PHYSICAL REVIEW E 80, 051502 共2009兲
Dipolar particles in an external field: Molecular dynamics simulation and mean field theory Ran Jia and Reinhard Hentschke* Fachbereich Mathematik und Naturwissenschaften, Bergische Universität, D-42097 Wuppertal, Germany 共Received 30 July 2009; published 5 November 2009兲 Using molecular dynamics computer simulation we compute gas-liquid phase coexistence curves for the Stockmayer fluid in an external electric field. We observe a field-induced shift of the critical temperature ⌬Tc. The sign of ⌬Tc depends on whether the potential or the surface charge density is held constant assuming that the dielectric material fills the space between capacitor plates. Our own as well as previous literature data for ⌬Tc are compared to and interpreted in terms of a simple mean field theory. Despite considerable errors in the simulation results, we find consistency between the simulation results obtained by different groups including our own and the mean field description. The latter ties the sign of ⌬Tc to the outside constraints via the electric field dependence of the orientation part of the mean field free energy. DOI: 10.1103/PhysRevE.80.051502
PACS number共s兲: 64.70.F⫺, 68.35.Rh, 87.10.Tf, 64.75.Gh
I. INTRODUCTION
Over many years the phase behavior of dipolar liquids has created significant interest 共see, in particular, the following reviews 关1–4兴兲. Aside from the practical importance many apparently simple model systems have been studied both theoretically and via computer simulation because of the rich and difficult phase behavior due to the addition of dipoledipole interaction to simple short-ranged fluid potentials. Interesting questions are, for instance, the discrepancy between the “easy” prediction and computer observation of the isotropic-to-ferroelectric transition in models of dipolar fluids and their elusiveness in real low molecular weight liquids possessing rather large molecular dipole moments 共e.g., 关5,6兴兲. Another question is whether dipole-dipole interaction alone, i.e., without another contribution to particle-particle attraction, can lead to vapor-liquid coexistence 共e.g., 关7兴兲. Here we focus on the gas-liquid 共g-l兲 critical point shift, i.e., ⌬Tc, in a model dipolar liquid caused by an external electric field. The number of simulation studies computing the field-dependent shift of the g-l critical parameters for dipolar fluids is relatively small 关8–11兴 共see also 关12兴兲 even though the much larger number of theoretical works addressing this phenomenon attests to its general and sustained interest 共selected examples are 关13–22兴兲. In particular the authors of the last of these references, i.e., Ref. 关22兴, emphasize an apparent inconsistency regarding the sign of ⌬Tc in the literature discussing the g-l critical temperature shift under the influence of an electric field. In this work we present simulation results for the critical point shift in a Stockmayer fluid in response to an external electric field if the reduced dipole moment is small. We also look in detail at the available results obtained by other groups via computer simulation and compare them to a simple mean field model based on Onsager’s description of liquid dielectrics, which, in our opinion, resolves the inconsistency mentioned above.
*Author to whom correspondence should be addressed; http://constanze.materials.uni-wuppertal.de;
[email protected] 1539-3755/2009/80共5兲/051502共9兲
The paper is structured as follows. Section II discusses the molecular dynamics 共MD兲 method. Important details can be found in the Appendix. Section III presents a simple mean field theory useful for the conceptual understanding. The results are compiled in Sec. IV. Section V is the conclusion. II. SIMULATION METHOD
We carry out MD computer simulations of the Stockmayer fluid in an external electric field. The total potential energy of the system is 1 1 pជ 2i 1 ជi U = ULJ − m −m ជ iTijm ជj+ ជ i · Eជ ext ជi·M i − gm 2 2␣ 2
共1兲
共making use of the summation convention兲. Here ULJ is the sum over all Lennard-Jones 共LJ兲 pair potentials, N
1 4共r−12 − r−6 ULJ = 兺 ij 兲 + ULJ,lrc , 2 i,j=1共i⫽j,rij⬍rcut兲 ij
共2兲
between Stockmayer particles i and j separated by the distance rij. Notice that for the LJ parameters we assume = = 1. Only particles within the cutoff distance rcut interact explicitly. Interactions from beyond rcut in the case of ULJ are included via the usual long-range correction, ULJ,lrc −3 = −共8 / 3兲Nrcut , where is the particle number density. The remaining terms in Eq. 共1兲 are due to the interaction between point dipole moments, m ជi= ជ i + pជ i ,
共3兲
located on every LJ site i. The dipole moments are in units of 冑4⑀o3 共⑀o: vacuum permittivity兲. Here ជ i is a permanent point dipole moment and pជ i is an induced point dipole moment computed via
ជ i, pជ i = ␣E
共4兲
where ␣ is a point polarizability associated with every LJ site. Even though our present results are almost exclusively for ␣ = 0, we include the case ␣ ⫽ 0 for sake of latter referជ i is the total electric field experienced by the particle ence. E located at site i given by 051502-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 051502 共2009兲
RAN JIA AND REINHARD HENTSCHKE
ជ i = Tijm ជ E ជ j + Eជ ext i + gM i .
共5兲
TABLE I. ⌬Tc at constant E共⬁兲 or Eext: simulation results and mean field theory.
