PHYSICAL REVIE% A VOLUME 13, NUMB EB 3 Self-diffusion in simple dense fluids P. Carelli Laboratorio di Flettroniea dello Stato Solido of the Consiglio Nazionale delle Ricerche, Via Cineto Romano, 42 Rome, Italy A. De Santis Istituto di Fisica, Facolta di Chimica Industriale, Uni)/ersita di Venezia, Vene~ia, Italy I. Modena Laboratorio di Elettronica dello Stato Solido of the Consigho Itlazionale delle Ricerehe, Via Cineto Romano, 42 Rome, Italy F. P. Rien Istituto di Fisica 'G, Marconi, ' Universita degh Studi di aroma and Gruppo Nazionale di Struttura della J()/lateria of the Consiglio Nazionale delle Ricerche, aroma, Italy (Received 14 October 1975) The behavior of self-diffusion in Kr and CH~ at intermediate densities is discussed in comparison ~ith the Lennard-Jones and the hard-sphere Quid. Additional measurements of self-difFusion in Kr at low densities are reported. I. INTRODUnION Recently the main, progress on the theory of dense fluids has been furthered by the results of computer "experiments. " These experiments have been performed using either realistic inter- atomic potentials, such as the Lennard-Jones 12- 6, ' or the hard-sphere potential. ' These experi- ments are vex'y useful as a check on the theories; however, it is also important to have an idea of the relationship between these models and the real flu- ids. Such an evaluation can be obtained by compar- ing the model results with the macroscopic experi- mental data of the real fluids. This comparison could be carried out for both the equilibrium prop- erties and the transport properties, in the latter case for mainly the self-diffusion coefficient D. Following this point of view, some years ago we measured the self-diffusion coefficient in krypton at T=220K as a function of the density p in the range 0. 35 «p «2. 0 g/cm'. From our work' we drew the followmg p/eisppg DlQJ'p conclusion: 80th a Len nard-Jones and a hard-sphex e fluid show only qual- itative agreement with the experimental behavior of & in Kr as a function of density. The reasons why we eall these previous conclusions' "preliminary" are the following: (a) In the case of the Lennard-Jones fluid we used only one set of interatomic potential constants (e and &) to translate the reduced quantities to the Kr ease. Moreover, we compax'ed the experimental results with the computer experiments' which do not consider the influence on the value of D of the long time tail in the velocity autocorrelation func- tion (VAF). (b) In the case of the hard-sphere model we lim- ited our analysis to a comparison with the Aldex- %ainwright plot by using a single value of the ef- fective hard-sphere diameter derived from I'VT data. The present paper completes our previous analy- sis along the following lines: (a) In the case of the Lennard-Jones fluid we will perform the coxnparison by using not only one set of & and & values but all meaningful sets which are consistent with the limiting low-density value of the product Dp, i.e. , (Dp)~ 0. We also take into ac- count the correction' to D for the long time tail in the VAF. (b) We will test the validity of the hard-sphere model with respect to our data by investigating in detail the behavior of the effective hard-sphexe radius as a function of the density. (c) We extend 'the analysts done for Kr to the available experixnental data for CH~. Since as far as we know there are no available data for Kr at 226 K and at low densities in the lit- erature, we completed our previous measurements by collecting new data at T = 220 K in the range 0& p «0. 358 g/cm'. For the sake of completeness we also carried out some measurements of D at 7 = 2't3 K and 0 & p «0. 5 g/cm'. II. EXPERIMENTAL RESULTS The capillary method of Anderson and Saddington was used. The apparatus, as well as the expex'i- mental procedure, has been extensively described in our previous work. ' The capillary was 1. 5 mm in diametex and 48. 0 mm long. The experimental results ax'e given in Tables I and II. In the last column we show the Dp values at 220 K (for Table I) and 2'l3 K (for Table II). As ln our previous
Transcript
PHYSICAL REVIE% A VOLUME 13, NUMB EB 3
Self-diffusion in simple dense fluids
P. CarelliLaboratorio di Flettroniea dello Stato Solido of the Consiglio Nazionale delle Ricerche, Via Cineto Romano, 42 Rome, Italy
A. De SantisIstituto di Fisica, Facolta di Chimica Industriale, Uni)/ersita di Venezia, Vene~ia, Italy
I. ModenaLaboratorio di Elettronica dello Stato Solido of the Consigho Itlazionale delle Ricerehe, Via Cineto Romano, 42 Rome, Italy
F. P. RienIstituto di Fisica 'G, Marconi, '
Universita degh Studi di aromaand Gruppo Nazionale di Struttura della J()/lateria of the Consiglio Nazionale delle Ricerche, aroma, Italy
(Received 14 October 1975)
The behavior of self-diffusion in Kr and CH~ at intermediate densities is discussed in comparison ~ith theLennard-Jones and the hard-sphere Quid. Additional measurements of self-difFusion in Kr at low densities arereported.
