A Computer Simulation Study of Fluid Ammonia

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Fluid Phase Equilibria 37 1987) 305-325Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands 305

A Computer Simulation Study of Fluid Ammonia

K.A. Hansour And S. Hurad

Department of Chemical EngineeringUniversity of Illinois at Chicago

Chicago, Illinois 60680, USA.

ABSTRACTWe report results of computer simulation studies for fluid ammonia. Thei ntermol ecu1 ar potential model consists of a central Cennard-Jones part, towhich are added point dipoles, quadrupoles and also polarizability. The lat-ter introduces effective many body intermolecular interactions. The modelparameters were obtained from dilute gas and crystal lattice properties.Properties calculated include dimer. liquid and solid structure and energy,transport and thermodynamic properties. The simulation results have been com-pared with experimental data to demonstrate the adequacy of the model forwide range of properties over a wide range of state conditions.

INTRODU T ION

Computer simulations studies are useful for investigating intermolecularpotentials, because no statistical mechanical approximations are involved inits implementation. Horeover because of their unique status as an intermedi-ate between experiment and theory. simulations can be useful in two generalways. In their first role, simulations can serve as experiments againstwhich theoretical methods such as perturbation theories and integral equations

0378-3&312/87/$03.50 0 1987 Elsevier Science Publishers B.V.


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can be tested for model fluids. Since a wide range of both thermodynamic andtransport properties can be obtained using simulations, a potential model canbe tested and compared against a wide range of available experimental data

using simuiations. In this paper, we have used the method of molecular dynam-ics (MO) to investigate a potential model for ammonia.

We report the results of a computer simulation study of fluid ammonia. Thepotential model consists of a central Lennard-Jones (LJ) interaction, withadded point dipole and quadrupole interactions, as well as dipole polarizabil-ity. Polar fluids such as ammonia are of interest because of their biologi-Cal, chemical and engineering significance. In addition, such a study canprovide valuable insight into the nature of several orientation-dependentproperties.

Computer simulations were carried out using the method of molecular dynam-ics (both equi 1 ibrium and nonequi 1 ibrium) . The properties studied includethermodynamic properties, structure, self-diffusion coefficient, and shearviscosity. Wherever possible the simulation results have been compared withexperimental data. The agreement with experimental data is in general quitegood. demonstrating the adequacy of the potential model used for a wide rangeof properties, for which sufficient good quality data is available to make

INTERiOECUAR POTENTALAmmonia was mode led using a potential of the form:

(:.W,*~s)= U( :l2) =4,(12) + y2) +Ucl p( 12) +UQQt

+u i nd(121+ upol (12)

such comparisons meaningful.

1)

where 7 is the vector joining the centers of mass of molecules 1 and 2, and wiis the orientation of molecule i, Uo(12) is the central LJ potential,


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U,(r,,) = 4c C(u/r,1)2 - (o/rJ 1 (2)

and w UCrQ UQQ Uind, Upo1, are the dipole-dipole, dipole-quadrupole,quadrupole-quadrupole. induction and polarization contributions respectively.

The induction energy is the contribution due to polarizability, and the polar-

ization energy is the work expended in creating th induced dipole. This mod-

el accounts explicitly for the multi-body induction effects, which are calcu-

lated using a self-consistent technique (Gray and Gubbins, 1984; flurad, 1984).

We calculate the electric fields due to all the j molecules at the COH of mol-

ecule i by:

-P + 1E= 2i i j fi.. ; (2)ij 2 :J 3 i+j j : T (3)ij

wh r

T (4) = iTq l/ r I .ij ijThe induced-dipole for molecule i is then given by:

+ a +P II Q . Einduced, i i

(3)

(4)

where P is the polarizability tensor. In this study we used a scalar polariz-

ability represented by, a = (113) Cnxx + nyy + nzz I since the anisotropy of

the polarirability in ammonia is small. The total dipole moment used in cal-

culations is given by,

+ -t +c1 sz fi + Ptotal,i perm. i induced, i (5)

since equations (3) and (4) are coupled. an iterative Solution is necessary to

solve them. We found that 2 iterations were sufficient for the convergence o

the fields, with a tolerance of less than 10-4 .