The quantity Tij is the dipole tensor whose Cartesian components 共␣ ,  = 1 , 2 , 3兲 are T␣ ij = 3
r␣ijrij r5ij
− ␦␣
1
共6兲
r3ij
ជ ext 共i ⫽ j兲. E is the external field as it is felt at site i. The i external field is assumed constant throughout the volume V of the system. However, we keep the index i as a reminder ជ ext, e.g., genthat the relation between a true external field E erated by capacitor plates between which the system is placed, and Eជ ext i does depend on how long-range interactions are handled in the simulation. Notice that two scenarios are particularly relevant: 共i兲 fixed charge density on the capacitor plates; 共ii兲 fixed potenជ ext is tial. Case 共i兲 corresponds to Eជ ext held constant. Here E the electric field in a slit separating the dielectric from a ជ 共⬁兲 held constant. capacitor plate. Case 共ii兲 corresponds to E Here Eជ 共⬁兲 is the average or Maxwell field inside the dielectric. As before for the LJ interactions we use a cutoff rcut around each particle. Inside the cutoff all dipole-dipole interactions are computed explicitly. The electrostatic effects on dipole i due to dipoles beyond rcut are included in the reacជ i in Eq. 共1兲 tion field approximation via the terms − 21 gm ជ i·M ជ and gM i, the reaction field, in Eq. 共5兲, where g=
2共⑀ − 1兲 1 3 , 共2⑀ + 1兲 rcut
共7兲
ជ i is the total dipole moment inside the cutoff sphere and M surrounding particle i. ⑀ is the static dielectric constant in the system under given conditions computed via 共⑀ − 1兲共2⑀ + 1兲 ext 1 ជi = ជ E 3 具M i典. 9⑀ rcut
共8兲
The details of the above can be found in the Appendix. Additional quantities derived via the total potential energy are the force and the torque on particle i, i.e., N
ជ i = Fជ i,LJ + F
兺 j=1共j⫽i,r ⬍r ij
+ rជij共m ជi·m ជ j兲兴 −
cut兲
15 r7ij
再
3 r5ij
关m ជ i共rជij · m ជ j兲 + m ជ j共rជij · m ជ i兲
冎
共rជij · m ជ i兲共rជij · m ជ j兲rជij ,
共9兲
and
ជi = m N ជ i ⫻ Eជ i = i ⫻ Eជ i ,
共10兲
and the pressure P = PLJ − where P⑀ is given by
1 具m ជ iTijm ជ j典 + P⑀ , 2V
共11兲
E共⬁兲
⌬Tsim c
Ref.
⌬TcMF
0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 冑2 冑2 冑2 2.0 2.5 2.5 2.5
2.0 6.0 10.0 20.0 0.5 1.0 1.0 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.0 3.0 3.0 0.4 0.8 1.2 1.0 1.0 2.0 5.0
0.010 0.069 0.106 0.147 0.005 0.031 0.018 0.017 0.019 0.034 0.034 0.086 0.053 0.052 0.057 0.101 0.108 0.004 0.038 0.073 0.12 0.154 0.265 0.525
This group This group This group This group 关11兴 关8兴 关9兴 关11兴 This group 关9兴 关11兴 关8兴 关9兴 关11兴 This group 关8兴 This group 关9兴 关9兴 关9兴 关10兴 关8兴 关8兴 关8兴
0.010 0.063 0.105 0.159 0.002 0.018
0.010 0.039 0.080 0.132 0.128 0.358 1.059
Eext
⌬Tsim c
Ref.
⌬TcMF
0.5 0.5 1.0 1.0
2.0 4.0 1.0 2.0
−0.025 −0.084 −0.014 −0.058
P⑀ = −
3 V
冓 冉
This This This This
0.042 0.069
0.126
group group group group
−0.021 −0.083 −0.014 −0.056
冊
ជ i ⑀ m 1 ជi·M ជ i · Aជ i + 3 2 m 共2⑀ + 1兲 rcut
冔
共12兲
ជ i = Eជ 共⬁兲 if E共⬁兲 is held constant and Aជ i = −2Eជ ext if Eext is with A held constant instead. We note that potential energy, force, torque, and virial, in the absence of the external field, were worked out by Vesely 关23兴. We note also that our expression for the reaction field contribution to the pressure differs from the corresponding contribution in Vesely’s work because of the neglect of the density dependence of ⑀ in the aforementioned reference. Notice in this context that the internal virial may be obtained also via −3V U / V. Notice also that the Gibbs ensemble technique used by the other groups listed below in the context of Table I does not require the explicit calculation of the pressure. The translational motion of the particles is governed by ជ i, whereas the equation of motion governing the dipole rជ¨ i = F
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ជ i = I ជ¨ i. Here I is the moment of orientation follows via N inertia with respect to the momentary axis of rotation. The ជ i can be replaced by ជ˙ i angle of rotation vector ជ i using ជ˙ · ⫻ ជ = ជ˙ and ជ = 0. The resulting equation of motion for i
i
i
i
ជ i can be found in Sec. 8.2 of Ref. 关24兴 or in the Appendix of Ref. 关25兴. The equations of motion are integrated using the velocity Verlet algorithm. Temperature, which here is in units of kB / ⑀ 共kB: Boltzmann’s constant兲, is controlled via the weak coupling method of Berendsen et al. 关26兴. Notice that the rotational temperature is given by 共22兲−1具 ជ˙ 2典 = T. At equilibrium the rotational temperature must be equal to the translational temperature of course. Notice also that here we set the moments of inertia with respect to the major axes equal to one in LJ units. Induced dipole moments may be calculated at every MD step using the iteration scheme ជ i共pជ 共k兲 = ␣E pជ 共k+1兲 i i 兲. The dielectric constant is calculated via Eq. 共8兲. G-L phase coexistence curves are obtained by the same method as introduced previously 关27,28兴; i.e., phase coexistence is established using the Maxwell construction method applied to simulation isotherms at different temperatures. The long-range cutoff, rcut, varies between 4.5 and 5.5. III. SIMPLE MEAN FIELD DESCRIPTION
