JOURNAL OF CHEMICAL PHYSICS
VOLUME 114, NUMBER 11
15 MARCH 2001
Equation of state and structure of binary mixtures of hard d-dimensional hyperspheres M. Gonza´lez-Melchora) Departamento de Fı´sica, Centro de Investigacio´n y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000 Me´xico D.F., Me´xico
J. Alejandreb) Programa de Simulacio´n Molecular, Instituto Mexicano del Petro´leo, Eje Central La´zaro Ca´rdenas 152, Apdo. Postal 14-805, 07730 Me´xico D.F., Me´xico and Departamento de Quı´mica, Universidad Auto´noma Metropolitana-Iztapalapa, Apdo. Postal 55-534, 09340 Me´xico D.F., Me´xico
M. Lo´pez de Haroc) Centro de Investigacio´n en Energı´a, UNAM, Temixco, Morelos 62580, Me´xico and Programa de Simulacio´n Molecular, Instituto Mexicano del Petro´leo, Eje Central La´zaro Ca´rdenas 152, Apdo. Postal 14-805, 07730 Me´xico D.F., Me´xico
共Received 21 July 2000; accepted 21 December 2000兲 Computer simulations have been performed on binary fluid mixtures of hard hyperspheres in four and five dimensions. The equation of state and the radial distribution function have been obtained for a variety of compositions and size ratios. The simulation results for the excess compressibility factor and the contact values of the cross radial distribution functions in both dimensions are described rather accurately by a recent theoretical proposal for these quantities up to a reduced density where some features arise which are reminiscent of a fluid–solid phase transition. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1349094兴
I. INTRODUCTION
spheres of exhibiting a first order freezing transition. This transition occurs at a packing fraction f that, relative to the close-packing fraction cp , decreases monotonically with increasing d. Studies on mixtures of hard hyperspheres on the other hand, are to our knowledge rather scarce.13–15 In fact, we are not aware of the availability of simulation data for these systems and so the accuracy and reliability of the 共few兲 equations of state that have been proposed remain to be assessed. Therefore the major aim of this paper is to fill in the undesirable gap in the simulation results as well as to test the performance of a particular proposal for the equation of state that has been recently developed.15 The paper is organized as follows. In Sec. II, we give a brief account of the derivation of the equation of state for mixtures of hard hyperspheres in d⫽4 and d⫽5. Section III deals with the details of the simulation, including the validation of the method by comparison with the simulation data available for pure hard hypersphere fluids in these dimensions. Section IV contains the results for the compressibility factor and the values of the cross radial distribution functions at contact as obtained with the simulation and with the analytical approach. We close the paper in Sec. V with some concluding remarks.
Hard core fluids are model systems that have been used for understanding molecular fluid behavior for a long time. Some of them have been thoroughly studied and their results are reasonably well established. In particular in three dimensions, the Carnahan–Starling1 and the Boublik– Mansoori–Carnahan–Starling–Leland2 equations of state provide a rather satisfactory overall picture of the PVT behavior of the pure hard-sphere fluid and hard-sphere mixtures. Being semiempirical in nature, the accuracy of these equations of state has been mainly tested against numerical simulation studies, which have further served to also clarify their merits and limitations. A few years ago, Frisch and Percus3 pointed out that, when the dimensionality d of the system increases, the statistical mechanics of a fluid of particles interacting via a repulsive potential of finite range is considerably simplified. More recently,4 the same authors have also argued that high spatial dimensionality has a parallel with limiting high density situations 共which allow for performing explicit calculations兲 and so one would expect in dealing with systems in high spatial dimensionality to obtain clean caricatures of any thermodynamic phenomenology that extends to such dimensionality. This is precisely the case of pure fluids of hard hyperspheres (d⭓4) which have attracted the attention of a number of researchers over the last 20 years.5–12 Amongst other similarities, they share the property of hard disks and
II. THE EQUATION OF STATE OF A BINARY MIXTURE OF HARD HYPERSPHERES
In this section we consider a binary mixture of 共additive兲 hard hyperspheres in both d⫽4 and d⫽5 dimensions. In order to derive the equation of state 共EOS兲 for these systems, we will follow the procedure described in Ref. 15 for a multicomponent mixture of d-dimensional hard spheres.
