[6] P. W. Bridgman, Phys. Rev. 3, 153 (1914).
[7] S. M . Ross and J. H . Strange, M o l . Cryst. Liq. Cryst. 36, 321 (1976).
[8] M . J. P. Muringer, N . J. Trappeniers, and S. N . Biswas, Phys.
Chem. Liq. /4, 273 (1985).
[9] S. N . Biswas, Thesis, University of Amsterdam 1974.
[10] T. F. Sun, P. J. Kortbeek, N . J. Trappeniers, and S. N . Biswas, Phys. Chem. Liq. 16, 163 (1987).
[11] T. Takagi, T. Mem. of Faculty Ind. Arts, Kyoto Tech. Univ., Sci. and Tech. 25, 51 (1976).
[12] E . A. Block, Z . Phys. Chem. 82, 403 (1913).
(Eingegangen am 15. M a i 1987) E 6495
High Pressure NMR Study of the Molecular Dynamics of Liquid Methylfluoride and Deutero-Methylfluoride
E. W. Lang, F . X. Prielmeier, H . Radkowitsch9 and H . - D . Liidemann
Institut fiir Biophysik und Physikalische Biochemie, Universitat Regensburg, Postfach 397, D-8400 Regensburg Diffusion / High Pressure / Liquids / Spectroscopy, Nuclear Magnetic Resonance / Transport Properties
The 2D - , 1H - and wF-Spin-Iattice relaxation rates R1 have been investigated in fluid methylfluoride in the temperature range 150 K < T
< 450 K and at pressures up to p = 200 M Pa. In addition, the self-diffusion coefficient D has been obtained with N M R techniques in the same p, T-interval. — The latter are interpreted within the rough hard sphere model yielding a temperature independent R H S diameter d and a strongly temperature dependent rotation-translation coupling parameter A. The total 2D - , 1H - and 1 9F-relaxation rates in C D3F and C H3F arc decomposed into their respective quadrupolar, dipolar and spin-rotation contributions. Furthermore orientational and spin-rotational correlation times are estimated. It is shown that they are in good agreement with the predictions of the extended M - diffusion model (EDM). Inertial effects influence the molecular dynamics in the whole p, /"-range investigated. The self-diffusion coefficients
and orientational correlation times are in excellent agreement with MD-simulation data.
Introduction
A n increasing number of experimental techniques [1 —4]
like I R and R a m a n bandshape contours, light scattering, dielectric relaxation and absorption, neutron scattering and N M R relaxation have been applied i n recent years to the investigation of molecular dynamics i n low molecular weight liquids. F r o m high-frequency methods orientational corre- lation functions C\(t) m a y be deduced over a limited time range thus revealing the short time dynamics of orienta- tional fluctuations mainly. N u c l e a r magnetic relaxation studies have limitations i n p r o b i n g details of the m i c r o d y - namics since i n l o w viscosity liquids only the area of the relevant correlation function is n o r m a l l y determined, and not its shape. D y n a m i c parameters obtained from N M R methods include the self-diffusion coefficient D, the molec- ular orientation correlation time r0,i = 2an d the correlation time TM of the molecular angular velocity. T h e importance of simultaneous measurement of T(K2 and TW lies i n the model- dependent relationship between these microscopic parame- ters. T h u s together they m a y provide insight into the state of molecular m o t i o n .
T h e present investigation is a study, by N M R methods, of the small polar symmetric-top molecule C H3F a n d its deuterated analogue i n the neat l i q u i d phase over the tem- perature range 150 K to 450 K a n d a pressure range 0.1 to 200 M P a . T h e self-diffusion coefficient D is the only dynamic parameter that c a n be obtained directly from the experi- ment. T h e various correlation times (x0,2, O m a y be ex- tracted according to m o t i o n a l models from the measured spin-lattice relaxation rates caused by magnetic dipole-di- pole-, electric quadrupole- a n d spin-rotation interactions of the nuclear spin w i t h m o t i o n a l degrees of freedom of the molecules.
Experimental
Spin-lattice relaxation times were measured by the inversion- recovery method with alternating phase [5] on a Bruker MSL-300 multi-purpose solid-liquid N M R spectrometer operating at 300 M H z for 1H , 282 M H z for 1 9F and 46 M H z for 2D and on a Varian XL-100 high resolution spectrometer operating at 100.1 M H z for
1H , 94.07 M H z for 1 9F and 15.35 M H z for 2D . Self-diffusion coef- ficients D were determined by the spin-echo technique on a Varian XL-100 using a home-built quadrupole coil to apply a steady field gradient. The experimental procedure and high pressure set-up have been described elsewhere [6]. The pressure was measured with a Heise-bourdon gauge (Heise, Connecticut, U.S.A.) with a precision of ± 0.5 M P a . The temperature was altered by blowing precooled gaseous N2 around the sample using the modified variable temper- ature units of the two spectrometers. The temperatures were meas- ured with a 0.5 mm Chromel-Alumel thermocouple (Philips, Kassel, F.R.G.) and are considered reliable to ± 1 K . The self-diffusion coefficients and spin-lattice relaxation times are judged reliable to
+ 10%.