I. INTRODUnION
Recently the main, progress on the theory ofdense fluids has been furthered by the results ofcomputer "experiments. " These experimentshave been performed using either realistic inter-atomic potentials, such as the Lennard-Jones 12-6, ' or the hard-sphere potential. ' These experi-ments are vex'y useful as a check on the theories;however, it is also important to have an idea of therelationship between these models and the real flu-ids. Such an evaluation can be obtained by compar-ing the model results with the macroscopic experi-mental data of the real fluids. This comparisoncould be carried out for both the equilibrium prop-erties and the transport properties, in the lattercase for mainly the self-diffusion coefficient D.
Following this point of view, some years ago wemeasured the self-diffusion coefficient in kryptonat T=220K as a function of the density p in therange 0.35 «p «2. 0 g/cm'. From our work' we drewthe followmg p/eisppg DlQJ'p conclusion: 80th a Lennard-Jones and a hard-sphex e fluid show only qual-itative agreement with the experimental behavior of& in Kr as a function of density. The reasons whywe eall these previous conclusions' "preliminary"are the following:
(a) In the case of the Lennard-Jones fluid we usedonly one set of interatomic potential constants (eand &) to translate the reduced quantities to the Krease. Moreover, we compax'ed the experimentalresults with the computer experiments' which donot consider the influence on the value of D of thelong time tail in the velocity autocorrelation func-tion (VAF).
(b) In the case of the hard-sphere model we lim-
ited our analysis to a comparison with the Aldex-%ainwright plot by using a single value of the ef-fective hard-sphere diameter derived from I'VTdata.
The present paper completes our previous analy-sis along the following lines:
(a) In the case of the Lennard-Jones fluid we willperform the coxnparison by using not only one setof & and & values but all meaningful sets which areconsistent with the limiting low-density value of theproduct Dp, i.e., (Dp)~ 0. We also take into ac-count the correction' to D for the long time tail inthe VAF.
(b) We will test the validity of the hard-spheremodel with respect to our data by investigating indetail the behavior of the effective hard-sphexeradius as a function of the density.
(c) We extend 'the analysts done for Kr to theavailable experixnental data for CH~.
Since as far as we know there are no availabledata for Kr at 226 K and at low densities in the lit-erature, we completed our previous measurementsby collecting new data at T = 220 K in the range0& p «0.358 g/cm'. For the sake of completenesswe also carried out some measurements of D at 7= 2't3 K and 0 & p «0. 5 g/cm'.
II. EXPERIMENTAL RESULTS
The capillary method of Anderson and Saddingtonwas used. The apparatus, as well as the expex'i-mental procedure, has been extensively describedin our previous work. ' The capillary was 1.5 mmin diametex and 48.0 mm long. The experimentalresults ax'e given in Tables I and II. In the lastcolumn we show the Dp values at 220 K (for TableI) and 2'l3 K (for Table II). As ln our previous
work' we used the scale factor (7.'0/T)" to reducethe data, for small temperature differences, to thetemperature To. This scale factor was deduced'
from the Chapman-Enskog solution of the Boltz-mann equation'and from the CH4 experimental data'and is in quite good agreement with results of the
computer experiments. The experimental errorsresult mainly from the statistical errors in the
concentration measurements. ' The reproducibilityof the data is within the experimental error.