a兲
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© 2001 American Institute of Physics
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4906
Gonza´lez-Melchor, Alejandre, and Lo´pez de Haro
J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
Let the total number density of the mixture be , the set of molar fractions be 兵 x i 其 , and the set of diameters be 兵 i 其 , 2 i (i⫽1,2). The volume packing fraction is ⫽ 兺 i⫽1 d d ⫽ v d 具 典 , where i ⫽ v d i i is the partial volume packing fraction due to species i, i ⫽ x i is the partial number density corresponding to species i, v d ⫽( /4) d/2/⌫(1⫹d/2) is the volume of a d-dimensional sphere of unit diameter, and 2 x i di . Following previous work15 we will now 具 d 典 ⬅ 兺 i⫽1 propose a simple equation of state for the mixture, Z m ( ), consistent with a given EOS for a single component system, Z s ( ), where Z⫽p/ k B T is the compressibility factor, p being the pressure, T is the absolute temperature, and k B is Boltzmann’s constant. Our proposal consists of providing an approximation for the contact values of the radial distribution functions, g i j ( i j ), where i j ⫽( i ⫹ j )/2, the knowledge of which implies that of the EOS through the relation 2
Z m 共 兲 ⫽1⫹2
d⫺1
兺
2
d
ij x ix j d g i j共 i j 兲. 兺 具 典 i⫽1 j⫽1
共1兲
While some other approximations have been suggested13,14 in Ref. 15, g i j was approximated for all d as g i j共 i j 兲⫽
冋
册
1 1 具 d⫺1 典 i j ⫹ g共 兲⫺ . 1⫺ 1⫺ 具 d 典 i j
共2兲
This approximation was motivated by the form of g i j obtained through the solution of the Persus–Yevick 共PY兲 equation for hard spheres (d⫽3), 16 and its merits and limitations may be judged a posteriori. In Eq. 共2兲 the contact value of the radial distribution function for the single component fluid in d dimensions, g( ), is of course related to Z s ( ) through Z s ( )⫽1⫹2 d⫺1 g( ). When the ansatz 共2兲 is inserted into Eq. 共1兲, one gets Z m 共 兲 ⫺1⫽ 关 Z s 共 兲 ⫺1 兴 2 1⫺d ⌬ 0 ⫹
冋
册
1 1⫺⌬ 0 ⫹ ⌬ 1 , 1⫺ 2
共3兲
d⫺1
共 d⫹ p⫺1 兲 ! 具 d⫹p⫺1 典 ⌬ p⬅ 具 n⫺p⫹1 典具 d⫺n 典 . 兺 d 2 具 典 n⫽p n! 共 d⫹ p⫺1⫺n 兲 !
共4兲
In the one-dimensional case, Eq. 共3兲 yields the exact result Z m ( )⫽Z s ( ). Further, for d⫽2 and 3, it proved to be very satisfactory when a reasonably accurate Z s ( ) was taken.15,17 In order to use it also for d⫽4 and d⫽5, we require an accurate expression for Z s ( ) in these dimensions. It should be recalled that the virial series representation of the compressibility factor of a pure fluid of hard hyper⬁ b n⫹1 n , spheres in d dimensions reads Z s ( )⫽1⫹ 兺 n⫽1 where b n are 共reduced兲 virial coefficients. The exact values of the virial coefficients b 2 , b 3 , and b 4 for both d⫽4 and d⫽5 are known.6,7,10 In terms of these, although many others have been devised, perhaps the most accurate proposals to date for Z s ( ) in these dimensionalities are the semiempirical equations of state proposed by Luban and Michels.10 These authors first introduced a function 共兲 defined by
Z s 共 兲 ⫽1⫹b 2
1⫹ 关 b 3 /b 2 ⫺ 共 兲 b 4 /b 3 兴 , 1⫺ 共 兲共 b 4 /b 3 兲 ⫹ 关 共 兲 ⫺1 兴共 b 4 /b 2 兲 2 共5兲
and observed that the computer simulation data favor a linear approximation for 共兲. By a least-squares fit procedure they found ( )⫽1.2973(59)⫺0.062(13) / cp for d⫽4 and ( )⫽1.074(16)⫹0.163(45) / cp for d⫽5. Note that cp⫽ 2 /16⯝0.617 in d⫽4 and cp⫽ 2 &/30⯝0.465 in d ⫽5.10 Also, for future reference it is useful to recall that the freezing transition in the same systems occurs at a packing fraction f ⯝0.31 and f ⯝0.20, respectively.8 Substitution of Eq. 共5兲, using the above linear fits for 共兲, into Eq. 共3兲 produces the EOS of a mixture of hard additive hyperspheres in d⫽4 and d⫽5 dimensions, respectively, which is the main result of this section. The explicit expressions for these EOS follow trivially from the procedure just described above and will therefore be ommitted. Their accuracy will be examined later on. Although they will not be used in this paper, as a straightforward application of Eq. 共3兲, one can also easily get the 共approximate兲 virial coefficients B n of the binary mixture defined by Z m ( )⫽1 ⬁ B n n⫺1 in terms of the b n of the pure fluid. The ⫹ 兺 n⫽2 result is B n ⫽ v n⫺1 具 d 典 n⫺1 关 2 1⫺d ⌬ 0 b n ⫹1⫺⌬ 0 ⫹ 21 ⌬ 1 兴 , d which readily allows one to compute up to B 4 in these dimensions. Recently Enciso et al.18 have calculated B 3 and B 4 for d⫽4 and d⫽5 using Monte Carlo techniques and shown that the agreement with the results of the above formula is very good.