Substances
C H3F (99%) was purchased from J. T. Baker (GroB-Gerau, F.R.G.), C D3F (99% deuterated) was obtained from I. C Chemi- kalien (Munchen, F.R.G.). The substances were stored in glass flasks. Residual moisture was removed over molecular sieve 3 A . In addition the gaseous compounds were kept for approximately one day in contact with potassiumhydroxide resp. phosphorpentoxide in order to remove reactive impurities. After these treatments the
1H - , 2H - and 1 9F - N M R spectra showed no sign of any impurity.
Results and Discussion The Self-Diffusion Coefficient of Methylfluoride
Structure and dynamics of simple liquids are determined mainly by short-range, repulsive interactions [7]. A n ideal- ized simple l i q u i d is composed of smooth h a r d spheres. It's self-diffusion coefficient may be expressed as [8]
Ber. Bunsenges. Phys. Chem. 91, 1017-1025 (1987) - © V C H Verlagsgesellschaft m b H , D-6940 Weinheim, 1987.
0005-9021/87/1010-1017 $ 02.50/0
3 / kT V '2
with d the hard sphere diameter a n d Q the number density.
T h e coefficients of a n e m p i r i c a l p o l y n o m i a l P(od3) i n the p a c k i n g fraction of the molecules have been obtained by various authors [8] v i a fitting E q . (1) to the M D - s i m u l a t i o n results o f A l d e r et al. [ 9 ] .
Table 1
Molecular Constants in C H3F and C D3F
1) distances and angles (CH3F) [31, 32]
rC H = (1-106 ± 0.001) • 10-1 0 (m) (HCH) = 109° 59' ± 3' /VF = (1.3852 ± 0.00005) • 10-'n (m) (HCF) - 108, 95 rH H = 1-812 • U)-l 0(m)
rH, = 2.034- 10 1 0 (m) 2) dipole moment (CH3F) [24]
// = 1.8585 (Dy)
3) principal moments of inertia
C H3F [33] C D3F
/ , = 34.82 • IO-4 7 (kg • m2) 45.96 • 10~47 (kg • m2) /„ - 5.47 • 10 4 7 (kg • m2) 10.94 • 10 4 7 (kg • m2)
<'>-(i..U)'
= 12.5 • 10 4 7 (kg-nr) 22.2 • 10~47 (kg • m2) 4) principal spin-rotation tensor components [24]
C H3F C D3F
,1>F: Cx = 4.0 + 1.9 (kHz) 3.03 (kHz) C1, = -51.1 ± 1.3 (kHz) -25.55 (kHz)
1H: C1 = 1 ( C \ + Cy) - 0.8 (kHz) C1 = 14.66 (kHz)
R o u g h hard spheres provide the simplest model l i q u i d representing real, non-spherical molecules or spherical m o l - ecules with anisotropic intermolecular interactions. In these liquids a c o u p l i n g o f rotational a n d translational fluctua- tions may occur leading to a decrease o f Z)SHS- T h i s R - T - c o u p l i n g may be accounted for simply by introducing a R - T - c o u p l i n g parameter A such that [10]
DCXP ^ DRHS = A • Z )S H S • (2)
Table 2 collects the experimental self-diffusion coefficients D together with the mass-density Q. T h e latter have been measured between 300 K a n d 600 K a n d at pressures u p to 300 M P a by B o z d a g a n d F r a n c k [11]. These data have been extrapolated linearly to lower temperatures i n accord w i t h k n o w n densities at s.v.p. [12].
