨r extraterrestrische Physik, D-85741 Garching, Germany
(Dated: January 15, 2013)
It is demonstrated that thecoordinates of thefcc-bcc-fluid triple point of various model systems
are located in a relatively narrow region, when expressed in terms of the two proper variables,
characterizingthesoftnessandstrengthoftheinteractionforceatthemeaninterparticleseparation.
Thiscanberegardedasaconsequenceofthe“correspondingstatesprinciple”forstronglyinteracting
particle systemswe haveput forward recently [S.A.Khrapak,M. Chaudhuri,and G.E. Morfill, J.
Chem. Phys. 134,241101(2011)]. Therelatedpossibilitiestopredicttheexistenceandapproximate
3
location of the fcc-bcc-fluid triple point for a wide range of pair interactions with variable softness
1
areillustrated. Relation oftheobtainedresultstoexperimentalstudiesofcomplex(dusty)plasmas
0
are briefly discussed.
2

n PACSnumbers: 64.60.Ej,64.70.D-,52.27.Lw
a
J

4 Manysystemsofstronglyinteractingparticlesinther- havior of two different systems of strongly interacting
1 modynamicequilibriumareknowntoformface-centered- particles can be expected when the interparticle force
cubic (fcc) or body-centered-cubic (bcc) lattices in the anditsfirstderivative(evaluatedatthemeaninterparti-

]
solid phase. Normally, for sufficiently soft interactions cledistance)inonesystemareequaltothoseintheother
t
f there is a tendency to forma bcc solid. Onthe contrary, system. The simplisticargumentsbehindthis conjecture
o
s steep interactions favor the fcc structure. When the in- are as follows: In a strongly coupled state the particles
. teractionpotential exhibits variable softness,a polymor- form a regular structure where large deviations of the
t
a phic phase transition between fcc and bcc solids can be interparticle separation from its average value are very
m
present and the fcc-bcc-fluid triple point emerges on the seldom. Consequently, the state of the system should
- respective phase diagram. be virtually insensitive to the exact shape of the inter-
d
action potential at short distances. If, in addition, the
n Finding the location of fcc-bcc phase transition re-
potentialdecayssufficientlyfastfordistancesbeyondthe
o mains a difficult and resource demanding (computa-
c tional) task since usually a very tiny difference between mean interparticle separation, its long-range asymptote
[ isoflittleimportanceforthethermodynamicsofthesys-
thefreeenergiesofthecorrespondingphaseshastobere-
tem [4]. The behavior of the potential at the averagein-
2 solved. In this Letter we discuss a very simple approach
v toapproximatelylocatethefcc-bcc-fluidtriplepoint. The terparticle separation plays a dominant role. Physically,
9 approach is based on analyzing the shape of the pair in- the force and its first derivative are the two quantities
2 teractionpotentialonlyanddoesnotrequireanykindof which should mainly matter. Using the IPL potential as
6 areference system,we identifiedthe twoscaledvariables
numerical simulations.
3 which conveniently characterize the state of a strongly
. The qualitative similarities in the phase behavior of
5 coupled system of particles interacting via the pair po-
modelsystems withsoftrepulsiveinteractionshavebeen
0 tential U(r). These are the generalized softness param-
2 discussedpreviouslyfromvariousperspectives. Inpartic- eter s = [−1−U′′(∆)∆/U′(∆)]−1 and the reduced force
1 ular,Prestipino et al. [1, 2] discussedthe commontopol- ′
F =−U (∆)∆/T,where∆isthemeaninterparticlesep-
: ogy of the phase diagrams of the Gaussian core model
v aration and T is the temperature (in energy units). The
(GCM),inverse-power-law(IPL),andYukawamodelpo-
i approximate “corresponding states principle” postulates
X tentials in the vicinity of the fcc-bcc-fluid triple point.
that two different systems of strongly interacting parti-
r In an attempt to relate the phase behaviors of the GCM
a cles having the same values of s and F behave alike [3].
andYukawapotentialstothatoftheIPLmodel,theyre-
Asatestofthevitalityofthisprinciple,wedemonstrated
quiredthe logarithmicderivatives ofthe three potentials
thatthe meltingcurvesofvariousmodelsystemssuchas
to match, for separations close to the average nearest-
IPL, Yukawa, conventional Lennard-Jones (LJ), n − 6
neighbor distance. As a result, the effective exponents
LJ, exp−6 and (slightly less successfully) GCM are es-
fortheGCMandYukawainteractionsatthe triplepoint
sentiallycollapsingona single curvewhen plottedin the
wereshowntolienottoofarfromthecorrespondingIPL
(s,F) plane and, hence, the “universal melting curve”
value. This was a useful observation, although perhaps
emerges [3].
insufficient for direct practical applications.