III. SIMULATION DETAILS
In this section we present the simulation details to study binary mixtures of hard spheres systems in four and five dimensions. The method is based on that used to simulate one component systems of hard spheres in three,19 four, and five8 dimensions. The position of the N particles in the initial configuration is chosen to be in a periodic cell, the so-called d-type lattice. In three-dimensional space this lattice is identical with the fcc structure.8 The systems in four and five dimensions dealt with N⫽8n 4 and N⫽16n 5 particles, respectively, where n is an integer. In the present work we used n⫽3 for d⫽4 and n⫽2 for d⫽5. Some mixtures were studied using n⫽3 for d⫽5 to analyze system size effects on the results. The velocities were assigned according with a Maxwell distribution. The simulation cell is a hypercubic cell with side L and volume V⫽L d . Clearly the number density is ⫽N/V. The minimum image convention and periodic boundary conditions in all directions have been applied in the same way as is done in three dimensions.20 During the simulation only binary collisions are taken into account while collisions between three or four particles are ignored. A collision occurs when the distance between the particles is equal to i j ⫽( i ⫹ j )/2. The collision time for every pair of particles is calculated and the minimum value is obtained. All the particles are moved during this time at constant velocity. The pair of particles that suffers a
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J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
Binary mixtures of hyperspheres
TABLE I. Excess compressibility factor, ⌶⫽Z⫺1, and contact values, g 12( 12), for d⫽4 obtained with molecular dynamics 共MD兲 simulations and using Eqs. 共3兲 and 共2兲 共label th兲, respectively. The size ratios are 2 / 1 ⫽0.25 and 2 / 1 ⫽0.33.
2 / 1 ⫽0.25 x1
d1
⌶
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.50
0.75
⌶
TABLE II. Excess compressibility factor, ⌶⫽Z⫺1, and contact values, g 12( 12), for d⫽4 obtained with molecular dynamics 共MD兲 simulations and using Eqs. 共3兲 and 共2兲 共label th兲, respectively. The size ratios are 2 / 1 ⫽0.5 and 2 / 1 ⫽0.75.