F i g . 1 shows the isothermal density dependence of the self- diffusion coefficient. T h e lines d r a w n through the data are calculated w i t h E q s . (1) a n d (2). T h e rough h a r d sphere d i - ameter m a y be obtained from the isothermal density de- pendence of the self-diffusion coefficient D from a non-linear least-squares fit ( N A G E 0 4 F C F ) w i t h E q s . (1) a n d (2). A temperature independent hard sphere diameter d = (0.357
Table 2
Self-Diffusion Coefficient D and Mass-Density Q of CH3F (T[K], p [MPa], D [10-9 m2/s], Q [kg/m3])
Tp s.v.p. 20 50 100 150 200
153 178 204 230 256 284 314 345 375 405 440
°400 500 6 0 0 700 8 0 0 900 1000 1100
— g ( k g / m3)
Fig. 1
Isothermal density dependence of the self diffusion coefficient D
± 0.004) n m results for the fluid phase of C H3F . The R-T- c o u p l i n g parameter A is also obtained from the fit. It is temperature dependent a n d increases from A = 0.37 at low temperatures to A = 1 at the highest temperatures (see Table 3). Hence the c o u p l i n g o f rotational and translational de-
2.10 1.90 1.65 965 998 1027 3.70 3.20 2.75 917 950 990 5.70 5.10 4.40 860 905 952 8.50 7.30 6.20 800 855 910 12.30 10.50 8.30 720 805 870 14.50 11.00 755 833 20.00 15.00 675 785 28.00 19.00 570 730 41.00 25.00 490 677 52.00 30.00 390 630 74.00 37.00 295 570
Q 2.30 1.95 1.75 1035 1060 1087 3.60 3.05 2.60 1005 1033 1065 5.00 4.30 3.80 970 1005 1040 6.70 5.80 5.10 940 980 1015 8.60 7.30 6.20 905 950 990 11.20 9.00 7.40 867 920 960 13.50 10.80 9.20 830 885 930 16.00 13.00 11.00 792 857 905 20.50 15.50 12.50 755 832 880 23.50 17.50 14.50 715 797 850
grees of freedom decreases w i t h increasing temperature. Re- cently B o h m et al. [13] developed a new interaction poten- tial for methylfluoride molecules i n the l i q u i d phase and performed molecular dynamics calculations to deduce var- ious structural and d y n a m i c properties of the l i q u i d . Their self-diffusion coefficients are i n g o o d agreement with the results obtained i n this study (see Table 5). T h e present data offer the possibility for further testing their potential func- tion over a wide range of temperatures and pressures.
Table 3
Temperature dependence of the Rotation-Translation Coupling Parameter A r [ K ] 153 178 204 230 256 284 314 345 375 405 440 A 0.37 0.50 0.62 0.71 0.80 0.87 0.93 0.94 0.99 1.00 0.98
2D Spin-Lattice Relaxation Rates in C D3F
D e u t e r i u m nuclei possess a spin 1 = 1 and interact pre- dominantly v i a intramolecular electric quadrupole interac- tions. Presuming rigid molecules of spherical shape spin- lattice relaxation rates in the fast m o t i o n regime are given i n a laboratory-fixed frame (L) as [14]
(3) 21 + 3
I2( 2 1 - l )
If independent information from solid state N M R or m o - lecular beam studies is available o n the magnitude of the deuterium quadrupole c o u p l i n g constant, the integral o r i - entational correlation time x0,2 may be obtained directly from the measured deuterium relaxation rates. T h e only source of information available is an F T - N M R study of the spectrum of C D3F i n the nematic phase of a l i q u i d crystalline solvent [15]. T h e deuterium quadrupole coupling constant (referred to the C - D bond) has been obtained to 133 ± 1 k H z w i t h an asymmetry parameter n = 0.03 ± 0.03. Bhat- tacharyya and D a i l y [15] quote a private c o m m u n i c a t i o n by Griffith saying that a comparison of R a m a n and N M R data suggests a deuterium quadrupole c o u p l i n g constant of 143 k H z . A n ab initio calculation [16] of the efg yielded a coupling constant of 212 k H z , however. A l s o a comparison with k n o w n quadrupole c o u p l i n g constants of similar c o m - pounds (see Table 4) suggests that the rather low value of 133 k H z obtained i n a l i q u i d crystalline solvent may not be appropriate to the neat l i q u i d phase [37]. A s w i l l be ex- plained later, a value of 143 k H z w i l l be used. It should be noted that the integral correlation time T(K2 must be consid- ered as an effective correlation time. This is because of the asymmetric mass distribution i n the symmetric top, a l - though its shape is almost spherical (a = 4.5 A , b = 4.0 A ) , with concomitant different moments of inertia parallel and perpendicular to the symmetry axis of the molecule. Hence two different correlation times c o u l d arise for orientational fluctuations around the symmetry axis and an axis perpen- dicular to it. W i t h i n a diffusional model this w o u l d lead to [14]
where I is the spin of the quadrupole nucleus.
the quadrupole c o u p l i n g constant i n H e r t z with eQ the elec- tric quadrupole moment of the nucleus and eqzz the largest component of the electric field gradient (efg) tensor along the C - D b o n d . T h e efg asymmetry parameter rj = (qxx-qyy)/
q.. is generally small and w i l l be neglected.