A simple mean field description of the g-l transition may be constructed as follows. The orientation partition function, including all dipolar interactions, is given via QD ⬀
冉 冊 冋
N! ⌬⍀ ⌸ N ! 4
N
册
exp − T−1 兺 NuD共cos 兲 . 共13兲
Here the dipoles are sorted in equal size solid angle cones ⌬⍀ whose orientation in space is labeled via the index . uD共cos 兲 is given by Eq. 共A11兲 with ជ · Eជ cav = Ecav cos . Application of Stirling’s approximation leads to the orientation distribution of the dipoles given via f共兲 = lim⌬⍀→0共4 / ⌬⍀兲N / N ⬀ exp关K cos 兴 with K = Ecav / 关共1 − ␣g兲T兴. Thus, we obtain for the relevant free energy
冉
冊
1 sinh关K兴 ⌬FD − = − ln 共g2 + ␣E2cav兲. NT K 2共1 − ␣g兲T
the Debye model which differs from the Onsager model used here. In Debye’s model the local field includes a term proportional to 具m ជ 典 instead of the reaction field term proportional to m ជ as in Eq. 共A7兲 共for a discussion of this difference see Ref. 关29兴兲. We prefer Onsager’s approach because of its clearly identifiable approximations. The above-mentioned 具m ជ 典 proportionality in particular leads to an isotropic liquidto-ferroelectric liquid transition in the Debye model which is absent in the Onsager model. However, here we are interested in the effect of an electric field on the g-l transition under conditions for which there is no isotropic liquid-toferroelectric liquid transition, no association of dipoles into reversible aggregates, usually chains, and no inhomogeneity in the system. The effects of chain formation on the g-l transition were discussed previously in Refs. 关27,28兴. In general, Onsager’s model is a good description of dipolar interaction in simple systems if the parameter T / 共2兲 is around or larger than unity 关29兴 共assuming ␣ = 0兲. IV. RESULTS
Figure 1 illustrates the dependence of the static dielectric constant as obtained via mean field theory on the external ជ ext. Notice that rcut here refers to Eq. 共7兲. In field strength E the mean field approach rcut is the radius of the cavity containing a single dipole. Here rcut = 0.8, which we use throughout in all mean field calculations, turns out to yield the overall best agreement between mean field theory and simulation. However, Eq. 共7兲 is used also to estimate the long-range contribution to the electrostatic potential energy. In this case rcut is the radius of the much larger cutoff sphere in the simulation. Figure 2 shows a comparison of the effect of the ជ ext = const and Eជ 共⬁兲 = const. electric field for the two cases E Temperature and density are close to the respective critical values for = 1 and zero field. Figure 3 compares the density dependence of the static dielectric constant from the simulation to corresponding mean field results. Figure 4 shows the average dipole-dipole interaction at low densities as obtained by simulation with ␣ = 0. We may compare this to the leading contribution to 具uDD典 for weakly correlated dipoles in a weak field
具uDD典 ⬇ 4 2
共14兲 Notice that it is the first term which accounts for the entropy loss due to orientation in the field. Notice also that in order to evaluate ⌬FD we need ⑀ which we compute by solving Eq. 共A13兲 numerically. Our total free energy includes the nondipolar interactions via the van der Waals description, i.e.,
冉
冊
/共3oc 兲 9 /共3oc 兲 ⌬FvdW . = ln − 8 T/共3Toc 兲 NT 1 − /共3oc 兲 oc
⬇−
r⬎R
drr2具uDD典orient
冋
冉 冊册
4 4 1 Eext 1 + 1125 9 TR3 T
4
,
共16兲
where 具uDD典orient is the orientational average in the case of weakly correlated dipoles. An analogous calculation carried out for large Eext yields to leading order
共15兲
具uDD典 ⬇ −
Toc
and refer to the critical points of the pure LJ Here system. Thus, ⌬F = ⌬FD + ⌬FvdW is the free energy contribution governing g-l phase separation. A similar approach was used by Zhang and Widom to map out the global phase behavior of dipolar fluids 关17兴. These authors describe the dipolar interactions in terms of
冕
冋 冉 冊册
8 4 T 1− 15 TR3 Eext
4
.
共17兲
Both limiting laws are included in Fig. 4. For larger densities the above assumptions are no longer valid and the form of 具uDD典 quickly deviates from the above limiting laws. Figure 5 shows simulation results for 具uDD典 for selected conditions, in particular at higher densities, in comparison to corre-
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RAN JIA AND REINHARD HENTSCHKE 5
Ε
Ε
1.40 1.35
4
1.30 3
1.25 1.20
2
1.15
0
6
2
4
6
8
10
4 3 2
4
6
8
10
Eext
FIG. 1. Mean field dielectric constant ⑀ vs the external field Eext. Upper panel: = 0.3, = 2, ␣ = 0, and T = 2 , 4 , 6 共from top to bottom兲. Lower panel: = 0.3, T = 2, = 1 共lower curve兲, and = 2 共upper curve兲. Corresponding dashed curves with ␣ = 0.05 共rcut = 0.8兲 instead of ␣ = 0.