2 / 1 ⫽0.33
th
g MD 12
g th 12
⌶
0.063 0.133 0.210 0.295 0.389 0.496 0.612 0.742 0.885 ¯
0.063 0.133 0.211 0.297 0.392 0.498 0.616 0.746 0.892 1.054
¯ 1.02 1.05 1.09 1.14 1.17 1.22 1.27 1.31 ¯
1.04 1.08 1.12 1.16 1.21 1.26 1.32 1.37 1.44 1.51
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.181 0.407 0.695 1.043 1.485 2.038 2.758 3.650 4.766 ¯
0.181 0.408 0.692 1.048 1.495 2.057 2.767 3.669 4.820 6.298
1.01 1.08 1.17 1.25 1.37 1.47 1.62 1.75 1.93 ¯
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.368 0.892 1.639 2.682 4.173 6.388 9.493 10.784 19.331 ¯
0.368 0.891 1.635 2.693 4.206 6.390 9.584 14.330 21.533 32.743
1.05 1.16 1.31 1.48 1.66 1.95 2.21 2.69 2.63 ¯
MD
⌶
2 / 1 ⫽0.50 x1
d1
⌶
1.04 1.09 1.14 1.19 1.25 1.32 1.38 1.46 1.54 1.62
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
1.04 1.11 1.22 1.33 1.45 1.60 1.76 1.93 2.16 ¯
1.09 1.18 1.30 1.43 1.58 1.77 1.98 2.24 2.55 2.93
0.50
1.07 1.21 1.36 1.56 1.84 2.15 2.55 2.92 2.68 ¯
1.13 1.29 1.49 1.75 2.07 2.51 3.08 3.88 4.99 6.62
0.75
th
g MD 12
g th 12
0.075 0.157 0.250 0.352 0.468 0.591 0.738 0.892 1.066 ¯
0.075 0.158 0.250 0.352 0.468 0.596 0.740 0.899 1.078 1.279
1.00 1.04 1.08 1.13 1.18 1.23 1.29 1.35 1.41 ¯
1.07 1.16 1.25 1.37 1.51 1.65 1.82 2.03 2.29 2.59
0.195 0.437 0.749 1.129 1.611 2.219 2.983 3.969 5.200 ¯
0.195 0.440 0.748 1.134 1.620 2.234 3.012 4.002 5.268 6.900
1.11 1.25 1.42 1.64 1.91 2.27 2.75 3.40 4.30 5.6
0.378 0.915 1.680 2.777 4.333 6.574 9.835 11.208 19.992 ¯
0.378 0.917 1.684 2.776 4.340 6.599 9.908 14.833 22.320 33.995
MD
collision is treated according to the impulsive dynamics and the velocities are changed; in this step the hard collisional virial is calculated. The program we used was validated by reproducing the excess compressibility factor obtained by Michels and Trappeniers8 for one component systems with d⫽4 and d ⫽5, respectively. Binary mixtures with different diameter sizes and composition were prepared using the total number of molecules for each dimension. In the mixtures, one component has a diameter of 1 and the other varies from 0.25 to 1, the compositions of molecules with the bigger diameter being 0.75, 0.5, and 0.25. The total number densities are chosen to be in the fluid phase, below the density where the freezing phase transition of the component with the bigger diameter occurs. The maximum reduced density is 1.8 and 2.4 for systems with d⫽4 and d⫽5, respectively. The equilibration period in the simulations consists of 2.5⫻105 collisions and additional 2.5⫻105 collisions for averaging properties. The main results of the simulations are the equation of state and the radial distribution function 共rdf兲. The contact value, g( i j ) was obtained by fitting the points of the rdf with distance less than i j ⫹0.2 1 to a quadratic function.
4907
⌶
2 / 1 ⫽0.75
th
g MD 12
g th 12
⌶
0.114 0.245 0.393 0.559 0.753 0.974 1.225 1.504 1.833 ¯
0.114 0.244 0.393 0.562 0.755 0.975 1.227 1.515 1.845 2.222
1.09 1.12 1.18 1.25 1.33 1.42 1.53 1.62 1.75 ¯
1.06 1.13 1.21 1.30 1.39 1.49 1.60 1.72 1.85 2.00
0.256 0.578 0.990 1.504 2.134 2.961 3.962 5.270 6.923 ¯
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.236 0.540 0.924 1.413 2.038 2.830 3.845 5.199 6.883 ¯
0.237 0.540 0.926 1.419 2.048 2.858 3.892 5.234 6.979 9.267
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.408 0.990 1.829 3.044 4.786 7.286 10.968 12.140 22.387 ¯
0.408 0.996 1.836 3.041 4.781 7.315 11.057 16.679 25.316 38.945
MD
⌶ th