The spectral density j (a>) for the m o t i o n i n the lattice is the F o u r i e r - L a p l a c e transform of the relevant correlation function for orientational fluctuations of the molecular m a i n axis system ( M ) of the efg-tensor
j{a)) = J 0 ( O e - " " d t = J I
o o '» (4)
In the fast m o t i o n regime the spectral density is independent of the observing frequency a> and an orientational correla- tion time T,)2 may be defined through the relation
f g(t) f y < f l i & ( Q ) P l & ( 0 >
giving finally (I = 1)
m . 3 *2 (e2q:-.Q\2
( 5 )
(6)
T 0,2 — '
1/2-(3 cos26)- 6D±
I )2 3 s i n2 Ocos2O 3/4 s i n4 0
— + ^ ; — + -
5D± + Dl{ 2 Z ) _ L + 4 Z ) , | (7)
in terms of the components of the rotational diffusion tensor and the angle 0 between the symmetry axis and the C - D b o n d axis. However, even w i t h i n this m o t i o n a l model, two
Table 4
Deuterium quadrupole coupling constants in various halomethanes Compound e2qQ/h (kHz)
C D3F 133 + 7 [15]
143 [15]
212 [16]
CD3Br 173 + 4 [37]
177 + 1 [34]
178 + 3 [35]
CD3I 181.7 + 0.4 [36]
182 ± 5 [37]
C D2C l2 171.7 ± 0.8 [44]
CD2Br2 180 + 1 [43]
C D2I2 175 ± 2 [45]
C D F3 159 ± 5 [40]
170.8 ± 2 [41]
C D C l3 180 + 3 [39]
167 ± 1 [36]
166.9 ± 0.1 [38]
CDBr3 171.2 + 0.8 [42]
184 ± 1 [34]
quadrupole nuclei i n different positions w o u l d be neccessary to extract D± a n d Dlh which are not available i n C D3F . Intramolecular dipole-dipole interactions provide equiva- lent information about orientational fluctuations, if it is pos- sible to separate the intra-rate from the various c o n t r i b u - tions to the total rate measured for 1H and 1 9F nuclei. H o w - ever, i n either case is the intra-rate negligible and cannot be obtained w i t h any reasonable degree of accuracy. T h u s it is not possible to extract detailed information about the an- ticipated [46, 47] fast spinning m o t i o n a r o u n d the C3 axis a n d the t u m b l i n g m o t i o n of this axis. E q . (7) has been used frequently to deduce the diffusional anisotropy Q = D^/D ± in various halomethanes, methylcyanide and its halogenated derivatives although the small step diffusion m o d e l is cer- tainly not appropriate for the fast axial spinning m o t i o n . I n these investigations the largest m o t i o n a l anisotropy has been observed for molecules with large dipole moments a n d highly anisotropic molecular shape. A s a rule, a large inertial anisotropy I±/I\\ corresponds to a substantial diffusional an- isotropy D11ZD1. T h u s w i t h i n the series of halomethanes, C H3F is expected to exhibit the smallest m o t i o n a l aniso- tropy. Despite the above mentioned inconsistency, one can, at least at low temperatures, estimate the anisotropy g from E q . (7) because i n the r o t a t i o n a l diffusion model the relation DL = ( I id i c l) - -1
holds and Td i c l have been measured i n the temperature range Ti-^iripi at saturation v a p o r pressure [17]. A t the two lowest temperatures T= 153 K : ^ = 1.84 a n d T = 170 K:Q = 2.1 is obtained indicating a modest diffusional anisotropy only. Hence i n the following we w i l l only discuss effective integral correlation times. These integral correlation times as well as the self-diffusion coefficients are i n g o o d agreement w i t h the molecular dynamics results c o r r o b o r a t i n g the use- fulness of the potential functions developed by B o h m et al.
[13].
i.o-
lps)
t
0.5-0.1
9 = 1 0 0 0 kg/mK
5 6
— I O3/ T ( K "1)
Fig. 2
lsochoric temperature dependence of the orientational correlation time x0>2
F i g . 2 shows the isochoric temperature dependence of the correlation times x(K2. A t constant density the temperature dependence reflects the sole influence of the kinetic energy of the molecules u p o n the orientational fluctuations. Be- cause "collisions" occur more frequent, i.e. intermolecular torques fluctuate more rapidly, at higher temperature, ori- entational correlations decay o n a shorter time scale leading to decreasing correlation times. T h e latter are seen to be shorter than 1 ps i n the p, T-range investigated. Further- more, the isochoric temperature dependence reveals an Ar- rhenius-dependence with a density independent "activation energy" £ * = 2.48 (kJ/mol) corresponding to kT at room temperature roughly. A correspondingly low apparent ac- tivation energy of 3.77 k J / m o l has been obtained from die- lectric measurements along the saturation vapour pressure curve.
F i g . 3 shows the isothermal density dependence of the correlation times Tf u- A t constant temperature the correla-
1.0
(t e,2(g))T- (ps)
I
0 . 5 -0.1
3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1100 g ( k g / m3)
Fig. 3
Isothermal density dependence of the orientational correlation time
X 0,2
Ti CD3F
100 150 200 P(MPQ)
Fig. 4
Comparison of the p, T-dependence of the deuterium spin-lattice relaxation time T1 (2H) and the self-diffusion coefficient D
tion times increase i n a nonlinear fashion with density re- flecting the retarding influence of molecular torques upon the reorientation process at higher p a c k i n g fractions. The increase is the more pronounced the lower the temperature.