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Our potential mod el has two adjustable parameters the two LJ parameters).These were obtained from dilute gas and crystal lattice properties asexplained later. A sl ight f inal adjustment was made in r/k to obtain the cor-

rect saturation densities atso see the discussion of liquid thermo dynam icproperties). The parameters of the mod el as well as other e xperimental con-

stants used Kukol ich, 1971) are given in table 1.

Table I: Intermolecular potential parameters for amm onia.

c/k (K) 220.6o A) 3.400P esu-cm) 1.47 x 10-18Q esu-cm - 3.307 x lo-26CY cm21 22.6 x 10-25

IAULATION WETHODS

AII the properties reported were obtained using the usual mo lecular dynamicstechnique, MD in the microcan onical ensemble, except for shear viscosity whichwas obtained using nonequil ibrium moand Harris, 1984: Haile and tupta,

1ecular dynamics, NEHD Evans, 1983; Evans

983). For both types of simulations, weused a fifth-order Predictor-Corrector metho d Gear, 1971) for the solution ofthe translational motion , and a fourth-order metho d for the solution of therotational motion. The singularity free quaternion algorithm Evans and Hurad.


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1977) was used for solving the rotational equations. Most of the simulationswere started using the final positions and orientations of a similar previoussimulation to speed up the equilibration process. We typically rejected 1200

steps, and the production run consisted of 6000-~000 time steps depending onthe state point studied. Denser state conditions generally required longerruns.

The NEMO technique used time-varying non-orthogonal boundary conditionswith homogeneous shear for couette flow, where a 1 inear hydrodynamic velocityprofile was imposed on the system during the initial equilibration period,after which this constraint was removed. In addition, during the equilibra-tion period we resealed both the translational and rotational velocities, butduring the production run we only resealed the translational velocity. Al 1NEfiD simulations were run at a reduced strain rate of y* ~1.0, where y* =yo (c/H) -12. We found that the viscosity was independent of the shear rate upto shear rates of y* ~1.0. For both MD and NEMD, we used a reduced time step,At* = 0.00017. In addition, a cutoff of 30 was used. We also tested the con-tributions due to long range forces for these properties using the reactionf ietd method (Gray et al ., 1986) . Our results show that these contributionsare not signif icant ( i.e. within the statistical fluctuations) for the prop-erties calculated. However, we expect these contributions to be more signifi-cant for dielectric properties. Unfortunately, our simulations were not longenough to to provide a reliable estimate of the dielectric properties, whichperhaps need production runs in excess of 50,000 time-steps.

RESULTS0; lute Gas PropertiesThe potential parameters used in this study were first obtained by a simulta-neous non-linear least squares fit between the calculated values and availableexper imenta 1 values of the pressure second virial coefficient, dimer struc-ture. dimer energy, crystal lattice structure and energy.


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Pessure Second Vr ia Coefficients:

The pressure second virial coefficients were calculated using (Buckingham,1959) :

N 2B (T) = 4P C r,r dr c sin%, d%, c sin%, d%, f: db,,

- lJ(l2)/kT 6). (l-e 1

The multiple integral in (6) was evaluated using a nonproduct integrationalgorithm (Stroud, 197); Murad, 1978). The results obtained using the poten-tial are shown in figure 1, where they are also compared with available expcr-imental results (Haar and Gal lagher. 1978) , The uncertainty in the experimen-tal data is at least *9 cm3/mol., and we estimate the accuracy of ourcalculated values to be within 2 percent. Our results show better agreementfor the pressure virialies of ammonia (Sagarik

coef ficients than that obtained in previous such stud-et a I . , 1986; Klein et al., 1979).

