Density dependence of rotational relaxation of supercritical CF 3HSusumu Okazaki,a) Masayuki Matsumoto, and Isao OkadaDepartment of Electronic Chemistry, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku,Yokohama 226, Japan
Katsutoshi Maeda and Yosuke KataokaDepartment of Material Chemistry, College of Technology, University of Hosei, Kajino-cho, Koganei,Tokyo 184, Japan
Supercritical fluid continuously changes its density alongan isotherm from a gas-like density to a liquid-like one. Thisprovides a variety of environments to molecules in the fluidsince the state of molecular aggregation depends much uponthe fluid density. Structure and dynamics of the fluid can thusbe investigated as a function of molecular collectivity withthe kinetic energy kept constant. The observation maypresent a key for a deeper understanding of many-body ef-fects of molecules on the structure and dynamics of con-densed phase.
From this point of view, physico-chemical studies on thesupercritical fluids have intensively been done recently.1 Theinterest is slightly different from that in critical phenomenasuch as the critical exponents which have been thoroughlyinvestigated for more than three decades.2 In fact, the sys-tems of interest in recent investigations are molecular fluidsslightly far from the critical point such as carbon dioxide andwater, and sometimes mixtures. They are mainly concernedwith statics, e.g. cluster formation,3–5 short or long rangestructure,6–8 and thermodynamics.9
With regard to density dependence of rotational relax-ation, which is one of the most interesting dynamics of mol-ecules, depolarized Rayleigh scattering has been measured sofar just for supercritical CO2 ~Ref. 10! and N2 ~Ref. 11!.NMR spin-lattice relaxation time measurement has also beendone to obtain rotational relaxation time for supercriticalCF3D,
12 CD3F,13 N2 ,
14 and ClF,15 taking the advantage of itshigh sensitivity. However, no report has been found for thestudy based upon polarized and depolarized Raman scatter-ing. Compared with an abundance of the investigations bythese kinds of experiments for liquids,16,17 the studies for thesupercritical fluids are thus rather few, even though molecu-
lar dynamics calculation intensively performed for particularsupercritical molecular fluids such as N2,
18 CO2,19,20 and
H2O ~Ref. 21! presented useful information which is not ob-tained by experiments alone.
In general, rotation of molecules in neat liquids withstrong interactions may be described by several periods ofmodulating libration around a local minimum followed bycollective rearrangement of the rotor and the surroundingmolecules, i.e. transition to another minimum of thequenched structure or cage.22 On the contrary, in a dilute gaslimit, free rotation is predominant. In the fluid at intermedi-ate densities, rotation of molecules must depend much uponthe state of molecular aggregation, showing various behaviorin the functional form of the correlation function. A system-atic investigation of density dependence of rotational relax-ation may thus present valuable information on emergence ofmolecular collectivity in the condensed phase. For example,if the motion of molecule in the supercritical fluid is uncor-related to those of the surrounding molecules, i.e. if the sur-rounding molecules can be assumed to form a random envi-ronment, the rotation may be described by the free rotationfollowed by random collisions. This typical behavior is rep-resented by the J-extended diffusion model,23 where bothmagnitude and direction of angular velocity randomlychange on collision. In this sense, comparison of the experi-mental or MD correlation function with the J-extendedmodel is very interesting. The difference represents the cor-related collision; many-body effect in the molecular aggre-gation. This kind of examination is further attractive if be-havior of the function is discussed in relation to clusterswhich have been considered to form in supercriticalfluids.3–5
However, most of the above experiments have not pre-sented enough information on density dependence of rota-tional relaxations covering from gas-like fluid to liquid-likeone via the critical point. This is probably because the signala!Author to whom correspondence should be addressed.
of the spectra is weak for the very low density fluid. In orderto examine an influence of molecular collectiveness on themolecular dynamics, a wide density range of the fluid includ-ing the critical density region must be investigated system-atically for molecules acting with various interactions.