A comparison of the pressure and temperature dependence of the self-diffusion coefficient D a n d the 2H spin-lattice re- laxation time Tx (Fig. 4) reveals that orientational fluctua- tions are m u c h less hindered by compression or removal of thermal energy than are positional fluctuations. Together with the fairly short correlation times x(K2 < 1 ps and the low barrier to rotation E*e = 2.48 k J / m o l this weak de- pendence o n p, T indicates that the reorientational processes must proceed rather freely. T h i s raises the question whether inertial effects may be important. A n i n d i c a t i o n of the i m - portance of inertial effects u p o n orientational fluctuations may be obtained by c o m p a r i n g the orientational correlation
3 /</>
times Tfl,2 with the free rotor correlation time Tf = — ^p-
Ifthe ratio T^2A f > 1, reorientational processes are diffusive and inertial effects may be neglected. If T ^2/ rf ^ 1 inertial effects severely influence orientational time correlation func- tions. In C D3F at l o w temperatures the ratio T0 j 2A f yields values ~ 2 whereas at high temperatures values ~ 1 are found. Hence it must be concluded that the molecular dy- namics of methylfluoride do not correspond to a diffusive process over most of the p, T range investigated.
1H and 1 9F Spin-Lattice Relaxation Rates in C D3F , C H3F Protons a n d fluorine nuclei possess a spin 1 = 1/2, their spin-lattice relaxation proceeds m a i n l y via magnetic dipole- dipole interactions and spin-rotation interactions. T h e d i - pole-dipole interactions have to be decomposed into intra- molecular and intermolecular interactions. In the fast m o - tion regime the various d i p o l a r relaxation rates are given, again presuming rigid molecules of spherical shape, as [12]
a) dipole-dipole, intra, like spins 1 = 1/2
n intra „.4 3
* U d —
Y y
1 4K 1I-=I=I (9)
b) dipole-dipole, intra, unlike spins I = 1/2, S = 1/2
Dintra _ .,!,.if W>Y . A 2 Y _ - 6 . _
\4K; S
c) dipole-dipole, inter, like spins 1 = 1/2
(10)
8n
9 471, D 1 * 1 (11)
d) dipole-dipole, inter, unlike spins I = 1 /2, S = 1 /2 16*
27 4 T T D s 0IS
(12)
The expressions for the intermolecular d i p o l a r relaxation rate may be obtained with a diffusion equation w i t h reflect-
ing boundary conditions [ 1 8 , 1 9 ] . A n y off-center effects have been neglected as they contribute less than 1 0 % to the lead- ing term [20]. T h e relative diffusion coefficient has been replaced by twice the measured self-diffusion coefficient. T h e distance of closest approach a has been taken from radial pair distribution functions as obtained by molecular d y n a - mics simulations of B o h m et al. [13]. Otherwise a uniform distribution of the molecules has been assumed.
The spin-rotation relaxation rate is given for a spherical top by [21]
2kBT
< / > [ C2- T< 0 + 2 ( A C )2- TS R] (13)
w i t h </> = - £ / / the mean moment of inertia of the
molecule. C0 = y ( C1 1 4- 2 Cl) and A C = j ( C1 1 - C±) give the isotropic and the anisotropic part of the spin-ro- tation interaction tensor i n terms of the parallel and per- pendicular component of the tensor i n its m a i n axis system.
TW is the correlation time for fluctuations of the molecular angular velocity and Ts r is the correlation time for the an- gular velocity-orientational product correlation function which characterizes the anisotropic spin-rotational inter- actions. E q . (13) is equivalent to an expression given by H u b - b a r d [22, 23]
2k*T
< / > - - ( 2 c i + q )
(14)if due account is taken of the different definitions of TS R. E q . (14) has been used to calculate the correlation times TSR . The necessary molecular constants are compiled in T a b l e 1.
Separation of the 1 9F-Relaxation Rates in C D3F
The 1 9F spin-lattice relaxation rate is a sum of different
contributions according to K i (1 9F ) =
i W
9F )
+ U^ S(1 9F -1 9F )+ ^ inS(1 9F -2D ) + K f t g (1 9F -2D ) .
(15)
Because of the small gyromagnetic ratio of the deuterium nucleus, the last two terms contribute but little to the meas- ured relaxation rate and w i l l be neglected.
Inspection of F i g . 5 reveals that the spin-rotation relax- ation rate dominates the observed rate over almost the en- tire p, G r a n g e investigated. It may be obtained by subtract- ing from the total rate the intermolecular d i p o l a r rate
^ inS d (1 9F -1 9F ) as calculated with the help of E q . (11). Here the measured self-diffusion coefficient has been used together with a distance of closest approach a = 0.30 n m as taken from the M D simulations [13]. A t high temperatures the spin-rotation rate is practically identical with the observed rate. T o calculate correlation times TSR independent infor- m a t i o n about the components of the spin-rotation tensor is neccessary. These have been determined i n the case of C H3F from molecular-beam electric resonance spectra (see Table 1)
2 0 -
1 1 1 1 r - 2 3 4 5 6 7
— I O 3 / T ( K "1)
Fig. 5
Isobaric temperature dependence of the fluor-19 spin-lattice relax- ation times in C D1F . (O s.v.p., • 20 M P a , A 50 M P a , + 100
M P a , V 150 M P a , x 200 MPa)
[27]. The corresponding components of the tensor i n C D3F may be obtained from the relation [25]
Ca = 01 9 r • Xa • /i/(2 • /.) (16)
where gx% is the nuclear ^-factor, Ia the a-th component of the moment of inertia tensor (see T a b l e 1) a n d Aix a dimen- sionless quantity independent of the isotopomere.