The question we address here is whether this corre-
As we conjectured recently [3], similarities in the be-
sponding states principle can also be useful to estimate
(at least approximately) the location of the fcc-bcc-fluid
triple point. In order to answer it, we carefully ana-
∗Also at Joint Institute for High Temperatures, 125412 Moscow, lyze the available data on the fcc-bcc phase transition
Russia;Electronicmail: [e-mail] for three selected repulsivepotentials. We show that the
2
triplepointcoordinatesinthe(s,F)planeareindeedrel-
TABLEI:TheestimatedvaluesoftheparametersΓ,F ands
atively close for all the systems considered. In addition,
at the fcc-bcc-fluid triple point for various model interaction
the phase diagramsin the vicinity of the triple point are
potentials.
topologically equivalent.
We consider the IPL potential, U(r) = ǫ(σ/r)n, Potential ΓTP FTP sTP Ref.
the Yukawa (exponentially screened Coulomb) poten- GCM 2.47 24.08 0.129 [2]
tial, U(r) = ǫ(σ/r)exp(−r/σ), and the GCM potential, IPL 3.98 28.46 0.14 [2]
2 2
U(r) =ǫexp(−r /σ ). Here ǫ and σ are the energy and IPL 5.05 31.57 0.16 [16]
length scales, and r is the distance between the two in-
Yukawa 4.03 31.26 0.145 [21]
teracting particles. The IPL potential defines a system
Yukawa 3.50 27.65 0.142 [22]
of “soft” spheres which can serve as a model for a wide
Yukawa 3.11 27.09 0.128 [24]
range of physical systems, including the one-component
plasma(OCP)andsimplemetalsunderextremethermo-
dynamic conditions. At n → ∞ it approaches the hard
sphere (HS) limit. The Yukawa (Dedbye-Hu¨ckel) poten- triple pointaresummarizedinTable I.Thescatteringin
tial is extensively used in the context of colloidal sus- the reportedΓTP valuesisrelativelylarge,uptoafactor
pension, physics of plasmas, and complex (dusty) plas- of two (see the first column in Table I). On the other
mas [6–10]. The GCM potential [11] is frequently used hand, the scattering in the triple-point values of F and
in the context of soft matter physics, e.g. to describe s are considerably less pronounced. The difference be-
effective interactions in many body systems of polymer tween the corresponding values for different interaction
solutions [9, 12, 13] potentials is comparableto that fromdifferent studies of
a single interaction (see e.g. the data for the Yukawa
The phase state of the considered systems in ther-
potential). Some (weak) dependence of sTP and FTP on
modynamic equilibrium can be conveniently character-
ized by the reduced density ρ∗ = ρσ3 and temperature the shape of the interaction potential cannot be com-
pletely ruled out. At the same time, all the data points
T∗ = T/ǫ (For the IPL potential, the single variable
ρ∗T∗−3/n fully describes the thermodynamic state of the fFaTllPin≃th2e8r.e0la±tiv4e.0ly. nTarhrioswsruapnpgoertssTPth≃e 0“.c1o4rr±es0p.0o2ndainndg
system). Another possible choice of state variables for
states principle” conjecture formulated in terms of the
purely repulsive interactions uses the coupling parame-
equality between the values of the parameters s and F,
ter Γ = U(∆)/T and the screening parameter κ = ∆/σ,
characterizing different interacting particle systems. In
where ∆ = ρ−1/3 is the structure-independent interpar-
addition, the conditions
ticle distance. This choice is particularly suitable for
Yukawa interactions and is commonly used in the field s(ρ∗)≃0.14, F(ρ∗,T∗)≃28, (1)
of complex (dusty) plasmas [6, 7].