Dielectric Second Vria Coeff icients:We also used the potential to calculate the dielectric second virial coeffi-cient, B,(T) which is generally very difficult to measure experimentally.8, (T) is very sensitive to the non-central part of the potential and repre-sents the first deviation from the ideal behavior due to polarization (Buck-i ngham, 1959) , and is given by:

2xNA6 (T)= - 9 kT JZ r 2 dr ,z

2asin%, d%, ,I sin%, d%, I,t

-U(lZ)/kT-2&J ). e 7)

where


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100

-200

-300

-400

6 -6OCz50 -6cN

1

= THIS STUDY

I I 1 I 160 200 260 30 0 350 400

W

i g u r e I: Pressure Second Virial Coefficients.. The solid line is the exper-imental data (Haar and Gal lagher. 1978).


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The results obtained are compared with some estimates of B,(T) obtained fromexperimental dielectric constant values in table 2. The uncertainty in theseexperimental values (fiacRury and Steele, 1974) is therefore large, and in

view of this we feel that our results are satisfactory.

Table 2: Dielectric Second Virial Coefficients.

Temperature This Study Experiment

K Ei, cm6/mo12 )373 175 618398 145 545423 125 410

Dmer Sructure and Energy:

Results for the dimer are reported in table 3, which show good agreement withexisting experimental data and other calculated results (towder, 1970; Kuchit-su et al ., 1968; Oi 11 et al ., 1975; Allen, 1975; Brink and Glasser. 1981).As mentioned earlier, the dimer properties were fitted to two LJ parameters.Our results predict a linear geometry for the dimer as shown in figure 2. Wenote that a more reasonable range for the dimer energy would be 8.37 - 14.8KJ/mol, since the 18.83 KJ/mol value reported reported by Lewder 1970)appears unreasonable (Hinchliffe et al. 1981; Jorgensen and Ibrahim, 1980).


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igure 2: Dimer Structure.

Table 3: Ammonia Dimer

This Study Exp or QH19 (degr eel 71 670~ (degree) 0 0I B (degree) 0 0

r, z (A) 3.340 3.05-3.53

- E (KJ/mol) 11.74 8.37-18.83


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Several of the simulations for fluid ammonia were carried out along the co-existence curve, since most of the experimental data (Haar and Gallagher,

,978) available for ammonia is along the co-existence curve. As mentionedearlier, we found it necessary to rescale c/k, the energy parameter of ourpotential model, if we were to reproduce the pressure correctly. This is notunusual and has been found to be true by others as wel I since the pressure isvery sensitive to the potential model. The resealing procedure used is simi-lar to that used by Hurad, et al. 198O), except that we did not find it nec-essary to rescale u . The resealed value of c/k was found to be 220.6 K. Theuncertainty in the experimental data (Haar and Gallagher, 1978) is reported tobe less than f 1 % for the data represented in figures 3 and 4.

Thermodynamic PropertiesIn figure 3 we report results for the configurational internal energy. As canbe seen, the agreement with experiment (Haar and Gallagher, 1978) is verygood. We must point out here that what we have shown as error bars are infact only fluctuations in observables from HO simulations, such fluctuationsare size dependent. The true errors in ND simulations are very difficult toestimate, especially since they vary from property to property and depend uponseveral factors, among them are (a) round-off error resulting from the comput-er hardware, (b) reliability of the finite-difference algorithm used to solvethe differential equations, (c) characterization of equilibrium or steadystate (depending on the type of property under study, e.g. equilibrium or ordynamic) and the length of the segments over which averages are accumulated tocalculate the different properties.In figure 4 we present results for the co-existence curve between the tripleand critical points of ammonia.


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200 225 250 275 300 325 350 375 400 425 4TEMPERATURE.K

3 5

0

Figure 3: Configurational Internal Energy. The solid line is the experimen-tal data (Haar and Gallagher, 1978).

60

55

G 500; 45

w 405g 35

5i 30

25

20

15 /

.+-

150 175 200 22s 250 276 300 325 350 375 400 425 450 4T WI

6

Figure 4: Thermodynamic Co-existence Curve. The solid line is the experimen-

tal data (Haar and Gal lagher, 1978) .