g MD 12
g th 12
0.256 0.580 0.988 1.501 2.147 2.961 3.988 5.289 6.943 9.058
1.09 1.25 1.43 1.63 1.86 2.17 2.49 2.89 3.35 ¯
1.13 1.28 1.46 1.67 1.91 2.20 2.54 2.95 3.45 4.06
1.10 1.20 1.34 1.49 1.68 1.88 2.15 2.47 2.81 ¯
1.11 0.362 0.362 1.24 0.855 0.857 1.39 1.532 1.534 1.57 2.451 2.457 1.79 3.710 3.719 2.05 5.442 5.455 2.36 7.801 7.858 2.73 11.154 11.220 3.20 14.386 15.985 3.78 ¯ 22.841
1.13 1.33 1.59 1.89 2.28 2.76 3.40 3.81 2.73 ¯
1.17 1.38 1.64 1.96 2.37 2.88 3.55 4.42 5.58 7.16
1.14 1.31 1.53 1.80 2.15 2.59 3.24 2.73 2.67 ¯
1.16 0.488 0.488 1.37 1.215 1.214 1.62 2.292 2.290 1.95 3.885 3.888 2.39 6.257 6.277 2.96 9.873 9.895 3.75 9.898 15.470 4.84 14.205 24.263 6.41 27.573 38.542 8.73 ¯ 62.561
1.18 1.21 1.44 1.49 1.77 1.85 2.23 2.33 2.86 2.98 3.55 3.89 1.98 5.17 1.29 7.05 1.35 9.90 ¯ 14.40
MD
The mean virial of forces between colliding particles is given by ⌶⫽⫺
1 N 具 v 2典 t
Nc
兺c 共 ri j •⌬vi 兲 ,
共6兲
where 具 v 2 典 is the mean velocity squared, t is the total simulation time, N c is the number of collisions, and ⌬vi is the change in velocity of the particle i on collision. Using conservation of energy and total momentum, ⌬vi is calculated according to ⌬vi ⫽⫺
bij
2i j
ri j ,
共7兲
with b i j ⫽ri j •vi j evaluated at the moment of impact. The distance, r i j , between particles is r i j ⫽ 共 x 21i j ⫹x 22i j ⫹¯⫹x 2di j 兲 1/2,
共8兲
with x Li j ⬅x Li ⫺x Lj ,x Lk being the Lth Cartesian coordinate of the kth particle 共L⫽1,2,...,d; k⫽1,2,...,N兲. From the mean virial 关cf. Eq. 共6兲兴, the equation of state is obtained as Z⫽⌶⫹1.
共9兲
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4908
Gonza´lez-Melchor, Alejandre, and Lo´pez de Haro
J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
The d-dimensional radial distribution function, g ␣ (r), of particles of species ␣ around one particle of species  is obtained through the calculation of the average number of particles of species , ⌬N ␣ , located at a distance between r and r⫹⌬r using the relation ⌬N ␣ ⫽
冕
␣ g ␣ 共 r 兲 dV
d
,
共10兲
where ␣ is the bulk number density of particles ␣ and dV is the volume element defined as dV d ⫽dx 1 dx 2 ¯dx d ⫽Jdrd d 1 ¯d d⫺2 .
Note that Eq. 共12兲 leads in the particular case of d⫽3 共where there is only one polar angle 1 兲 to the usual transformation J⫽r 2 sin 1 and thus dV 3 ⫽r 2 sin 1 drd1d. Another structural quantity that is not difficult to evaluate in the simulations and which will be useful later is the density profile ␣ (x i ) in a given direction x i . This density profile is obtained in every direction by dividing the length of the simulation cell by N s slabs of width ⌬x i ⫽0.05 1 . It is calculated during the simulation every 100 collisions as ⌬V
,
x1
⌶
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.50
0.75
共11兲
J⫽r d⫺1 sind⫺2 共 1 兲 sind⫺3 共 2 兲 ¯sin2 共 d⫺3 兲 sin共 d⫺2 兲 . 共12兲
具 ⌬N ␣ 典
2 / 1 ⫽0.25 d1
d
Here 1 , 2 ¯ d⫺2 correspond to the d⫺2 polar angles which define the d-space, is the azimuthal angle and J is the Jacobian which transforms the coordinates from Cartesian to hyperspherical given by
␣共 x i 兲 ⫽
TABLE III. Excess compressibility factor, ⌶⫽Z⫺1, and contact values, g 12( 12) for d⫽5, obtained with molecular dynamics 共MD兲 simulations and using Eqs. 共3兲 and 共2兲 共label th兲, respectively. The size ratios are 2 / 1 ⫽0.25 and 2 / 1 ⫽0.33.
共13兲
where 具 ⌬N ␣ 典 is the average number of particles of component ␣ with coordinates between x i and x i ⫹⌬x i and ⌬V ⫽L d⫺1 ⌬x i is the volume of the slab.