Ε 2.2 2.0 1.8 Eext const 1.6 1.4 1.2
E const 0
10
0.2
0.3
Ρ
0.4
FIG. 3. Static dielectric constant ⑀ vs dipole number density 共 = 0.5, T = 1.35兲. Symbols: simulation results 共squares: Eext = 0; up triangles: Eext = 1; down triangles: Eext = 2兲. Lines 共mean field theory兲 from top to bottom: Eext = 0 , 1 , 2.
5
2
Eext
Ε
0
sponding curves obtained by computing the first term in Eq. 共A11兲. Qualitatively the Onsager theory agrees with the simulation. Quantitatively we find growing deviation between simulation and the Onsager model as Eext increases. Figure 6 shows g-l coexistence curves obtained in this work via MD simulation. The values for ⌬Tc = Tc共 , E兲 − Tc共 , 0兲, where E = Eext or E = E共⬁兲, are listed in Table I. The overall comparison between the simulation results and the mean field theory is quite good. In particular we note that the sign of ⌬Tc does depend on whether Eext or E共⬁兲 is held constant. In the latter case the sign is positive whereas in the former it is negative. Figure 6 also reveals a slight shift of the critical density from lower densities for large E = E共⬁兲 to higher densities at large E = Eext. For = 0.5 this shift certainly is within the scatter of the results, but for = 1.0 the shift is clearly discernible. A detailed plot of the reduced mean field critical temperature for = 0.5 is shown in Fig. 7 共upper panel兲. We note again that the direction of the critical temperature shift depends on whether the field outside the dielectric is held constant 共fixed charge density兲 or whether the average field in the dielectric is held constant 共fixed potential兲 while density uDD 20
40 0.0005
60
80
100
Eext
0.0010
20
30
40
50
E
FIG. 2. Static dielectric constant ⑀ vs field strength 共 = 1.0, T = 1.45, = 0.3123兲. Symbols are simulation results. Crosses: E = Eext; circles: E = E共⬁兲. Note that ⑀ at E = 0 was computed using Eq. 共9兲 in Ref. 关29兴. The solid lines are computed via mean field theory.
0.0015
FIG. 4. Average dipole-dipole potential energy 具uDD典 vs Eext共 = 0.5, T = 10兲. Symbols: simulation results 共squares: = 0.02; up triangles: = 0.05; down triangles: = 0.1兲. Lines: approximations according to Eqs. 共16兲 and 共17兲 with R = 0.9 plotted to where the correction term exceeds 10% of the leading term.
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o
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0.6
or temperature changes. Notice that we do not include simulation results for E = Eext at large field strength. This is because simulation snapshots at this field strength appear rather inhomogeneous. Figure 7 共bottom panel兲 also shows a comparison between mean field theory and simulation in the case of the critical density shift, ⌬c, as function of external field strength, E, for = 1.0. For E = Eext the quantitative agreement is quite good, whereas for E = E共⬁兲 it is qualitative only. The sign of ⌬Tc depending on the field held constant may be understood as follows. Figure 8 shows the free energy as function of volume 共assuming N = 100 particles兲 as obtained
1.45
E
1.4
E
E
1.35
E E=0
1.3
ext
E =2
1.25
ext
E =4
1.2 1.55
E
1.5
E
1.45
E E=0
1.4
ext
E =1
1.35 ext
E =2
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
5
10
15
Ρc 0.04
0.02 0.01
0.01 0.02
E
E Eext
0.03
0.00
20
0.5
1.0
1.5
2.0
2.5
3.0
E
E E
0.03
FIG. 7. Top: mean field critical temperature in units of the LJ critical temperature vs E 共specified for each curve兲 for = 0.5. Symbols are simulation results. Circles: E = E共⬁兲; crosses: E = Eext. Bottom: mean field critical density shift vs E for = 1.0. Symbols are simulation results. The statistical error again is comparable to the size of the symbols. Circles: E = E共⬁兲; crosses: E = Eext. In all cases rcut = 0.8, oc = 0.3, Toc = 1.32, and ␣ = 0.
1.5
1.3 0.15
0.7
FIG. 5. Average dipole-dipole contribution to the potential energy 具uDD典 vs Eext for = 0.5 and ␣ = 0. Open circles: simulation for = 0.308 at T = 1.351; crosses: simulation for = 0.8 at T = 1.332; lines: corresponding mean field results with rcut = 0.8 given by the first term in Eq. 共A11兲.