In the present study, polarized and depolarized Ramanscattering measurements have been performed for supercriti-cal CF3H along an isotherm above the critical temperature byabout 6 K in order to clarify the density dependence of rota-tional relaxations. The molecule has a large dipole momentenabling us to examine a role of strong interaction in therotational relaxation. The present density range is wideenough to cover from gas-like density to liquid-like one;from about one eighth of the critical density to about twice ofit. High precision measurement for the very low density flu-ids was attained by the long time measurement. For example,the signal was accumulated totally for more than 1,000 s ateach wavenumber; more than 1,000 iterations of 1 s mea-surement. Further, the present method presents short timerotational autocorrelation function as a function of the den-sity, which is not obtained by NMR. The function is freefrom the intermolecular cross-correlation which is unavoid-able in the case of the depolarized Rayleigh scattering andinfrared spectrum. The intermolecular cross-correlation termcauses erroneous rotational relaxation time, the relaxationtime being twice as large as that from Raman spectrum andNMR for polar molecules16,24 such as the present CF3H.
Molecular dynamics calculations have also been donefor the present system to obtain molecular interpretations ofthe experimental rotational autocorrelation function. Thismay present a deeper understanding of the relaxation mecha-nism based upon actual rotational motion of the molecules. Adirect observation of state of molecular aggregation in thesupercritical fluid is also given at each density. Then, therotational relaxation can be discussed explicitly in terms ofthe molecular collectivity.
In the present paper, density dependence of rotationalrelaxations of molecules in the supercritical CF3H is reportedtaking an account of the preciseness intrinsic in the methods.The J-extended diffusion model is also examined to obtain ameasure of the many-body effect on the rotational motion ofthe molecule in the fluid. The experimental method and cal-culation are described in Sec. II. The rotational autocorrela-tion functions obtained from the experiment, the MD calcu-lation, and the J-extended diffusion model are presented anddiscussed in detail in Sec. III.
II. EXPERIMENTAL AND CALCULATIONS
A. Polarized and depolarized Raman spectra
In order to accommodate the supercritical fluid at highpressures up to 20 MPa at moderate temperatures, a pressure-stable sample cell made of stainless steel with fused-silicaglass windows was used. Critical constants of CF3H areTc5299.1 K,Pc54.84 MPa, andrc50.525 g cm23. Usinga piston compressor outside the cell, the supercritical fluidwas prepared by compressing the low pressure CF3H gas~Asahi Glass Co. 99.999 wt%! supplied from a cylinder. Po-larized and depolarized Raman spectra were measured for
the supercritical fluid at six pressures along an isotherm at305.0 K. The pressures monitored by a semiconductor straingauge~Druck PTX520! were 10.00, 6.37, 5.67, 5.10, 4.47,and 2.15 MPa, corresponding to 0.91, 0.68, 0.54, 0.36, 0.24,and 0.07 g cm23, respectively.25 The cell, the attached partsfor a high pressure line, and the optical table were placed inan air thermostat controlled within60.3 K. The resultantpressure fluctuation of the fluid in the closed cell was lessthan 0.03 MPa. As shown in Fig. 1, the present measurementcovers a wide density range from a gas-like density to aliquid-like one.
Arrangement of the spectrometer was similar to that pre-viously employed.26 The polarized~VV ! and depolarized~VH! measurements were done for the totally symmetricstretching mode (n1) of CF3H. The spectra measurementswere iterated until sufficient statistics of the data was at-tained; e.g. more than 1000 scans were done for the lowpressure fluids. The rotational autocorrelation functionCR(t) was evaluated according to16
CR~ t !5 12^3cos
2u~ t !21&
5* I VH~v!exp$2 i ~v2v0!t%dv
*$I VV~v!2 43I VH~v!%exp$2 i ~v2v0!t%dv
, ~1!
whereu(t) is the rotational angle formed byC3 axis ~C-Hbond! of the CF3H molecule att50 and that att5t, andv0 the peak frequency. Data corrections and analyzing meth-ods were the same as those described in a previous paper.26
Fourier transformation was performed by numerical integra-tion at every 0.5 cm21 from 2900 cm21 to 3200 cm21. Theintegration range of 300 cm21 was wide enough to obtainappropriate transformation.
B. Molecular dynamics calculations
Molecular dynamics calculations were performed inNVT ensemble using the Nose´ thermostat.27 The equation ofmotion was solved by the Gear predictor-corrector methodwith a time step ofDt52 fs. A unit cell contained 216
FIG. 1. ThePrT’s where the present Raman scattering measurements havebeen performed. A double circle represents the critical point.