⌶
2 / 1 ⫽0.33
th
g MD 12
g th 12
⌶
0.054 0.113 0.178 0.245 0.324 0.407 0.494 0.595 0.700 0.809
0.054 0.114 0.179 0.249 0.326 0.409 0.500 0.599 0.706 0.822
¯ ¯ 1.00 1.02 1.05 1.08 1.10 1.12 1.15 1.19
1.03 1.06 1.09 1.12 1.15 1.19 1.22 1.26 1.30 1.34
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.173 0.383 0.634 0.930 1.288 1.807 2.211 2.805 3.493 4.339
0.173 0.383 0.635 0.937 1.297 1.724 2.230 2.830 3.539 4.375
¯ 1.01 1.08 1.12 1.18 1.24 1.30 1.37 1.46 1.52
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
0.369 0.862 1.518 2.374 3.463 4.885 6.722 9.051 12.024 10.929 15.085 23.667
0.369 0.863 1.515 2.366 3.471 4.899 6.735 9.082 12.055 15.760 20.260 25.503
1.00 1.05 1.15 1.26 1.34 1.40 1.57 1.66 1.85 1.96 1.90 1.04
MD
⌶ th
g MD 12
g th 12
0.063 0.131 0.206 0.286 0.378 0.470 0.574 0.686 0.809 0.935
0.063 0.132 0.207 0.289 0.378 0.475 0.580 0.695 0.820 0.955
¯ ¯ 1.03 1.06 1.08 1.11 1.14 1.19 1.21 1.26
1.03 1.07 1.10 1.14 1.18 1.22 1.27 1.31 1.36 1.41
1.06 1.12 1.18 1.26 1.34 1.73 1.53 1.64 1.76 1.90
0.184 0.406 0.672 0.985 1.369 1.807 2.353 2.945 3.716 4.567
0.184 0.407 0.676 0.996 1.379 1.834 2.372 3.010 3.764 4.655
¯ 1.05 1.10 1.17 1.24 1.31 1.39 1.47 1.58 1.66
1.07 1.14 1.22 1.31 1.40 1.51 1.63 1.77 1.92 2.09
1.09 1.18 1.30 1.43 1.58 1.76 1.97 2.22 2.52 2.85 3.23 3.63
0.377 0.882 1.547 2.408 3.543 4.985 6.874 9.266 11.961 11.777 14.972 23.594
0.377 0.882 1.549 2.419 3.549 5.008 6.886 9.286 12.326 16.114 20.714 26.072
1.02 1.10 1.21 1.31 1.46 1.59 1.73 1.91 2.14 2.09 1.86 0.60
1.10 1.22 1.35 1.51 1.69 1.91 2.17 2.47 2.83 3.24 3.70 4.19
MD
IV. RESULTS
In this section we present the simulation results for the density dependence of the excess compressibility factor and contact values of the cross radial distribution functions for mixtures with different composition, as specified by the value of x 1 and different values of the diameter ratio 2 / 1 , both in d⫽4 and d⫽5. This is done in Tables I–IV, where we have also included the density dependence of the contact values of the radial distribution functions as well as the theoretical results computed with the aid of Eqs. 共1兲–共5兲. Similar information is displayed in a more pictorial fashion in Figs. 1 and 2, where ⌶ is plotted vs d1 for 2 / 1 ⫽0.5 in d⫽4, and for 2 / 1 ⫽0.25 in d⫽5, respectively. For the latter mixture, runs with a bigger N⫽3888 were performed to assess the equilibration of the system and the effect of system size, as well as an initial exploration of the region where an interesting feature was observed. In the stable fluid region the results for ⌶ are indistinguishable whether N ⫽512 or N⫽3888. In all instances examined, the agreement between the simulation results and the theoretical predictions is strikingly good for the excess compressibility factors up to a reduced density where, for some values of the parameters and as it is
apparent particularly in the figures, a likely fluid–solid phase transition takes place. This phase transition certainly happens for the pure one-component system as also obtained by Michels and Trappeniers.8 We will come back to this point later on. Note that the agreement between theory and simulation also extends to a portion of the metastable fluid branch. On the other hand, the agreement between the results of the ansatz in Eq. 共2兲 for the contact values of g 12 and simulation is rather reasonable, the major differences being of the order of ten per cent in the stable fluid region. Further evidence for the likely phase transition can be drawn from Figs. 3–7. In Figs. 3 and 5, we present the results for the radial distribution functions of two mixtures, one in d⫽4 and one in d⫽5, respectively, for a given reduced density. Clearly the structure we are obtaining is that typical of a fluid. However, if one goes past the reduced density where the likely phase transition occurs 共cf. Figs. 1 and 2兲, the structure in the same mixtures starts to show some ordering as incipiently exhibited in Fig. 4 and much more neatly in Fig. 6, where g 11 looks much more solidlike. Figure 7 shows the density profiles of both components in the x 3 direction for the mixture of Fig. 6, where the above-
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J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
Binary mixtures of hyperspheres
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TABLE IV. Excess compressibility factor, ⌶⫽Z⫺1, and contact values, g 12( 12), for d⫽5 obtained with molecular dynamics 共MD兲 simulations and using Eqs. 共3兲 and 共2兲 共label th兲, respectively. The size ratios are 2 / 1 ⫽0.5 and 2 / 1 ⫽0.75.