U.1 -1 1 i 1 1 1 1 1 r -
3 0 0 AOO 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1100 Q ( k g / m3)
Fig. 6
Isothermal density dependence of the reduced spin-rotational cor- relation time T *R
F i g . 6 shows the isothermal density dependence and F i g . 7 gives the isochoric temperature dependence of the reduced correlation times T*R = Ts r( / : 7 7 < / »1 / 2. W i t h increasing den- sity T*R decreases due to the stronger perturbing influence of molecular torques at higher p a c k i n g fractions. A l s o the decrease is the more pronounced the lower the temperature.
However, even at the lowest temperatures T*R does not fall below 0.1 w h i c h corresponds roughly to the upper limit of validity of the rotational diffusion model. Furthermore the
product T *2 * T | R > 6~\ even for T *R <^ T *2, indicating that the H u b b a r d relation T* • T*R = 6_ 1 is not obeyed. The isochoric temperature dependence of T *r deviates from an Arrhenius-dependence w i t h a weaker dependence on tem- perature at lower densities.
0.1-1 , , , , r
2 3 4 5 6 7
—m* 1 03/ T ( K "1)
Fig. 7
Isochoric temperature dependence of the reduced spin-rotational correlation time T*R
Table 5
Comparison of MD-results and experimental results for the self-diffusion coefficient D and the integral orientational correlation time T2,, of methyl-
fluoride
MD Expt.
T IK] 192 132 192 132
Z) [ 1 0-9I n2S -1] 4.7 1.2 4.8 U5a)
TlJ [ P S ] 0.45 1.2 0.44 1.1a)
*) Obtained by slight extrapolation.
Separation of the Relaxation Rates in C H3F
The total 1 9F spin-lattice relaxation rate i n methylfluoride is composed of
K i (1 9F ) = ^s r(1 9F ) + ^ aa(1 9F -1H ) (n)
+ R f t j (1 9F -1H ) + K f teJ (1 9F -1 9F ) .
Isobars of the total rate are shown i n F i g . 8. Again the spin- rotation relaxation dominates over most of the p,T-range investigated, especially at l o w pressures. F i g . 9 gives the tem- perature dependence of the various contributions to the to-
tal 1 9F relaxation rates at three pressures. The intermolecular
d i p o l a r relaxation rates were calculated using a distance of closest approach a ( F - F ) = 0.3 n m and a (F-H) = 0.22 nm as explained i n the case of deuteromethylfluoride. At the lowest temperatures measured the intermolecular dipolar rates provide the m a i n contribution to the total rate meas- ured. Subtracting these contributions one is left with the conclusion that the intramolecular dipolar rate contributes but little to the total rate at a l l temperatures and pressures.
This entails a large uncertainty i n the estimate of the intra- molecular d i p o l a r rate and i n the concomitant effective cor-
relation time TD D 2 w h i c h c o u l d not be estimated with any reasonable degree of accuracy. Hence at high temperatures the total rate is to g o o d a p p r o x i m a t i o n equal to the spin- rotation relaxation rate. W i t h the components of the spin- rotation interaction tensor [24] the correlation times T"R may be calculated. A c o m p a r i s o n w i t h the correlation times T Pr as obtained i n C D3F allows an estimation of the isotope effect o n the correlation times Ts r . They scale with the square root of the mean moment of inertia. T h i s is to be expected with motions influenced by inertial effects. It is to be noted that the experimental ratio [ R1 ( C H3F ) / R , ( C D3F ) ] 1 9 [. ~ 1.6, which is practically equal to [ R ?R( C H3F ) / R ?R( C D3F ) ]1 9 h at high temperatures, may only be obtained if the mean m o - ment of inertia i n E q . (14) is calculated as given i n T a b l e 1 [26] and T *r independent of the isotopomere.
2-| , , , , , , , 1 1 2 0 2 5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
— I O3/ T ( K '1)
Fig. 8
Isobaric temperature dependence of the fluor-19 and proton (1H) spin-lattice relaxation times TX in C H3F
C H3F / 1 9F
SVP 100 MPa 200 MPa
/
/ /*
4 5 6 3 4 5 6 3 4 5
— I O3/ T (K"1)
Fig. 9
Temperature dependence of the various relaxation rates contrib- uting to the total fluor-19 spin-lattice relaxation rate At(wF ) in
C H3R (+ x Rt\0(R?J8+R?))