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CF3H molecules in the periodic boundary condition. The po-tential function proposed by Bo¨hm et al.19 was adopted.
In general, a critical point estimated by MD calculationdepends much upon the potential function used. Further,structure and dynamics of the supercritical fluid are very sen-sitive to how far the fluid is from the critical point. There-fore, in this study, the critical point was first evaluated for thepresent potential function from thePrT-diagram. This wasdone by executing several tens of MD runs along the iso-therms. After determining the critical constants, six MD runswere performed at a reduced temperatureT*5T/Tc51.02and reduced densitiesr*5r/rc51.73, 1.30, 1.03, 0.69,0.46, and 0.13 of the real system. Each calculation was com-posed of 14 ps equilibration steps and 18 ps data accumula-tion steps.
III. RESULTS AND DISCUSSION
A. Raman spectra
The observed isotropic and anisotropic Raman spectra ofthe totally symmetric stretchingn1 mode of supercriticalCF3H at each density are shown in Figs. 2 and 3, respec-tively. Peak frequencies and full widths at half height~FWHH! are also listed in Table I. The isotropic spectrumshows that~i! the peak position shifts to a higher wavenum-ber with increasing density,~ii ! the band becomes wider, and~iii ! while the shape of the spectrum is almost symmetric forhigh density fluids, i.e. Gaussian-Lorentzian as usually foundin ordinary liquids, an asymmetric character appears for thefluid at low densities, leading to a peak split for the fluid atthe lowest density. This behavior of the isotropic spectra maybe understood essentially in terms of the interaction of theoscillator with the surrounding molecules in various states ofaggregation. The blue shift can be explained by consideringan increase in the repulsive forces in the fluid at high pres-sures and using the perturbation theory for the vibrational
transition frequency.28 The field on the oscillator has continu-ous and broad distribution in the disordered dense fluid,which gives rise to a broad band, i.e. inhomogeneous broad-ening. At low densities, on the other hand, the number ofstates must be small. In the fluid at the lowest density, forexample, the state becomes sparse enough for the spectrumto be discrete. This will be easily understood by observing aperspective picture of the fluid from the MD calculation,which will be given later in Fig. 8. Isolated molecules, pairedmolecules, and some clusters are found, which can yield dis-crete or asymmetric spectra if their life time is not very shortcompared with the vibrational period. Contribution by a hotband may be neglected, because at this temperature the ratioof the number of molecules in the first excited state to that inthe ground state is as small as 631025%. No Fermi reso-nance is found nearn1 , either. The spectrum of the fluid atintermediate densities belowrc is in the course of the mer-gence. It should be noted that the changes in the spectrastated above are caused just by the change in density, i.e. thestate of aggregation, but not by temperature.
With respect to anisotropic spectra, the shape changes asa function of density. It is Gaussian-Lorentzian for the fluidat the three highest densities. On the other hand, not only asharp central peak but also apparently broad branches on
FIG. 2. Isotropic Raman spectra for then1 totally symmetric stretchingmode of CF3H in the supercritical state at~a! 10.00,~b! 6.37, ~c! 5.67, ~d!5.10, ~e! 4.47, and~f! 2.15 MPa.
FIG. 3. Anisotropic Raman spectra for then1 totally symmetric stretchingmode of CF3H in the supercritical state. The alphabetical is the same as inFig. 2.
TABLE I. Peak frequencyv0 and full width at half maximumG for theobserved isotropic and anisotropic Raman spectra at various densities.
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both sides are found for the fluid at the lowest density. Thisclearly represents a relatively free rotation of the moleculesin the rarefied gas-like fluid. For the fluid at intermediatedensities, the spectra are of triangular forms. The functionalforms of the anisotropic spectra are directly related to theshape of rotational autocorrelation function. The actual func-tional form will be discussed in detail in the following sub-section. The spectrum seems to be asymmetric for the fluid atlower densities. This may be regarded as a convolution of anasymmetric isotropic spectrum presented in Fig. 2 and asymmetric rotational spectrum. The discrete anisotropicspectrum as well as the asymmetric one might affect thedeconvolution in Eq.~1! to calculate the rotational autocor-relation function. In the present work, the error will be esti-mated inclusively of other factors by comparing the experi-mental rotational autocorrelation function with thatcalculated from MD simulation.