The present results are in good agreement withthose available in the literature, as is clearlyshown in Fig. I, where our data at 273 K are com-pared with those of Durbin and Kobayashi' and withthose of Trappeniers and Michels. ' All these datahave been scaled at 273 K by using the scale factor
FIG. 1. Dp values of Kr at 273 K as a function of den-sity. G: data of this vrork ~ ~ data of Ref. 6 ex-trapolated value of (Spp)& 0 of Ref. 7. The line repre-sents the best fit to Eq. (2).
(213/T) . In Fig. 2 we show all of the availabledRtR of the dlffuslon 1n Kl Rt 220 K by RdQHIg to thepxesent data those data px'eviously obtained. '
The low-density experimental data were fittedwith the equation Dp =A + &p. As has been alreadysuggested, ' the linear equation can be considereda good fit to the experimental data if the standarddeviation of the fitted points does not show any sys-tematic trend; the normalized g is always lessthRn 1. The coeff1clents of the f1tted equationswere found to be independent of the size of the den-sity interval within the experimental error.
The l1neRx' equRtlon for Dp wRs found to SRt1sfythe previous conditions at T =220 K fox 0&p&1.01g/cm' (the following intervals were chosen: 0& p&0.2, 0& p&0. 2'l5, 0&p&0."t91, and 0«p&1. 01 g/cm') and at T =2'l3 K (where the intervals were0& p&0. 2, 0&p&0.25, 0&p&0.3'73, and 0& p&0. 47g/cm').
ments on a I.ennard-Jones fluid' with the xeal ex-periments is the choice of the intermolecular po-tential parameters ~ and 0 to translate the reducedquantities into the real quantities related to ourparticular substange. In our previous papex' thevalues for e and cr were obtained from the criticaland triple point, i.e. , e/Es = 165 K and o =3.65 A,whexe K~ is the Boltzrnann constant. In the presentpapex we want to extend the comparison in order toinclude all of the meaningful couples of q and 0which are consistent with the (Dp)z„, value. Theprocedure for obtaining e and o from (Dp)„o iswell known. ' The values of (Dp), , for Kr avail-able in the literatuxe" have been considered to-gether with the values obtained from Eqs. (1) and(2) of the present work. AH of these data agreewithin the experi, mental error with the values forq and 0 contained in the band shown in Fig. 3. Inparticular we chose the sets reported in Table III.
The second problem encountered is that of takinginto account the contribution of the long tail of theVAF to the D values obtained from the computerexper1ments. Th18 col x'ection» AD& has beenshown theoretically'0 to be
4 Kg l'[4 (f1+ &)]3/a 0
where v 18 the klnematlc vlseoslty» to ls the timeat which the asymptotic behavior t ' ' of the VAFis already reached, i.e., for t & to the VAF is welldesex'ibed by
2EB Tm [4v(fl+v)j'"
where I is the mass of the molecule. " In Eq. (3)f o was chosen to be 400 h =400 x 0.032(neo'/48e)'as pxeviously suggested. '0 D and ~ are the experi-
Dgxi0 9cm sec
As we can see from Eqs. (I)»d (2) «»oPe ofBp as a function of p is clea.rly positive for T =220K, while there is an indication„although completelywithin the expex'imental ex'x'ox', that it becomesnegative at T =273 K. This last result is in agree-ment with that found at T =298 K.'
L
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III. DISCUSSION
A. COAlp8flSoA %F181 thC LCflASFd"JofleS flUKl
As already mentioned in Sec. I, the first problemencountex'ed %'hen comparing the computer expex'1-
FIG. 2. Dp of Kr at 220 K as a function of density.data of the present vrork; k: data of Ref. 3. The
dotted line represents the best fit to Eq. (1).
CABELL', DE SANTIS, MODENA, AND RICCI
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mental values for Kr." The comparison betweenthe Lennaxd-Jones fluid and the Kr data is shownin Table III.
The density and temperature ranges under can-sldex'ation were taken to be as wide as possible,in agreement with the values of the reduced densityp and the temperature 7.