ThephaseportraitsoftheIPL,Yukawa,andGCMsys- are expected to yield reasonable approximations for ρ∗
temsarerelativelywellstudied[1,2,14–27]. Concerning and T∗ at the fcc-bcc-fluid triple point of many other
interacting particle systems (we will verify this towards
thefcc-bcc-fluidtriplepoint,AgrawalandKofke[16]esti-
−3/n the end of this Letter).
mateditslocationfortheIPLsystemasρ∗T∗ ≃2.173 In order to illustrate the relation between the corre-
(in the fluid phase) and 1/n ≃ 0.16. They noted, how-
sponding states principle and the behavior of the in-
ever, that this estimate is subject to significant uncer-
teraction potentials around the mean interparticle sep-
tainty. Later, Prestipino et al. [2] performed more de-
aration, we have plotted the considered potentials in
tailed study of the IPL phase diagram near the triple
Fig 1. The potentials are normalized by the temper-
−3/n
point. Their result is ρ∗T∗ ≃ 1.787 (in the fluid ature, u(x) = U(x)/T, and the reduced distance is
phase)and1/n≃0.14. Inthesameworkthetriplepoint x=r/∆. The parameters of each potential are uniquely
temperature of the GCM system has been estimated as determined from the conditions s = 0.14 and F = 28
T∗ ≃ 0.0031. The corresponding fluid phase density is (which are taken as characteristic for the fcc-bcc-fluid
ρ∗ ≃ 0.093. Dupont et al. [21] reported the following triple point). We observe that there is a non-negligible
triple point coordinate for the Yukawa model: κ = 6.75 difference in the potentials around x = 1, which is ob-
andT∗ ≃0.43×10−4. Later,Hamaguchietal.[22]deter- servable even on the logarithmic scale of Fig 1. At the
minedthe locationofthe triple-pointinYukawasystems sametime,thefirsttwoderivativesarealmostcoinciding
as κ = 6.90 and Γ ≃ 3.50. Finally, in a recent paper by in some vicinity around x=1.
HoyandRobbins[24],thetriplepointofYukawasystems Nextweusenumericalresultsrelatedtothe fluid-solid
has been located at κ = 7.70 and Γ ≃ 3.11. For consis- and fcc-bcc phase transitions of the considered model
tency, we take the fluid density at the triple point where systems and plot the corresponding phase boundaries in
appropriate. For the data related to the Yukawa poten- Fig. 2. In this figure a portion of the unified phase di-
tial, the triple-point density has a single value, defined agram near the fcc-bcc-fluid triple point is shown in the
by the estimated value of κ. plane of reduced variables s/sTP and F/FTP. The nu-
ThecorrespondingvaluesofΓTP,FTP,andsTP atthe merical data are taken from Refs. [2, 15] (IPL), [22, 24]
3

103 1u’(x)04 3.0 Freezing IPL
102 Freezing Yukawa
Freezing GCM
2.5
FCC-BCC IPL
1 100 FCC-BCC Yukawa
10 FCC-BCC GCM
u(x) 104u"(x) 10-20.5 1.0 x 1.5 2.0 FF/TP2.0 FCC BCC
10-1 102 IPL 1.5
Y
100 G ukaw
C a
10-20.5 1.0 x1.5 2.0 M 1.0 FLUID
-3
10
0.5 1.0 1.5 2.0 0.8 1.0 1.2 1.4 1.6 1.8
x s/s
TP
FIG. 1: (Color online). IPL, Yukawa, and GCM interaction
FIG.2: (Coloronline). PhaseboundariesoftheIPL,Yukawa,
potentialsinthevicinityofthemeaninterparticleseparation.
and GCM systems near the fcc-bcc-fluid triple point in the
Potentials are normalized by the temperature, distance is in
units of the mean interparticle separation, x = r/∆. The plane of reduced parameters s/sTP and F/FTP. Solid sym-
parameters of the potential correspond to s=0.14 and F = bolscorrespondtothefcc-bccphasetransition,opensymbols
mark the fluid-solid (freezing) phase transition. The data
28 (see the text for details). The upper (lower) inset shows
for the IPL system (circles) are from Ref. [2] (freezing) and
thefirst (second) derivative of thepotentials.