Finally, the so called non-coincident effect29 is foundbetween the isotropic and anisotropic spectra, the peak posi-tion of the latter being higher than the former by as large as3 cm21. This is found only for the fluid at higher densitiesthan rc . The phenomenon might be explained29 by moreordered structure of the high density fluid than of the lowdensity fluid. However, the detailed analysis is beyond thepurpose of the present study.
B. Rotational relaxation
Rotational autocorrelation functionsCR(t) obtainedfrom Fourier transformation of experimental polarized anddepolarized Raman spectra are plotted logarithmically in Fig.4 for the fluid at each density. Small oscillations ofCR(t)smaller than 0.05 or 0.1 are Fourier ripples and have nophysical meaning. Since correct values must be at the centerof the artificial oscillations, most probable lines were drawnin the figure. A broken line represents a free rotor rotational
autocorrelation function for an oblate symmetric top30 withmoments of inertia I x5I y58.12310246 kg m2 andI z514.83310246 kg m2 for CF3H; the molecular coordinatewas set so that thez-axis coincides with theC3 symmetryaxis or C-H bond and that the values of moments of inertiaare consistent with a molecular model adopted in the MDcalculation in view of the experimental values.31
As clearly found in the figure, the functional form hasstrong density dependence. First, the molecule in the fluid athigher densities thanrc shows parabolic decay at smallt andan almost linear one at larget with a point of inflectionbetween them, which is qualitatively the same as that foundin the ordinary liquids. Second, different types of decay arefound for the fluid at lower densities thanrc . In particular, atthe lowest density, the functional form is very similar to thatof the free rotor. However, a difference is surely found quan-titatively between them because of still remaining correlationbetween the rotors. The molecules at this density are not yetpurely free. At two densities below the critical density, thefunction shows intermediate character, i.e. oscillatory decayat smallt. Thus, rotational relaxation mechanism changes itscharacter step by step from a liquid-like diffusional one to afree-rotor-like one. Roughly speaking, the fluid shows liquid-like relaxation at densities aboverc and gas-like one belowit. This was partly reported by Versmold10 in his depolarizedRayleigh scattering measurement for CO2 fluid, although theRayleigh autocorrelation function was not presented for thefluid near the critical density.
The rotational relaxation timetR was estimated for thefluid at each density by direct integration ofCR(t) up to anend of the line drawn in Fig. 4, beyond which an exponentialextrapolation was assumed forCR(t) to obtain a correctiontime to be added. The correction time was as small as about3%–10% of the integrated one except for the fluid at thelowest density. The relaxation time in reduced unit,tR*5AkT/I x3tR (52.283tR for tR in ps!, is listed in TableII and plotted in Fig. 5 as a function of the reduced densityr* . The figure includes plots of the reduced rotational relax-ation time obtained by NMR measurement by Langet al.12
for the supercritical CF3D at 330 K and that from Ramanscattering by DeZwaanet al.32 for CF3H in the liquid state atlow temperatures.
As clearly seen in the figure, our result based upon Ra-man scattering measurement is excellently consistent withthat obtained by the NMR measurement in spite of the quitedifferent principles; they are smoothly connected. This sup-ports the validity of the present analysis. However, the plot
FIG. 4. Experimental rotational autocorrelation functionCR(t) for CF3H inthe supercritical state atr5 0.91 g cm23 ~closed circle!, 0.68 g cm23
~closed triangle!, 0.54 g cm23 ~closed square!, 0.36 g cm23 ~open circle!,0.24 g cm23 ~open triangle!, and 0.07 g cm23 ~open square!. The brokenline represents the function for the free rotor.
TABLE II. Integrated rotational relaxation timetR* in reduced unit(tR*52.283tR for tR in ps!.
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for the liquid state seems to be slightly different. This sug-gests that the relaxation mechanism in the supercritical fluidat high temperatures may be different from that in the liquidstate at low temperatures.