' given in Ref. j. and theexperimental values of 8 available in the l.iterature.Fox' p*=0.3, the D values were determined byinterpolation; for p* & 0.5 and 0.7 & T & 5 we calcu-lated D„o from Eq. (49) of Ref. 1, which should becorrect within +5/z. ~ All Dwo have been consider-ed to be affected by an error of ~5%. To estimatethe best pair of « /K and o we constructed a qual-ity factor
5
~=-5 Q Ii~ I,whexe the index i x'anging from I to 5 covers all ofthe experimental points we considered for eachpair « /K and c. I' is defined in Table 111 and ithas the meaning of normalized deviation. FromTable DI it can be seen that the best choice of «/Kand o would be can=3. 70 A., «/K=174 K. However,in this case too, we must say that the behavior ofD vs p for Kr is cleaxly diffexent from that of I en-nard-Jones fluid since the deviations from it havea well-defined trend as a function of density; thisis also shown in Fig. 4. Since extensive data for8 as a function of p are available in the literatuxe'only for CH4, we repeated the present analysis inthis case also. The results are summarized inTable IV. This taMe clearly shows that the bestpair o and «/K' is o =3.60 A and « /K =187 K.Moreover, contrary to what happens for Kr, theI' values do not show any systematic trend as afunction of density, as is clearly shown in Fig. 4.
Ne then conclude that certainly a I.ennard-Jonesfluid cannot describe a xare gas as far as the be-havior of the diffusion coefficient is concerned,
PIG. 4. I' vs p/pc. : Kr; h. :
since we were not able to find a pair « /K and owhich could act as effective interatomic potentialparameters for a wide range of densities. Qn thecontrax'y, the behavior of CH4 does not seem todisagree with that of the Lennard-Jones fluid, atleast within the exrors of the available results ofthe computer experiments.
B. NBA'-SphCfC flUld
Next we want to verify that the behavior of D forKr as a function of p along an isotherm is well represented by that of a hard-sphere fluid. This canbe accomplished by verifying that the experimentalvalues satisfy the corrected" Enskog theory (CET)which gives the following x'elatioQship for B:
D EfAw
g(od'
whex'e D~p ls graven by experimental D values» Dg
is the D value calculated from the original Enskogtheory at low density (i.e., the Chapman-Enskogsolution of the Boltzmann equation for hardsphel es' ), &p ls the hard-sphere radius, g(vg) 18
the equilibrium radial distribution function at con-tact, and f„„is the Alder-Wainwright correctiongiven in the last column in Table I of Ref. 2. Toverify Eq. (4) it is necessary to specify the cr„valueAnother way to look at Eq. (4) it is that it defines,for every thermodynamic state, the g„value need-ed to match the expex'imental values and the CET.Qbviously saying that the real data are well rep-resented by the hard-sphere system is equivalentto saying that g„ is constant over an appreciaMerange of density, at constant temperature.
Ne performed this analysis for Kr at 220 K inthe density range 0& p&2p, ; the results are shownin Fig. 5. From this figure it is clear that forself-diffusion in Kr, at 7'/7', =1.055, && isstxongly density dependent and, moreover, thisdependence is not monotonic. Therefore in this
1136 CARE LLI, DE SA NTIS, MODE NA, AND RIC CI
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case the use of the hard-sphere model does notmake sense in any appreciable range of density, atleast for 0 & p & 2p„ if we want an agreement with
experiment better than 20%%u~. This conclusionagrees with the general expectations in the case oflow densities, p& p, , while for higher densities itopposes what has been suggested for other fluids. "In fact it was believed that when the density be-comes higher the sum of the attractive parts of thetwo-body interatomic potentials generates a nearlyflat potential surface. The obj ection could be raisedthat 220 K for Kr is a low temperature, T/T,= 1.055, and therefore the molecule could also besensitive to any small roughness of the potentialsurface. However, we think that this objection isincorrect since if we apply Eq. (4) to the case ofself-diffusion in CH, at 194.75 K, ' i.e., T/T,=1.025, we obtain the o~ values shown in Fig. 6.When examining this figure it becomes clear thatthe hard-sphere model makes sense in the regionp, & p. Therefore in the same range of reduceddensity (i.e., 1&p/p, &2) the hard-sphere model isa rather good approximation for the description ofself-diffusion in CH4. It is, however, a poor ap-proximation in the case of Kr, although the Kr dataare obtained at a reduced temperature higher thanthe CH4 data.