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Li qui d st r uctureFi gur e 5 compares our HO results for the nitrogen-nitrogen radial distributionfunction, GNN (r) with the experimental x-ray diffraction data (Narten, 1977) -Also shown are some of the other reported MC/MD studies. tn figure 5, thesolid curve (this study) represents a state point of T= 280 K. V= 26.51cm3/rnol. The experimental data (Narten, 1977) was reported at T= 277 K, V=26.92 cm3/mo1. The dashed I ine represents the MC results of Jorgensen andlbrahim (1980) at T= 240 K, V=24.97 cm3/moi ., the chain-dash represents the MDdata of Hinchl iffe et al. (1981), at T=271.3 K, V=26.49 cm3/mol, while thechain-dot curve represents the data of McDonald and Klein 0976) at T=l96 K,V=23.30 cm3/mol. The first neighbor peak from our study is at 3.57 i with amagnitude of 2.5 while Nartens data peaks at 3.37 i\ with a magnitude of 2.06.In addition, Nartens data shows a clear shoulder at 3.7 A, while our resultsshow a weaker shoulder at 3.8 i. Narten attributes the shoulder to the hydro-gen bonded neighbors around the central molecule. To our knowledge, no otherH or MC study (Hinchliffe et al., 1981; Jorgensen and Ibrahim. 1980; lmpeyand Klein, 1984; Kincaid and Scheraga, 1982; Sagarik et al ., 1986) has beenable to reproduce such a shoulder (see also figure 5). We fee) this is sig-nificant, especially since we did not include any explicit hydrogen bond forcein our potential. However, important differences between all simulationresults and Nartens data continue to exist. This is presumably because themodels do not show hydrogen bonding to be as strong as those shown by Nartensexperiment, al though doubts have also been raised about the accuracy of Nar-tens data (Sagarik et al ., 1986; Kincaid and Scheraga. 1982).

We also calculated the coordination number nNN up t0 the first minimumafter the the first peak using:

minnNN = 47rP I, CNN(r) r2 dr8)


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Figure 5: Radial Oistribution Function.. The sol id line represents the HDresults, om is the experimental data (Narten, 1977) .

where r,,,i,, is approximately 5.1 A (see Figure 5). The calculated value of nNNwas found to be 11.6 which compares favorably with the experimental value of12.0, although in view of the disagreement between our ghR and the experimen-tal values, this agreement could be Fortuitous.

Shear Vscosit y and SeJf-DffusionAnother test of a potential model lies in its abi 1 ity to predict transport(nonequilibrium) properties accurately. The self-diffusion coefficient, D wasobtained from the slope of the mean square displacement using the equation


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(ticQuarr ie, 1976 :

9)0 I I imt+- 6AtThe MD simulation results along the co-existence curve are compared in figure6 with the available experimental data (OReilly et al., 1973). We note thatthe dashed line in the figure is an extrapolation of the experimental data,since the experimental data are only available between 200- 298 K. Theexperimental data carry an uncertainty of f 7.5 %, while our values are prob-ably accurate to within IO percent. In the figure, we also report hD esti-mates of others (Impey and Klein, 1984; Sagari k et al., 1986 . In general,our results show Arhenuis-type behavior which is also evident in OReillysdata, and the agreement is rather satisfactory.

Most of the NEHD studies carried out previously have been for the transportproperties of nonpolar fluids and fluid mixtures (Evans, Ig8j;Hanley andEvans ; 1981; Heyes, 1983: Hoover et al ., 984). WE are aware of only one pre-vious study for a polar fluid (Murad, et al .,lg&). An NEMO estimate of vis-

cosity can be a useful test of the intermolecular potential model for a polarfluid. In figure 7 we present our NEMO viscosity results along with theavailable experimental data (Krynicki and Hennel, 1963; Alei et al., 1972) forState conditions on the co-existence curve. The error bars in the figure areindicative of the fluctuations in the calculated viscosity, and are within f4%. The experimental uncertainty is also reported to be within f 4 % for both

data sets.e would like to note that Some of our simulations are not exactly on the

co-existence curve as is usual in MO. However, we have made no attempt tocorrect the NEMO results for such small temperature differences. although ingeneral they would improve the agreement with experimental results. We also


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20figure 6:

nI I 1 1 532

ld/ T C-K44

Self-Diffusion Coefficients.. o our HD results,data (ORei lly et al ., 1973). n Sagarik et al.and Klein (1984).