Thus, density dependence of the rotational relaxationtime may be examined in a very wide range of the density. Itis remarkable that no singular behavior occurs but a shallowminimum is found around the critical density. This indicatesthat the rotational relaxation does not have a direct correla-tion with the density fluctuation which diverges to infinity atthe critical point. The relaxation time abruptly increased atvery low density. The zero-density limit of the relaxationtime must be infinite since a free rotor at this density doesnot lose correlation even at infinitet.
C. Molecular dynamics calculations
Molecular dynamics calculations have been performedin order to interpret on a molecular level the density depen-dence of the rotational autocorrelation function given abovefrom the Raman scattering measurement. As stated before,the state of the fluid is so sensitive to the potential functionadopted that usually MD calculation does not reproduce theexact value of the critical temperature or density. To avoidthe influence of not well-established potential function pro-posed so far, the correlation function was compared betweenthe fluids whose reduced temperature and density are thesame. The critical point of CF3H derived from the MD simu-lation with the potential function given by Bo¨hm et al.19
was determined by several tens of plots of pressures on thePrT-diagram along the isotherms. The result wasTccal5320610 K ~c.f. experimental 299.1 K! and
Pccal57.261.0 MPa. Since, in the present calculation, sig-
nificant difference was not found between the calculated andexperimental critical density, the experimental value ofrc50.525 g cm23 was adopted.
The calculated reduced pressuresPcal* 5Pcal /Pccal were
2.00, 1.48, 1.06, 0.98, 0.90, and 0.33 for the fluid atT*51.02~326 K! andr*51.73, 1.30, 1.03, 0.69, 0.46, and0.13, respectively. These are in good agreement with the ex-perimental valuesP*52.07, 1.32, 1.17, 1.05, 0.92, and 0.44,respectively. This certifies the validity of the comparison be-tween experimental and calculated rotational autocorrelationfunctions, although it should be limited to qualitative oneconsidering the error in the calculated critical point and theassumed extension of the principle of the corresponding stateto the rotational motion. The experimental and calculatedrotational autocorrelation functions are logarithmically plot-ted in Fig. 6 as a function of reduced timet*5AkT/I x3t(52.283t and 2.353t for t in ps at 305 K and 326 K,respectively! in order to reduce the kinetic effect caused bythe difference in their absolute temperatures. The agreementbetween them is satisfactory on the whole. The change fromthe free-rotor-like rotation to liquid-like diffusional and libra-tional one was well reproduced by the MD calculations.Comparing the functions at each density, the calculated func-tion shows slightly more rapid decay than the experimentalone. This phenomenon was found in other molecular liquidsof the rigid rotor model,33,34too, although the reason remainsto be clarified. In any case, the difference in the functions inFig. 6 is very small. Thus, both the experimental and calcu-lated rotational autocorrelation functions are sufficiently re-liable for further discussions.
The large change in the functional form ofCR(t) foundin Figs. 4 and 6 indicates a change in the relaxation mecha-nism in the fluid. At high densities, the relaxation mechanismmust be similar to that of liquids; molecules rotationally li-brate for a while with small angles around a local minimumof quenched structure and less frequently reorientate as aresult of simultaneous collective rearrangements of mol-
FIG. 5. Experimental reduced rotational relaxation timetR* evaluated bydirect integration ofCR(t) as a function of reduced density. Closed circle:this work for the supercritical fluid, open circle: the work by Langet al. inRef. 12 based upon NMR, and square: the work by DeZwaanet al. in Ref.32 for the liquid. FIG. 6. Experimental~solid line! and calculated~broken line! rotational
autocorrelation functionCR(t* ) for the fluid atr*5 ~a! 1.73, ~b! 1.30, ~c!1.03, ~d! 0.69, ~e! 0.46, and~f! 0.13.
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ecules involving the rotor and surrounding molecules.22,33,35
In fact, the function is composed of short-time parabolic de-cay reflecting the libration and linear decay at largert causedby the reorientational diffusive relaxation. On the contrary,the molecule rotates more freely at low densities. The relax-ation mechanism approaches the one-particle mechanismwhere the function decays by uncorrelated random binarycollisions. The starting point of this change may be the criti-cal density; if the fluid is cooled, the fluid at densities higherthanrc will be liquidized and for the fluid at densities lowerthan this the gas state will be generated.