Refs. [2, 15] (fcc-bcc transition). The data for the Yukawa
system (triangles) are from Refs. [22] (fluid-solid transition)
and [24] (fcc-bcc transition). The data for the GCM system
(Yukawa), and [2] (GCM), for details see caption of (stars) are from Ref. [2] (both the freezing and fcc-bcc tran-
Fig. 2. To evaluate the parameters s and F along the sitions).
fluid-solidandfcc-bccphasetransitionsinIPLandGCM
systemsweusedthe densitiesofthefluidandfccphases,
respectively. FortheYukawasystem,thedensitygapbe- triple point parameters to the input value of sTP. In
tween fcc and bcc phases as well as between fluid and this case, the criterion (1) is not very useful. Taking the
solid phases was not resolved in Refs. [22, 24]. The val- lower-limit value sTP = 0.12 suggested above, results in
ues of sTP and FTP employed are given in Table I. The ρ∗ ≃ 3.4. From the generic equation relating T∗ and ρ∗
black solid curve corresponds to the universal melting ofthe n−6 LJ fluids at freezing (Eq. (3) fromRef. [30])
equation F(s)≃106s2/3 [3] in the regime of sufficiently we get T∗ ≃32.3. This lowerestimate of the triple point
soft interactions (0.1<∼s<∼1) [28]. temperature falls in the interval predicted in [29]. Note
Figure 2 demonstrates that the phase diagrams of the thatthe7-6LJpotentialistheonlymemberofthen-6LJ
IPL,Yukawa,andGCMsystemsaretopologicallyequiv- family which is expected to form a stable bcc structure.
alentin the vicinity of the fcc-bcc-fluidtriple point. The Another model system displaying interplay between
triple point separates the region where the fluid freezes solid fcc and bcc phases is the system of particles in-
directlyintothefcccrystallinestructure(regimeofsmall teracting via the exp-6 potential of the form U(r) =
s)fromthatwhere the intermediatebcc phaseis present ǫ{ 6 exp[α(1−r/σ)]− α (σ)6} beyond the hard-core
α−6 α−6 r
(regime of large s). Another observation from Fig. 2 is at r = r0, such that U(r) exhibits maximum at r0. We
that the regionof stability of the bcc phase in the (s,F) concentrate here on the case α = 13, which was consid-
plane widens when going from the IPL to Yukawa and ered in several studies. Still, the exact location of fcc-
further to GCM interaction. This suggests that the sta- bcc-fluid triple point is a controversialissue. In Ref. [31]
bility of the bcc phase is promoted by the short-range triple point temperature has been estimated as T∗ ≃11,
character of the interaction. whileinalaterstudy[32]ahighervalueT∗ ≃20hasbeen
Let us now give several examples of a practical ap- reported. Applying the criterion (1) yields for the triple
plication of the obtained results. Recently, it has been point density and temperature: ρ∗ ≃ 4.4 and T∗ ≃31.3.
suggestedthatthe Lennard-Jones-type7-6potential,de- Significantscatteringinthetriplepointtemperaturesin-
fined as U(r) = Aǫ[(σ/r)7−(σ/r)6] (where A = 77/66), dicatesthatfurtherstudiesareprobablywarrantedtoac-
exhibitsfreezingintoastablebccstructureatsufficiently curatelylocatethefcc-bcc-fluidtriplepointinthismodel.