The total1 H spin-lattice relaxation rate i n methylfluoride is composed of
R1(1H ) = RK S R(1H ) + R1S (1H -1H ) + R f f i (1H -1 9F ) (lg)
+ R f t j (1H -1H ) + R f t J (1H -1 9F ) .
T h e intermolecular d i p o l a r relaxation rates may again be estimated with the k n o w n self-diffusion coefficients and the
distance of closest approach a ( H - F ) = 0.22 n m and a ( H - H ) = 0.22 n m as obtained from M D simulations. A t l o w temperatures they provide the m a i n c o n t r i b u t i o n to the total rate measured ( F i g . 10). A t the highest temperatures the lat- ter reaches a m a x i m u m (see F i g . 8) indicating the importance of spin-rotation relaxation at these temperatures. A s the components of the spin-rotation tensor for the protons have been determined also by Wofsy et al. [24], Ri s r(1H ) may be calculated using the correlation times Ts r derived from the
1 9F relaxation. A g a i n intramolecular d i p o l a r relaxation rates are small and m a y not be obtained with any reasonable degree of accuracy. If one scales the orientational correlation times T(K2 as obtained from the deuterium relaxation rates i n C D3F , with the square root of the mean moment of i n - ertia, intramolecular relaxation rates of the right order of magnitude a n d correct temperature dependence may be de- rived. This, again, is an i n d i c a t i o n of the importance of i n - ertial effects o n the orientational fluctuations of methylfluo- ride.
10 H 1 } 1 1 1 —i 1 , , , 1 , 1 I - J — . 1 , , 1 1-
4 5 6 3 4 5 6 3 4 5
— — IO3ZT ( K "1)
Fig. 10
Temperature dependence of the various relaxation rates K1 con- tributing to the total proton spin-lattice relaxation rate Ki(1H) in C H3R (+ RfJS, x Rf, V ( K fP - (RiIffi + K?R)), O (RiOS +
R1TJS + RsS))
Orientational ( r2) - and Spin-Rotational ( Ts r) Correlation Times in C D3F and C H3F
In this study only integral orientational correlation times could be determined. The orientational correlation func- tions, however, as obtainable by high-frequency methods over limited time spans show characteristic features of o r i - entational fluctuations. A t very short times purely kinetic (inertial) effects dominate as a simple consequence of the time reversal symmetry of classical autocorrelation functions [4]. A t intermediate times and lower temperatures l i b r a - t i o n a l oscillations show up characteristic for motions i n a cage [2]. The long-time tail of orientational correlation func- tions is almost always found to be exponential. T h i s M a r - k o v i a n behaviour is a simple consequence of the long-range isotropy of the l i q u i d [1]. A l l these characterisitic features are included in the orientational correlation times being the time integral of the corresponding orientational autocorre- lation function i n an unspecific way. Therefore it is difficult to draw detailed conclusions about the nature of a reorien- tation process because integral correlation times always ac- centuate the M a r k o v i a n nature of the orientational fluctua- tions. Spin-rotation relaxation relates to fluctuations of the
angular velocity and of the orientation of the molecules.
A g a i n only integral spin-rotational correlation times Ts r of the angular velocity-orientational product correlation func- tion can be determined from an N M R experiment. A l s o it is only w i t h i n certain m o t i o n a l models that these bivariate correlation functions have been obtained. Because orienta- tional correlation functions may also be obtained w i t h i n the realm of these models, a comparison of both T2 and Ts r may prove useful i n p r o v i d i n g insight into the state of molecular m o t i o n . The two most often used m o t i o n a l models for this purpose are the extended diffusion model ( E D J a n d E D M ) [27] and the F o k k e r - P l a n c k - L a n g e v i n m o d e l ( F P L ) [21, 23, 29]. In the two models applied to spherical tops a single parameter, To j, is changed to produce the limits of free ro- tation and rotational diffusion. B o t h models assume suc- cessive uncorrelated instantaneous collisions, but differ i n the strength of these collisions. In the F P L - m o d e l the an- gular impulse of each collision is so small that the angular momentum changes infinitesimally. In the E D J - m o d e l col- lisions are strong with large angular impulses which ran- domize the angular m o m e n t u m at each step whereas i n the E D M - m o d e l only the direction of the angular m o m e n t u m is randomized. However, r a n d o m uncorrelated collisions cannot cause a reversal of the angular m o m e n t u m as is ob- served in most molecular liquids a n d indeed has been ob- served i n methylfluoride also [28]. Hence x(a must be con- sidered a lower limit to the "lifetime" of the angular velocity correlation function. Because of these shortcomings of both models when applied to high-torque liquids their usefulness has been questioned recently [30].