At this point it is interesting to investigate thedirect comparison between the experimental dataof Kr and CHd. In Fig. 7 we show pD/(pD)~ d vsp/p, . Owing to the use of the reduced quantities,if their behavior were the same Kr and CH4 shouldcoincide since the difference in the reduced tem-
SELF-DIFFUSION IN SIMPLE DENSE FLUIDS
oh (A1also, we do not make use of the potential param-eters to build reduced quantities.
)))) I,
I l
0.1 0.2 0.3 0.4 q gfcm'
FIG. 6. Pseudo-hard-sphere diameter 0 z as a func-tion of reduced density p/p, for CH4 at 194.75 K. a
& hasbeen dellved froxn Eg. (4) using the expellxnental data ofBef. 4 for D». The bars represent the uncertainty de-rived from the uncertainty in D». The solid line repre-sents a fit to the points.
peratures is very small. From Fig. 7 we can seethat the only noticeable difference is for p & p, . Inthe density region around p, the Kr data show abump while the CH4 data are very flat; moreover,in the high-density region the slope
pc s(Dp)(Dp) -.
is higher fox Kr than for CH4, i.e., in reducedquantities the diffusion in methane decreases withincreasing density more slowly than in the case ofKr. VVe want to remark that the comparison madein Fig. 7 is better than that of Ref. 3 since in thepresent case we use the real experimental valuesfor D in CH4 instead of the interpolation formula;
C. Enskog theory
As we have seen above it is unrealistic to use theoriginal Enskog theory to describe self-diffusion inKr at 220 K, because it is not possible to xepre-sent a Kr molecule witha hard sphere for an appre-ciable range of densities (see Fig. 5). However„ inthe case of viscosity and thermal conductivity ofAr, 0» and P-H» some authors" pointed out thatthe modified version of the Enskog theory (MET)gives a good description of the experimental datain the range 0&p&2p, . It is then worthwhile to in-vestigate the success of the MET to describe theself-diffusion data in Kr at 220 K. The MET givesthe following relationship for D:
D (I/V)(b + T db/dT)D, (V/nr„Z, ){sP/sT), - I '
where 0, is the Chapman-Enskog solution of theBoltzmann equation for the real potential, "V isthe molar volume, N~ is Avogadro's number, and& is the second virial coefficient of our substance.We take b and (SP/S T)„ from the Kr P V T dataavailable in. the literature. " %e take Do from theexperimental value (Dp) ~, of Kr at 220 K so thatD, =2.43&&10 '/p cm'/sec as from Eq. (1). In thisway we avoid the problem of the choice of the rightinteratomic potential and we satisfy the conditionD/D, -1 for p-0. The results are shown in Fig.8. It is interesting to note that Eq. (5) gives a gooddescription of the experimental data. In fact, thequalitative behavior is well reproduced and thequantitative discrepancy, also increasing with in-creasing density, remains less than 8% at p, and
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I'IG. 7. p&/(pD}& 0 vs p/p, . *: CH4 data at T =194.75K from Bef. 4; 0: Kr data at 220 K from Bef. 3 andfrom the present paper.
FlG. 8. D/D0 vs p/p~ fox" Kr at 220 K. The solid curverepresents Eq. (5}; 0: experimental data of this workand of Bef. 3. The solid curve has been drawn asdashed for p & 0.578 g/cm3 since the I' VT data availableref. 19}do not permit evaluation of (8P/8T)z at lowerdensities. The dashed portion is a reasonable extrapo-lation.
CARELLI, DE SANTIS, MODENA, AND RICCI
less than 16@at 2p, .We performed the same analysis for CH4 at
194.75 K and the results are shown in Fig. 9.Looking at Fig. 9 it is clear that the MET stronglydisagrees with CH, data, again showing a contra-diction with the behavior of Kr at the same reducedtemperature. 1.0
De(t) e)q„o
IV. CONCLUSIONS
We have shown that the behavior of the self-dif-fusion coefficient of Kr can be described neitheras that of a Lennard-Jones fluid nor as that of ahard-sphere fluid. On the contrary, as far as thediffusion is concerned, CH4 seems to be a betterexample for both these models. These facts are,in our opinion, surprising since if one had tochoose a fluid more similar to a Lennard-Jones orto a hard sphere one would certainly choose arare-gas fluid rather than a polyatomic one. Anotherpoint of interest is the different degree of agree-ment we obtain by using the MET in the cases ofKr and CH, . It would be important to understandin terms of O„values what change is introduced bysubstituting the second virial coefficient of a hardsphere with b+ T Sb/ST and the pressureP withthe thermal pressure T(SPja T)„
Recently the diffusive behavior of Kr and CH, hasbeen tested with a m-6-8 fluid. ' '" In this casethe discrepancy with p between D» and D» ismore random than inthe case of the Lennard-Jones
FIG. 9. D/&0 vs p/p~ for CH4 at 194.75 K. The solidline repx esents Eq. (5). +: experimental points of Bef.4. The I' VT data are taken from H. D. Goodwin, Nat.Bur. Stand. Beport No. 10715 (1971).