2 / 1 ⫽0.50 x1
d1
⌶
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.50
0.75
⌶
2 / 1 ⫽0.75
th
g MD 12
g th 12
⌶
0.094 0.197 0.309 0.435 0.571 0.723 0.886 1.064 1.260 1.482
0.094 0.197 0.312 0.438 0.576 0.729 0.896 1.079 1.280 1.501
1.08 1.08 1.11 1.15 1.19 1.24 1.29 1.35 1.39 1.47
1.05 1.10 1.15 1.20 1.26 1.32 1.39 1.46 1.53 1.61
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.220 0.485 0.808 1.194 1.660 2.190 2.845 3.641 4.531 5.593
0.220 0.488 0.811 1.201 1.667 2.224 2.889 3.679 4.618 5.731
1.06 1.13 1.21 1.29 1.40 1.52 1.63 1.78 1.92 2.09
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
0.405 0.945 1.659 2.592 3.804 5.366 7.407 9.990 11.880 12.134 15.214 23.661
0.403 0.945 1.661 2.598 3.818 5.397 7.432 10.038 13.343 17.464 22.460 28.257
1.09 1.19 1.32 1.49 1.63 1.86 2.07 2.40 2.61 2.02 1.60 0.41
MD
⌶ th
g MD 12
g th 12
0.223 0.487 0.804 1.164 1.594 2.091 2.671 3.335 4.111 5.017
0.224 0.489 0.802 1.169 1.598 2.098 2.679 3.353 4.133 5.036
1.05 1.14 1.26 1.37 1.50 1.66 1.81 1.98 2.17 2.41
1.10 1.20 1.31 1.44 1.57 1.72 1.88 2.07 2.26 2.49
1.09 1.18 1.29 1.41 1.54 1.68 1.85 2.03 2.24 2.48
0.339 0.768 1.319 2.004 2.853 3.897 5.191 6.782 8.706 11.130
0.339 0.773 1.319 2.003 2.855 3.910 5.213 6.819 8.791 11.199
1.08 1.22 1.38 1.55 1.75 1.99 2.28 2.58 2.90 3.29
1.13 1.28 1.45 1.65 1.87 2.13 2.42 2.77 3.16 3.62
1.13 1.28 1.45 1.65 1.90 2.18 2.52 2.93 3.40 3.95 4.57 5.22
0.481 1.144 2.036 3.221 4.797 6.881 9.608 10.124 10.203 12.446 16.560 24.339
0.481 1.143 2.035 3.226 4.803 6.881 9.601 13.132 17.645 23.263 29.963 37.422
1.10 1.29 1.51 1.76 2.08 2.44 2.86 2.37 1.51 1.04 0.78 0.73
1.17 1.37 1.61 1.90 2.24 2.66 3.16 3.76 4.46 5.27 6.16 7.03
MD
FIG. 1. Excess compressibility factor, ⌶⫽Z⫺1, for three binary mixtures with a fixed size ratio 2 / 1 ⫽0.50 in d⫽4. Continuous lines are the theoretical results using Eq. 共3兲 and symbols are computer simulation data of this work. Open circles are results for the pure component of diameter 1 , which are consistent with those reported in Ref. 6. Filled circles, filled triangles, and filled squares are for compositions x 1 ⫽0.25, x 1 ⫽0.50, and x 1 ⫽0.75, respectively. The discontinuous line joining the simulation results is a guide to the eye indicating the likely presence of a phase transition not described by the theoretical equation of state.