T
0.8
0.08
T
Tc Tc 1.2 1.1
Eext
0.5
FIG. 6. Coexistence curves in the temperature-number densityplane based on simulation data. Top: = 0.5; bottom: = 1.0. In both cases ␣ = 0. Stars denote critical points. The statistical error is comparable to the size of the symbols.
via mean field theory. The solid 共dashed兲 line corresponds to E共⬁兲 共Eext兲 held constant. The dotted line is the zero field result. The temperature is below the critical temperature and we observe a van der Waals loop. The two lower panels in Fig. 8 show the attendant comparison for the orientation contribution to the free energy according to the first term in Eq. 共14兲, f orient, and the contribution due to dipolar interaction described by the second term in Eq. 共14兲, f DD. We notice that the latter virtually remains unaltered whereas f orient accounts ជ ext = ⑀Eជ 共⬁兲 we find that constant for the difference. Due to E ext E implies a reduction in E共⬁兲 when the density is increased because ⑀ increases. Notice also that at this small field strength ⑀ is basically independent of field strength. Because ជ 共⬁兲 and Eជ cav, which governs oriof relation 共A15兲 between E entation in the field, the quantity K in Eq. 共14兲, and therefore the orientation, is reduced when E共⬁兲 is reduced. This means that for constant Eext the free energy contribution f orient increases with increasing density, which in turn diminishes the van der Waals loop in comparison to the zero field case; i.e., ជ 共⬁兲 is held con⌬Tc is negative. The reverse is true when E stant. Here f orient decreases with increasing density, which in
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0.35
2.0
E const
0.25
0.20
0.30
2.5
3.0
0.15
500
1000
1500
2000
2500
3000
1500
2000
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Eext const
V 0.00
0.05
0.10
0.15
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Ρ
forient 500
1000
V
cos Θ 0.9524
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0.9522 0.9520
0.10 0.15
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0.20
E const
0.9512 0.9510 0.00
fDD 500
1000
1500
2000
2500
3000
V
0.1 0.2 0.3
Eext const
0.05
0.10
0.15
0.20
Ρ
FIG. 9. Top: Average orientation of the dipoles relative to the electric field direction, 具cos 典, vs density, , for = 1 and T = 1.2. Solid lines: mean field theory; open circles: simulation result for E共⬁兲 = 1; crosses: simulation result for Eext = 1. Bottom: same as above but for = 0.5, E共⬁兲 = 50, and Eext = 50.
It is interesting to compare the electric field dependence of the critical temperature discussed above with the textbook expression in Ref. 关14兴, which is given by
0.4 0.5
⌬Tc ⬇
0.6
FIG. 8. Top: f = ⌬F / 共NT兲 vs volume V; middle: f orient, according to the first term in Eq. 共14兲, vs V; bottom: f DD, according to the second term in Eq. 共14兲, vs V. In all cases = 1.0, T = 1.2, rcut = 0.8, oc = 0.3, Toc = 1.32, and ␣ = 0. Solid lines: E共⬁兲 = 1 is held constant; dashed lines: Eext = 1 is held constant; dotted lines: zero field.
turn enhances the van der Waals loop in comparison to the zero field case; i.e., ⌬Tc is positive. Figure 9 shows this effect in terms of the average projection of ជ / onto the direction of the electric field for = 0.5, 1.0 and T = 1.2. We note that the agreement between the mean field theory and the simulation is quite reasonable. We also note that in genជ ext held eral the scatter of the simulation results is larger for E constant than for Eជ 共⬁兲 held constant. In both cases each point is based on 1.5⫻ 106 MD steps 共of which 106 were used for averaging兲 and a step width of 3 ⫻ 10−3.
共2⑀/2兲T=Tc,E=0,E 1 c,E=0E2 2 . 8 共 P/ T兲Tc,E=0,E=0
共18兲
Provided that ⌬Tc is small this result is rather general. The field E in this case is E共⬁兲. In order to compare with the result in Fig. 7 we compute the denominator via van der Waals theory, i.e., 共2 P / T兲Tc,E=0,E=0 = 9 / 4 共this number is roughly half what one obtains based on the actual simulation data兲. The derivative in the numerator is computed using Onsager’s expression for ⑀ in Eq. 共A14兲 共with ␣ = 0兲. The result is the dashed line in Fig. 10 shown together with the ជ 共⬁兲 taken from Fig. previous mean field result for constant E 7. The deviation on the high field side occurs because in the Onsager theory ⑀ itself depends on E. This is explicitly excluded in the derivation of Eq. 共18兲. Computer simulations of dipolar fluids in an external field Eជ o usually are based on the following general approach. The field couples to the system via −兺i ជ i · Eជ o in the Hamiltonian 关cf. Eq. 共1兲兴. The sum extends over all dipole moments
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T 0.15
0.10
0.05
2
4
6
8
10
E
FIG. 10. Critical temperature shift ⌬Tc vs E共⬁兲 for = 0.5. Solid line: mean field theory developed in this work; dashed line: ⌬Tc as derived in Ref. 关14兴.