hightemperatures[29]. Inparticular,preliminaryresults One more potential, which is used to model the re-
from Ref. [29] indicate that the fcc-bcc-fluid triple point pulsive interactions in systems of soft particles, is the
temperature should be located somewhere in the inter- so-called Hertzian potential. It is defined as U(r) =
val 5.0 <∼ T∗ <∼ 100. Since the softness of the LJ-type ǫ(1−r/σ)5/2 for r < σ and U(r) = 0 otherwise. This
7-6 potential tends to s=0.143 from below in the high- potentialisbounded(likeGCM)and,additionally,hasa
temperature limit, it is very likely that the triple point finiteinteractionrange. Adetailedstudyofthephasedi-
does exist. However,veryweakdependence ofs onρ∗ in agramofHertziansphereshasbeenperformedbyPa`mies
this regime implies extreme sensitivity of the estimated etal.[33]. Acomplicatedpicturewithmultiplere-entrant
4
melting transitions along with first-order phase tran- useful tool for the preliminary order of magnitude esti-
sitions between crystals with different symmetries has mates. In particular, it can be helpful when exploring
been reported. Nevertheless, the low density part of this the unique advantages of complex plasmas, associated
phasediagramdisplaysfreezinguponcompressioneither with diversity and variability of interaction mechanisms
into a fcc or bcc crystal, the fcc-bcc phase transition, between the macroparticles [6–8, 37], in order to visit
and the fcc-bcc-fluid triple point. Applying the crite- different regions of their phase diagram. Present results
rion (1) to the Hertz interaction results in the following would be particularly relevant to situations when the
estimate for the triple point density and temperature: possibilities to tune and design interactions are present.
ρ∗ ≃ 1.7 and T∗ ≃ 4.6×10−3. These values are some- One of the possible methods that has been recently dis-
what lower than the triple point parameters ρ∗ ≃ 1.96 cussed[38]isbasedontheapplicationofexternalelectric
and T∗ ≃7.69×10−3 reported in Ref. [33], but still give fields of various polarizations. The physical idea is that
an acceptable first guess, which can then be further im- the applied field induces ionflow, which generateswakes
proved by detailed numerical simulations. downstream from the particles, and thus change inter-
Our last example is related to the Yoshida-Kamakura particle interactions (of course, the frequency should be
(YK) interaction law, U(r) = ǫexp[a(1−r/σ)−6(1− much higher than that characterizing particle dynamics,
r/σ)2ln(r/σ)], where a is a positive parameter control- so that the particles do not react to the field). Dif-
ling the softness of the repulsion. Phase diagram of the ferent anisotropic interaction classes as well as conven-
YK model with a = 2.1 has been reported in Ref [34] tional isotropic interactions with short-range repulsion
and with a=3.3 in Ref. [35]. Again, the reported phase and long-range attraction can be obtained [38].
diagramsare quite complex, with multiple re-entrantre- In order to realize these active manipulations of the
gions and reach solid polymorphism. The low-density interparticle potential a “PlasmaLab” project has been
part, however, is marked by the presence of the fcc and initiated [39]. This is a possible followup of the PK-3
bccsolidphases,phasetransitionbetweenthem,andthe Plus and PK-4 laboratories [40, 41] for complex plasma
fcc-bcc-fluid triple point. The formulated criterion (1) investigationsundermicrogravityconditionsonboardthe
yieldstriple-pointdensityandtemperatureρ∗ ≃0.24and InternationalSpaceStations. Twopossibleplasmacham-
T∗ ≃ 0.038 for a = 2.1, and ρ∗ ≃ 0.28 and T∗ ≃ 0.033 bers are under active investigations. One chamber has
for a = 3.3. These estimates are in fairly good agree- a traditional cylindrical geometry with a flexible in-
ment with the numerical results from Refs. [34] and [35] ner geometry and electrode system. The other has a
(namely, ρ∗ ≃ 0.21 and T∗ ≃ 0.040 for a = 2.1 and spherical-like geometry with twelve independently con-
ρ∗ ≃0.25 and T∗ ≃0.037 for a=3.3[36]). trolledelectrodes. Bothchambersareexpectedtoextend
Thus, the existence and approximate location of the the plasma parameter space available for investigations
fcc-bcc-fluid triple point can be often predicted from the and support interparticle force manipulations. In this
knowledgeofthe firsttwoderivativesof the pair interac- contextpresentresultscouldbequitehelpfulforprelimi-
tion potential, evaluated at the mean interparticle sepa- naryestimatesoftheexperimentalparameterregimesre-
ration. Although the method is not expected to provide quiredto visita desiredphasestate ofacomplex plasma
very high accuracy, its extreme simplicity can make it a under investigation.
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