fluid. Moreover, it seems that for the m-6-8 mod-el the agreement in the description of Kr and CH,is about the same, in opposition to what happens inthe Lennard-Jones case. However, what is unsat-isfactory in the m-6-8 case is that Kr and CH4 havethe same type of interatomic potential"'" (both,m =3, y = ll) although they do not obey the princi-ple of corresponding states using the proposed"'"z and & parameters. Therefore in order to estab-lish in a definite way whether the m-6-8 fluid candescribe equally well CH~ and Kr, it would be nec-essary to lower the statistical errors in the com-puter experiments on the m-6-8 fluid.
~D. Levesque and L. Verlet, Phys. Bev. A 2, 2514 (1970).2D. J. Alder, D. M. Gass, and T. E. Wainwright, J.
Chem. Phys. 53, 3813 (1970).3P. Cax'elli, I. Modena, and F. P. Bicci, Phys. Bev. A
7, 298 (1973); 8, 1657(E) (1973).4P. H. Oosting and N. J. Trappeniers, Physica (Utr, )
51, 418 (1971).5J. O. Hirschfelder, C. F. Curtiss, and H. B. Birds,
Molecular Theory of Gases and Liquids (Wiley, New
York, 1954).L. Durbin and H. Kobayashi, J. Chexn. Phys. 37, 1643(1962).
YN. J. Trappeniers and J. P. J. Michels, Chem. Phys.Lett. 1, 1 (1973).
H. J. M. Hanley, H. D. McCarty, and J. V. Sengers,J. Chem. Phys. 50, 857 (1969).
T. M. Heed and K. G. Gubbins, App/ied StatisticalMechanics (McGraw-Hill, New York, 1973), p- 378.
~oD. Levesque and W. T. Ashurst, Phys. Hev. Lett. 33,277 (1974).
~~For & values we use the interpolation formula sug-gested by N. J. Trappeniers, A. Botzen, C. A. Tensel-dam, and H. P. Van Oosten [Physica (Utr. ) 31, 1681(1965)l.
Heference 1 and D. Levesque {private communication).3J. Naghizadeh and S. A. Hice, J. Chem, Phys. 36,2710 (1962).
~4"Corrected Enskog theory" means the Enskog theorycorrected for the hypothesis of "molecular chaos. "This correction can be done exactly using the hard-sphere computer experiments {Bef.2).
=I,3 19X10 5 (&M) 2/po ] cm/sec, where M is themolecular weight.
~~J. H. Dymond, Physica (Utr. ) 75, 100 (1974).H. J. M. Hanley, B.D. McCarty, and E. G. D. Cohen,Physica (Utr. ) 60, 322 (1972).
~SD =I3 19X10 5 p"M)~~2/p0. 2Q"'~'@T/e)j, where M isthe molecular weight, 0 and c are the parameters ofthe intexatoxnic potential, and O~~*~~ is the relatedcollision integral.
SELF-DIFFUSION IN SIMPLE DENSE FLUIDS 1139
For Kr the PVT data are taken from F. T. Theewesand R. J. Bearman, J. Chem. Thermodyn. 2, 171(1970), and %. B. Street and A. L. K. Staveley, J.Chem. Phys. 55, 2495 (1971).
H. J. M. Hanley and R. O. Watts, Physica (Utr. ) 79A,351 (1975).H. J. M. Hanley and R. O. Watts, Mol. Phys. 29, 1907(1975).