yond the scope of the present work. We are currently undertaking such a detailed study which we hope to report in the near future. One could wonder to what extent the use of the Luban and Michels EOS 共Ref. 10兲 关cf. Eq. 共5兲兴 for the one component system is justified, specially in the case of d⫽5, where analytic results of Freasier and Isbister5 are available for the
mentioned characteristics of both components, the bigger one solidlike and the smaller one fluidlike, are also observed. V. CONCLUDING REMARKS
In this paper we have obtained the first 共to our knowledge兲 simulation results for the equation of state and structure of binary mixtures of hard hyperspheres in d⫽4 and d ⫽5. These results have allowed an assessment of a theoretical proposal for the equation of state of such mixtures, namely, Eq. 共3兲 关in conjunction with Eqs. 共5兲, 共4兲 and the explicit expressions for 共兲兴, which confirms the accuracy of such proposal that had been noted in lower dimensions. An interesting feature of the simulation data, not captured by the theoretical equation of state, is the appearance of a likely fluid–solid phase transition in these mixtures at high enough densities. We find evidence that the bigger component starts to order at a particular density while the smaller one tends to maintain a more fluidlike structure at the same density. Nevertheless, only a preliminary exploration of this region was carried out since a thorough study of the phase transition requiring the consideration of bigger system sizes was be-
FIG. 2. Excess compressibility factor, ⌶⫽Z⫺1, for three binary mixtures with a fixed size ratio 2 / 1 ⫽0.25 in d⫽5. Symbols have the same meaning as in Fig. 1. The open diamonds are results for x 1 ⫽0.75 and N ⫽3888. We have also included the results of this work for the single component case.
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J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
FIG. 3. Radial distribution functions for a mixture in d⫽4 with x 1 ⫽0.75, 2 / 1 ⫽0.50 and d1 ⫽1.4.
Gonza´lez-Melchor, Alejandre, and Lo´pez de Haro
FIG. 5. Radial distribution functions for a mixture in d⫽5 with parameters x 1 ⫽0.75, 2 / 1 ⫽0.25, and d1 ⫽1.4.
compressibility and virial EOS in the PY approximation. Of course it can be argued that, as we have already stated, Eq. 共5兲 represents the most accurate proposal even for d⫽5 up to the present day. Nevertheless, we have carried out computations using the virial EOS of Freasier and Isbister instead of Eq. 共5兲 in Eq. 共2兲 and confirmed that it leads to poorer agreement with the simulation data of the compressibility factor specially for the higher densities. For instance, for x 1 ⫽0.5, 51 ⫽2.0, and 2 / 1 ⫽0.33 we get ⌶ FI ⫽4.177, the label FI refering to Freasier and Isbister, which is to be compared with ⌶ MD⫽4.567 and ⌶ th⫽4.655. On the other hand, since the EOS of Luban and Michels is more accurate for the one component system, the contact values g 11 and g 22 obtained with the prescription of Eq. 共2兲 using the g( ) corresponding to Eq. 共5兲 are far superior to the ones obtained using the g( ) of Freasier and Isbister.5 Surprisingly, however, those of g 12 obtained from the EOS of Freasier and Isbister are in
better agreement with the results of simulation as provided in Tables III and IV. Nevertheless such a ‘‘better’’ agreement must be taken with reserve because one sometimes gets lower values than the simulation results and in other inFI MD stances, the values of g 12 are higher than those of g 12 . In th the case of the Luban and Michels EOS, g 12 always overestimates the simulation results but are otherwise reasonably accurate. It seems therefore that the overall performance of the Luban and Michels EOS for the one component system in connection with the present theoretical scheme supports the notion that its use is appropriate.
FIG. 4. Radial distribution functions for the same mixture as in Fig. 3 but for d1 ⫽1.6.
FIG. 6. Radial distribution functions for the same mixture as in Fig. 5 but for d1 ⫽2.0.
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J. Chem. Phys., Vol. 114, No. 11, 15 March 2001
Binary mixtures of hyperspheres
4911
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FIG. 7. Density profiles in the x 3 direction corresponding to the mixture in Fig. 6.
ACKNOWLEDGMENTS
M.G.M. and J.A. thank CONACyT for financial support. M.G.M. also thanks I.M.P. for a partial scholarship and V. S. Manko for his useful suggestions. The work of M.L.H. has been partially supported by DGAPA-UNAM under Project No. IN-117798. We also want to acknowledge useful discussions with A. Santos and S. B. Yuste on the subject matter of this paper.
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