ជ i. Ewald summation is used in conjunction with a continuum 共dielectric constant ⑀out兲 surrounding the sphere of explicit summation. The average field in the liquid 共Maxwell ជ , and Eជ o are related via Eជ = Eជ o − 4 P ជ / 共2⑀out + 1兲 共see field兲, E 关30兴兲. The polarization of the liquid of dipolar particles is ជ = 共⑀in − 1兲Eជ , where ⑀in is tied to the Maxwell field via 4 P ជ o = 共2⑀out the dielectric constant of this liquid and thus E ជ . In the literature discussed in the following + ⑀in兲 / 共2⑀out + 1兲E paragraph this is the method of choice. The following is, to the best of our knowledge, a complete list of simulation studies addressing the critical point shift in a pure dipolar liquid. Stevens and Grest 关8兴 have used Gibbs ensemble Monte Carlo simulations of the Stockmayer fluid in an applied field. They use tin-foil boundary conditions, which according to the preceding paragraph means ជ o = Eជ = Eជ 共⬁兲. In Ref. 关9兴 the authors study the influence of E static electric fields on the vapor-liquid coexistence of dipolar soft-sphere and Stockmayer fluids both via GubbinsPople-Stell perturbation theory and simulation. Again the applied field corresponds to E共⬁兲 because the simulations were performed in the reaction field geometry with conducting boundary conditions, where the spherical sample is embedded in a continuum with infinite dielectric constant. In Ref. 关10兴 the authors carry out Gibbs ensemble simulations of the Stockmayer fluid in an applied field Eជ o. The authors ជ o = Eជ , ⑀out = ⑀in, compare the cases ⑀out = ⬁ 共tin-foil兲, i.e., E −1 ជ ជ ជo i.e., Eo = 3⑀in共2⑀in + 1兲 E, and ⑀out = 1 共vacuum兲, i.e., E ជ = 共⑀in + 2兲 / 3E. Even though the authors do not compute Tc explicitly it is still possible to compare the relative order of Tc for the three boundary conditions based on their Fig. 2; i.e., Tc decreases in the order of discussion. Moreover Tc共⑀out = ⬁兲 is certainly above the Tc in absence of a field, whereas Tc共⑀out = ⑀in兲 is close to Tc in absence of a field. This is consistent with our Fig. 7, where the curves labeled E = E共⬁兲 and E = Ecav correspond to these two cases. Tc共⑀out = 1兲 appears to be below the zero field critical temperature. This we expect according to our discussion of Fig. 7 because E is decreased with increasing density. In Ref. 关11兴 the authors study hard-core dipolar Yukawa fluids using the NPT-Monte Carlo plus test particle method in conjunction
with reaction field long-range corrections and conducting boundaries for the dipole-dipole interactions. Their results ជ o = Eជ = Eជ 共⬁兲. In Table I we list the for ⌬Tc correspond to E simulation results for the shift of the critical temperature, ⌬Tsim c , for various dipole moments and field strengths extracted from these references, supplemented by our own results, which all use the Stockmayer potential 共with the exception of Ref. 关11兴兲. In addition we compute the corresponding ⌬TcMF from our simple mean field theory, which overall is in fair agreement with the literature simulation data. Notice that ⌬Tc is small compared to Tc and that the errors involved in the determination of Tc are often enough comparable to ⌬Tc. The number of theoretical studies of g-l coexistence in dipolar fluids under influence of an external electric field is far more numerous. To our knowledge the first work dealing with this problem in detail, aside from the above textbook formula, is Ref. 关16兴. Using density-functional theory the authors consider a sample of fluid in the field constrained to have a spherical shape—apparently surrounded by vacuum. They find a negative ⌬T upon increasing Eo in agreement with our above discussion and the ⑀out = 1 result in Ref. 关10兴. Other studies 关19兴, 共Gubbins-Pople-Stell perturbation theory兲 关20兴, 共density-functional theory兲 关21兴, 共modified mean field density-functional theory兲 关22兴, 共self-consistent theory based on mean spherical approximation for the dipolar Yukawa fluid兲, are more complicated, with the exception of Ref. 关17兴 already mentioned and usually less transparent but do not yield obvious improvements over the simple theory used here. The other simple theory, i.e., Ref. 关17兴, which is based on the Debye model, seems to perform less well than the one used here, which is based on Onsager’s refinement.
V. CONCLUSION
Using the MD technique we have computed g-l phase coexistence curves for the Stockmayer fluid in an external field. We observe a field-induced shift of the critical temperature ⌬Tc. Whether the critical temperature is increased or decreased depends on whether potential or surface charge density are kept constant assuming that the dielectric material fills the space between capacitor plates. Our own as well as previous literature data for ⌬Tc are compared to and interpreted in terms of a simple mean field theory. Despite considerable errors in the simulation results, we find that there is consistency between the simulation results obtained by different groups and the mean field description. The latter ties the sign of ⌬Tc to the outside constraints via the electric field dependence of the orientation part of the mean field free energy.
APPENDIX
The terms in Eq. 共1兲 describing explicit interactions between point dipole moments as well as their interaction with the external field, UDD, may be cast in various equivalent forms. The probably most transparent one is
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1 1 UDD = − ជ iTij ជj− ជ i · Eជ ext ជ i · Eជ 共o兲 i − p i . 2 2
Here the first and second terms describe the interaction of permanent dipoles with each other and with an external field. The last term describes the total decrease in energy if a polarizable medium is brought in an electrical field with fixed sources 共e.g., Sec. 4.7 in Ref. 关31兴兲, i.e., the permanent dipole moments at fixed orientation plus the external field due to likewise fixed sources. Here Eជ 共o兲 ជ j + Eជ ext i = Tij i .
共A2兲
Equation 共A1兲 may be rewritten as 1 1 UDD = − m ជ i · Eជ 共o兲 ជ i · Eជ ext i − i 2 2
ជ in = gm E ជ + Eជ cav .