For further insight into a microscopic picture of the ro-tational motion, an angular velocity autocorrelation functionCv(t) 5 ^v(t)v(0)&/^v(0)2& was calculated from MD tra-jectories at each density, wherev5(vxvy)
t is closely re-lated to the rotation of theC3 symmetry axis. Using cumu-lant expansion technique, a simple expression of therotational autocorrelation function in terms of the angularvelocity autocorrelation function was obtained byLynden-Bell22 for linear molecules and spherical top mol-ecules. Several fundamental characteristics of the rotationalautocorrelation function for liquids have been excellently ex-plained based upon this theory. The qualitative discussionthere may be applied to the present system of the symmetrictop molecule, too, although the algebraic formula for thelatter must be much more complicated than that for theformer. The result is shown in Fig. 7.
Furthermore, in order to visualize the states of molecularaggregation in the fluid, perspective pictures of the fluid atarbitrary time are presented for all the densities in Fig. 8.Each figure contains 216 molecules, where the basic cell wasso drawn as to have the same dimension on the figure and theatomic radii were set to be half of the Pauling diameters.Molecules in the fluid at higher density are, thus, apparentlygreater in size than those in the fluid at lower density.
Figure 7 shows that decay of the angular velocity auto-correlation function becomes slower with decreasing density.
The function crosses zero at aboutt50.4 ps for the fluid atthe highest density, while very slow decay without crossingzero is observed for the fluid at the lowest density. Theseclearly represent liquid-like librational motion for the formerand gas-like nearly free rotational one for the latter. Themotions are intuitively understood well corresponding to thestate of aggregation of molecules or packing of the mol-ecules in the fluid shown in Fig. 8. The snapshot of the fluidat the highest density cannot be distinguished from that ofthe liquids. In contrast to this, the packing state is very dif-ferent in the fluid at the lowest density. The fluid is com-posed of a lot of isolated molecules, dimers, trimers, ..., andclusters of several molecules. The isolated molecules maymake the main origin of the free-rotor-likeCR(t) shown inFig. 6~f!. Clustering of the molecules or collision preventsthe function from being of purely free-rotor one.
The molecules in the fluid at the intermediate densitiesare more constrained than those at the lowest density but lessthan those at the highest density, as is understood from theperspective pictures in Fig. 8. The state of aggregationchanges continuously from a gas-like one to a liquid-likeone. The fluid at densities lower thanrc may be described bya concept of clusters which is one of the most often usedmodels for supercritical fluids.3–5 The clusters are certainlyfound in the present instantaneous picture of the fluid at den-sities lower thanrc . These partly constrained molecules orcorrelated motions may result in the rotational autocorrela-tion functions shown in Figs. 6~d! and 6~e!. In Fig. 8, how-
FIG. 8. Snap shot of the fluid at arbitrary time. The alphabetical is the sameas in Fig. 6.
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ever, the fluid at the intermediate densities higher thanrcseems to be percolated as in the case of the fluid at thehighest density; there are not many free rotors or clusters ofsmall number molecules. Almost all the molecules appear toform one large cluster. This is closely related to the nearlyexponential liquid-like decay of the rotational autocorrela-tion function for these fluids. Thus, the gas-like relaxationand its continuous change to the liquid-like diffusional onewith libration for the fluid at higher densities thanrc viaintermediate oscillatory decay at smallt for the fluid at den-sities lower thanrc are well understood.
The timetCv50 whenCv(t) crosses zero represents halfof the averaged period of libration or head-on collision whenthe molecule changes its angular momentum to the oppositedirection, i.e. by more thanp/2. It may be related to thecollision frequency which has been widely used to describedynamics of gases and liquids. The values read from thefigure are 1.0, 1.5, 2.0, 2.2, and 2.4 in the reduced unit forr*51.73, 1.30, 1.03, 0.69, and 0.46, respectively. In the caseof the linear molecules and spherical top molecules, the timecoincides with the point of inflection inCR(t).