56

- experimental1986). x lmpey


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I I 1 I 1.04 30 0.55 0.60 0.65 0.70DENSITY (WiAM/CC) 0.75 0.10Figure 7: Shear Viscosity.. ee are the NEHD results, the solid I ine repre-

sents the experimental data (see text).

performed simulations at lower and higher shear rates. Our results confirmthat ammonia is Newtonian in the shear range used here. We also found ourresults for viscosity and self diffusion coefficients satisfy closely theStokes-Einstein relationship (S-E) :

kTD - 4r II llh (10)


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where Ah (= 1.83 A(1973) . For exampqS-E= 0.230 CP, whqS_E 0.179 cP.

1 I is the hydrodynamic radius as given by OReilly et al.l e. at p= 0.6834 gm/cm3 and T= 235.4 K. ~~, ~~~=0. 245 CP andile for p= 0.6332 gm/cm3, T= 276.2 K, VNEMO= 0.181 cP and

Crystal lattice

321

As mentioned earlier, we used the properties of the solid phase (energy andstructure) to obtain the potential parameters of our model, The calculationswere carried out using the technique suggested by Haymet et al. (1981). Theresults obtained for crystal lattice properties are shown in table Ir alongwith the available experimental data (Olovsson and Templeton, 1959; Reed andHarris, 1961; Shipman et al ., 1976) and and other reported studies (Righiniand Klein. 1978; Ouquette et al ., 1978; Righini et al., 1978: Klein et al..1979) . The uncertainty in the experimental estimate of the lattice energy(Shipman at al., 1976) is reported to be f 4.184 KJ/mol. The agreement is ingeneral satisfactory.

Table : Crystal Lattice Structure and Energy

This Study Exp or QHa Ci) 5.12 5.084b (;I 5.12 5.084c (A, 5.12 5.084P (g/cm 0.843 0.861-0.922

- E (KJ,hoi) 32.22 28.43-41.81


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CONCLUSIONSWe have developed an intermolecular potential model for ammonia which explic-tly includes multi-body induction effects, and has been found to reproduce a

wide range of thermodynamic and transport properties. Although ammonia hasbeen studied before. previous potential models have not been tested for aswide a range of properties and state conditions as has been done here. Theprevious models have also not included multi-body induction contributionswhich are important in ammonia.

List of Symbols

*hB,

BPDEiGNNn

NAQTT* -JU(12)

LJOuc(&PQQQ

a. b, ck

NN

hydrodynamic radius, equation (10)second dielectric virial coefficient

second pressure virial coefficientself-diffusion coefficientelectric field at molecule i due to all other j molecules

nitrogen-nitrogen radial distribution functionmolecular weightAvogadros numberquadrupole momenttemperaturemolecular tensor as defined in equation (3)pair interaction energy between molecules 1 and 2central Lennard-Jones interaction energy, equation (2)dipole-dipole interaction energydipole-quadrupole interaction energyquadrupole-quadrupole interaction energydimensions of the unit cell (crystal lattice), angstromsBoltzmanns constantcoordination number, defined by equation (8)


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r

mint

Ar*

intermolecular separation vector between centers of massthe location of the first minimum in the GNR(r)time

mean square displacement

Greek symbolspolarizabilityreduced strain rateenergy parameter of the intermolecular potentialshear viscosityshear viscosity obtained from NEMI simulationshear viscosity obtained from the Stokes-Einsteinrelationship, equation (10)dipole momentnumber density, N/V, where N is the number of molecules,and V is the volume of the cell.reduced densi ty, pa 3length parameter of the intermolecular potentialmolecular orientation anglesorientation vector of molecule i

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A Computer Simulation Study of Fluid Ammonia

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