共A1兲
共A3兲
The first term is the reaction field. Notice that in this case rcut in Eq. 共7兲 is the radius of the cavity occupied by the dipole m ជ only. Eជ cav, the second term, is the so-called cavity field given by
ជ cav = E
共A4兲
where Eជ i is given by Eq. 共5兲 omitting the last term, or 1 1 pជ 2i UDD = − m −m ជ iTijm ជj+ ជ i · Eជ ext i 2 2␣
共A5兲
关cf. Eq. 共1兲兴, making use of Eq. 共4兲 of course. In the context of computer simulation of dielectric constants in soft condensed-matter systems UDD usually appears as 1 1 ext ជ ជ ext UDD = − ជ iTijm ជ o,j − m ជ o,i · Eជ ext i − Ei 共AijE j 兲, 共A6兲 2 2 where m ជ o,j = 共␦ij − ␣Tij兲−1 ជ i and Aij = ␣共␦ij − ␣Tij兲−1. Note that here m ជ i= ជ i + ␣共Tijm ជ j + Eជ ext ជ j=m ជ o,j i 兲 is solved for m ext ជ i 关32,33兴. + AijE ជ ext Thus far the meaning of the external field, E i , is clear. It is the external field felt at the position of dipole i in the system. The need for specific boundary conditions, however, makes it necessary to precisely specify the relation of Eជ ext i to 共⬁兲 ជ the average field inside the dielectric material E or to the external field outside the dielectric material, Eជ ext. This is particularly relevant for the comparison to experiments or other simulation results. In the present case, the reaction field approach to longrange interaction, we assume that every dipole moment interacts explicitly with all other dipole moments within a spherical shell of radius rcut. Beyond rcut there is a dielectric continuum characterized by a dielectric constant ⑀. A simplified version of this situation has been studied by Onsager in his famous paper on the electric moments of molecules in liquids 关34兴. It is instructive to compare key expressions of this continuum approach to the analogous expressions used in the simulation. Onsager considers a single dipole moment m ជ akin to m ជ i in Eq. 共3兲. The dipole moment m ជ feels an electric field
3⑀ 共⬁兲 ជ , E 2⑀ + 1
共A8兲
where ⑀ is the dielectric constant of the surrounding medium and Eជ 共⬁兲 is a homogeneous electric field at large distances from the cavity due to external sources. The calculation of Eជ in on the basis of Maxwell’s equations in dielectric media can be found in Appendix A.2 of Fröhlich’s book on the theory of dielectrics 关35兴 as well as in the original paper of course. In analogy to Eq. 共A3兲 the potential energy of the dipole at the center of the cavity is
or 1 1 UDD = − ជ i · Eជ i − m ជ i · Eជ ext i , 2 2
共A7兲
1 1 uD = − m ជ · 共g ជ + Eជ cav兲 − ជ · Eជ cav . 2 2
共A9兲
ជ in, i.e., Eq. 共4兲, we may rewrite this into Using pជ = ␣E 1 1 pជ 2 uD = − m −m ជ ·g·m ជ+ ជ · Eជ cav . 2 2␣
共A10兲
This result corresponds to Eq. 共A5兲. Rewriting m ជ as m ជ = 共1 −1 −1 ជ − ␣g兲 ជ + ␣共1 − ␣g兲 Ecav yields yet another expression, i.e., uD = −
1 1 ␣ 1 g ជ2 . E 2 − ជ · Eជ cav − 2 1 − ␣g 2 1 − ␣g cav 1 − ␣g 共A11兲
This is the analog of Eq. 共A6兲, which generalizes Onsager’s calculation to more than one dipole inside the cavity 关32,36兴. It is worth noting that this form of uD is particular suitable to compute 具m ជ 典 because we merely need to compute 具 ជ典 using the Boltzmann weight exp关共1 − ␣g兲−1 ជ · Eជ cav / T兴. The result is 具m ជ 典 = L共x兲
␣ ជ cav , eជ z + E 1 − ␣g 1 − ␣g
共A12兲
with L共x兲 = coth共x兲 − x−1 and x = 共1 − ␣g兲−1Ecav / T. Here eជ z is a unit vector parallel to the cavity field. Using the relation ជ = 共⑀ − 1兲Eជ 共⬁兲 / 共4兲, where P ជ is the polarization, to具m ជ 典=P gether with Eq. 共A8兲 we obtain the result 1 共⑀ − 1兲共2⑀ + 1兲 ␣ ជ cav = L共x兲 ជ cav , E eជ z + E 3⑀ 4 1 − ␣g 1 − ␣g 共A13兲 and in the limit E共⬁兲 → 0,
冉
1 共⑀ − 1兲共2⑀ + 1兲 1 = 3⑀ 4 3T 1 − ␣g
冊
2
+
␣ . 1 − ␣g 共A14兲
This formula obtained by Onsager relates the microscopic dipole moment, , and polarizability, ␣, to the macroscopic
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dielectric constant ⑀ 关34兴. Its interesting relation to Debye’s earlier equation 关37兴 is discussed in Ref. 关29兴. If we now consider a spherical cavity occupied by a number of dipole moments m ជ i we may directly obtain the reacជ i in Eq. 共5兲. Notice tion field corrections to U in Eq. 共1兲 and E ជ i, where M i that the reaction field now becomes gM = 兺 j苸Vcavm ជ j 共including m ជ i兲. Thus, comparing Eq. 共A5兲 to Eq. 共A10兲 leads to Eq. 共1兲 and comparing the first two terms in Eq. 共5兲 to Eq. 共A7兲 leads to the full Eq. 共5兲. ជ ext ជ cav. corresponds to E In particular we also note that E i ext 共⬁兲 ជ . We now Via Eq. 共A8兲 we may therefore relate Eជ i to E consider a 共small兲 cavity embedded in a homogenous slab of
Unless stated otherwise we will use this relation throughout. Notice in particular that Eq. 共8兲 follows via 3 ជ i典 / 共4rcut ជ 共⬁兲 / 共4兲 in conjunction with Eq. / 3兲 = 共⑀ − 1兲E 具M 共A15兲.