22 In thepresent system of symmetric top molecules, too, the zero-crossing point inCv(t) roughly agrees well with the inflec-tion point in CR(t), although the rigorous relation must bemore complicated than that for the linear molecules andspherical top molecules. The function for the fluid atr*50.13 did not cross zero, being almost free from libra-tional motion, which is in good accordance with the free-rotor-like behavior of the shape of rotational autocorrelationfunction. Except for this,tCv50 and inflection time inCR(t) decrease continuously with increasing density.
D. J-extended diffusion model
The calculation was done by direct integration methodfor the oblate symmetric top.23,30 The present molecule ischaracterized by a parameterj5(I x /I z)21520.4528. Therotational autocorrelation function from the MD calculation
was fitted by that of the J-extended diffusion model by ad-justing only one parameterb* , averaged reduced collisionfrequency. The best fit is presented in Fig. 9. The change inthe shape of the function from the free-rotor-like one to theliquid-like-diffusive one is roughly reproduced. The agree-ment is particularly excellent for the fluid at the lowest den-sity. As stated above, the angular velocity autocorrelationfunction of this fluid is of almost exponential form withoutzero-crossing. This suggests that the relaxation process at thelowest densities may be explained by random collisionswhich accompany random changes in both magnitude anddirection of the angular velocity.22 As clearly shown in thefigure, however, they do not agree well with each other quan-titatively for the fluid at the higher densities. Rigorouslyspeaking, the functional form itself could not be reproduced;for example, the experimental parabolic decay at smallt andrather slow linear relaxation at larget with a point of inflec-tion between them were reproduced by no more than thecurve without an inflection point and with a considerablylarger value at larget if the slope of linear decay was pref-erentially fitted. The situation is similar for the fluid at theintermediate densities, too. It has been reported10 so far thatthe Rayleigh autocorrelation function for carbon dioxidecould be fitted well by the J-extended diffusion model, al-though the agreement between memory functions was poorat high densities. As stated above, however, the rotationalautocorrelation function from Raman scattering measure-ment or MD calculation for the present CF3H fluid could notbe reproduced. This indicates that, at high densities, the re-laxation mechanism of the molecule interacting with thestrong dipole moment with high orientational correlations isdifferent from that assumed by the J-extended diffusionmodel. In fact, as shown in Fig. 7, the functional form ofangular velocity autocorrelation function of the fluid at thesedensities is not exponential and take a negative sign crossingzero at a relatively short time. This contradicts the assump-tion made for the J-extended diffusion model. That is, themodulating libration and the less frequent reorientation ac-companied by the collective rearrangement of moleculesaround the rotor is predominant in the real fluid at thesedensities.
IV. CONCLUSION
In the present work, polarized and depolarized Ramanscattering measurement and molecular dynamics calculationhas been performed for supercritical CF3H at various densi-ties along an isotherm higher thanTc by about 6 K in orderto investigate the density dependence of rotational relax-ation. The rotational autocorrelation functions obtained fromboth methods were in satisfactory agreement with each other.They showed liquid-like diffusional decay for the fluid atdensities higher thanrc . The function changed in shape con-tinuously to a nearly free-rotor-like one at the lowest densitygoing through the oscillatory decay at intermediate densities.The detailed analysis of the MD trajectories has been done inorder to clarify the relaxation mechanism at each density.Applicability of the J-extended diffusion model was also ex-
FIG. 9. Best fit of the MD rotational autocorrelation functionCR(t* ) ~sym-bols! by the J-extended diffusion model~broken lines!. The symbols are thesame as in Fig. 7.
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amined. They showed that the density dependence of therotational relaxation may be explained in terms of the statesof molecular aggregation in the fluid.
ACKNOWLEDGMENTS
The authors thank Professor M. Nakahara at Kyoto Uni-versity for valuable discussions and suggestion. They arealso grateful to the computer centers of the Institute for Mo-lecular Science and the High Energy Physics Institute for theuse of NEC SX-3 and FUJITSU VPP-500 supercomputers,respectively. This work was supported in part by the Grant-in-Aid for Scientific Research on Priority Area ‘‘Supercriti-cal Fluids’’ and ‘‘Many-body Chemistry’’ from the Ministryof Education, Science and Culture, Japan.
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J. Chem. Phys., Vol. 103, No. 19, 15 November 1995
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