关1兴 C. Holm and J.-J. Weis, Curr. Opin. Colloid Interface Sci. 10, 133 共2005兲. 关2兴 J.-J. Weis and D. Levesque, in Advanced Computer Simulation Approaches for Soft Matter Sciences II, Advances in Polymer Science Vol. 185, edited by C. Holm and K. Kremer 共Springer, New York, 2005兲. 关3兴 S. H. L. Klapp, J. Phys.: Condens. Matter 17, R525 共2005兲. 关4兴 B. Huke and M. Lücke, Rep. Prog. Phys. 67, 1731 共2004兲. 关5兴 M. A. Pounds and P. A. Madden, J. Chem. Phys. 126, 104506 共2007兲. 关6兴 D. P. Shelton and Z. Quine, J. Chem. Phys. 127, 204503 共2007兲, and references therein. 关7兴 G. Ganzenmüller, G. N. Patey, and P. J. Camp, Mol. Phys. 107, 403 共2009兲. 关8兴 M. J. Stevens and G. S. Grest, Phys. Rev. E 51, 5976 共1995兲. 关9兴 D. Boda, J. Winkelmann, J. Liszi, and I. Szalai, Mol. Phys. 87, 601 共1996兲. 关10兴 K. Kiyohara, K. J. Oh, X. C. Zeng, and K. Ohta, Mol. Simul. 23, 95 共1999兲. 关11兴 I. Szalai and S. Dietrich, J. Phys.: Condens. Matter 20, 204122 共2008兲. 关12兴 We note that Lado et al. 关38兴 have determined phase coexistence curves for a classical Heisenberg spin fluid with shortrange interactions in an external magnetic field. They observe an increase in the critical temperature with increasing field strength. 关13兴 P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155 共1965兲. 关14兴 L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media 共Pergamon Press, New York, 1984兲, Vol. 8. 关15兴 J. R. Quint, J. A. Gates, and R. H. Wood, J. Phys. Chem. 89, 2647 共1985兲. 关16兴 C. E. Woodward and S. Nordholm, J. Phys. Chem. 92, 501 共1988兲. 关17兴 H. Zhang and M. Widom, Phys. Rev. E 49, R3591 共1994兲. 关18兴 A. Onuki, Europhys. Lett. 29, 611 共1995兲.
关19兴 D. Boda, I. Szalai, and J. Liszi, J. Chem. Soc., Faraday Trans. 91, 889 共1995兲. 关20兴 B. Groh and S. Dietrich, Phys. Rev. E 53, 2509 共1996兲. 关21兴 V. B. Warshavsky and X. C. Zeng, Phys. Rev. E 68, 051203 共2003兲. 关22兴 I. Szalai, K.-Y. Chan, and Y. W. Tang, Mol. Phys. 101, 1819 共2003兲. 关23兴 F. J. Vesely, J. Comput. Phys. 24, 361 共1977兲. 关24兴 D. C. Rapaport, The Art of Molecular Dynamics Simulation 共Cambridge University Press, Cambridge, 1995兲. 关25兴 W.-Z. Ouyang and R. Hentschke, J. Chem. Phys. 127, 164501 共2007兲. 关26兴 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak, J. Chem. Phys. 81, 3684 共1984兲. 关27兴 R. Hentschke, J. Bartke, and F. Pesth, Phys. Rev. E 75, 011506 共2007兲. 关28兴 J. Bartke and R. Hentschke, Phys. Rev. E 75, 061503 共2007兲. 关29兴 J. Bartke and R. Hentschke, Mol. Phys. 104, 3057 共2006兲. 关30兴 This formula is derived in Appendix A.2 of Ref. 关35兴. Note, however, that the dielectric constant ⑀2 in A.2 is unity in the present case because it refers to an “extra” underlying continuum and should not be confused with ⑀in. 关31兴 J. D. Jackson, Classical Electrodynamics 共John Wiley & Sons, New York, 1975兲. 关32兴 M. Mandel and P. Mazur, Physica 24, 116 共1958兲. 关33兴 E. L. Pollock, B. J. Alder, and G. N. Patey, Physica A 108, 14 共1981兲. 关34兴 L. Onsager, J. Am. Chem. Soc. 58, 1486 共1936兲. 关35兴 H. Fröhlich, Theory of Dielectrics 共Oxford University Press, London, 1958兲. 关36兴 J. G. Kirkwood, J. Chem. Phys. 7, 911 共1939兲. 关37兴 P. Debye, Polare Molekeln 共Hirzel Verlag, Leipzig, 1929兲. 关38兴 F. Lado, E. Lomba, and J. J. Weis, Phys. Rev. E 58, 3478 共1998兲.
dielectric material 共like in the case of an infinite plate capacitor filled with dielectric material兲. Assuming a constant field ជ ext outside the dielectric and perpendicular to the adjacent E ជ ext = ⑀Eជ 共⬁兲. Thus, we find slab surfaces we have E
ជ ext E i =
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共A15兲