bard-relation T2 T *
TS R = 6 1 is shown corresponding to the
— * ~ LS R
Fig. 11
Reduced orientational correlation times T J versus reduced spin- rotational correlation times T*R. Full curve gives T*(T*R) according to the EDM-model. Straight line represents the Hubbard relation T* • t JR = 6"1. ( O s.v.p., • 20 M P a , A 50 M P a , + 100 M P a , V
150 M P a , x 200 MPa)
F i g . 11 shows a plot of T2 = T2
IkT
versus TS R = TS R
</> . The full curve gives the dependence of T * ( T *R)
Debye rotational diffusion model. The latter is contained in the E D J - m o d e l a n d the F P L - m o d e l i n the limit T*R 4 T*
However, the experimental data clearly show that the H u b b a r d relation is not obeyed i n the limit T* <^ T*. Only the E D M - m o d e l allows for a deviation from the Hubbard relation i n this limit. O r i g i n a l l y the orientational correlation times T2 had been calculated with a deuterium quadrupole coupling constant of 133 k H z with the result that the data fell above the theoretical curve for the E D M - m o d e l . Chang- ing the coupling constant to 143 k H z as suggested by a comparison of R a m a n and N M R data [15] brings the ex- perimental data i n coincidence with the theoretical curve predicted by the E D M - m o d e l . F o r the H u b b a r d relation to be fullfilled at least approximately i n the limes T*R <^ t*
one w o u l d have to change the coupling constant to 188 kHz.
This seems to be too large if compared with coupling con- stants of similar compounds (see Table 4) although the the- oretical estimate (212 k H z ) is even larger. F o r comparison F i g . 12 gives the graph T2 versus T *r with T2 calculated with a c o u p l i n g constant of 188 k H z . A l s o the prediction of the F P L - m o d e l [23,29] and the Debye rotation diffusion model are shown. The agreement with the F P L - m o d e l is less con- vincing, hence the E D M - m o d e l offers the most satisfactory description, within the realm of these models, of the molec- ular dynamics of methylfluoride.
2 . 0 -
T *
t
1.0 0.9 0.8 0.7 0.6
0.5
0.4
0.3
0.1 02 0 3 0.4 0.5 0.6
1S R
according to the E D M - m o d e l [27]. Furthermore the H u b -
Fig. 12
Reduced orientational correlation times x* versus reduced spin- rotational correlation times T*R. Full curve represents the predic- tion according to the Fokker-Planck-Langevin model, T* has been calculated with (e2qQ/h) (2H) = 188 (kHz). Symbols as in Fig. 11.
The straight line gives the Hubbard relation x* • T JR = 6_ 1
The expert technical assistance by M r . R. Knott, E. Treml, S.
Heyn and G . Wiihrl made this work feasible only. Prof. E. U.
Franck is thanked for providing us the densities obtained in his laboratory. Financial support by the Deutsche Forschungsgemein- schaft, the Fonds der Chemischen Industrie and the Friedrich- Ebert-Stiftung is gratefully acknowledged.
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(Eingegangen am 19. Miirz 1987, E 6467 endgultige Fassung am 18. M a i 1987)
High Pressure NMR Study of the Molecular Dynamics of Liquid Fluoroform and Deutero-Fluoroform
E . W . Lang, F . X . Prielmeier, H . Radkowitsch, and H . - D . Liidemann
Institut fur Biophysik und Physikalische Biochemie, Universitat Regensburg, Postfach 397, D-8400 Regensburg
Diffusion / High Pressure / Liquids / Spectroscopy, Nuclear Magnetic Resonance / Transport Properties
The 2D - , 1H - and , 9F-spin-lattice relaxation rates R1 have been investigated in fluid fluoroform in the temperature range 150 K < T <
450 K and at pressures up to p = 200 M Pa. Previous measurements of the self-diffusion coefficient D have been supplemented to cover the same p, T-interval. Within the rough hard sphere (RHS) approximation a temperature independent R H S diameter d and a strongly temperature dependent rotation-translation coupling ART are obtained. Both parameters arc also compared with those obtained in a series of related halomethanes. The total 2D - , 1H - and 1 9F-relaxation rates in C D F3 and C H F3 are decomposed into their respective quadrupole, dipolar and spin-rotation contributions and orientational and spin-rotational correlation times are extracted from these rates. It is shown that they are in good agreement with the predictions of the Fokker-Planck-Langevin model. Inertial effects influence the molecular dynamics at high temperatures and low densities. The agreement of self-diffusion coefficients and orientational correlation
times with MD-simulation data is very satisfactory.
Introduction
T h e m o l e c u l a r d y n a m i c s of l i q u i d fluoroform have been studied i n recent years by a variety of experimental tech- niques such as N M R , dielectric r e l a x a t i o n and a b s o r p t i o n , R a m a n scattering, a n d IR-spectroscopy [1 — 10]. F r o m the high frequency methods o r i e n t a t i o n a l c o r r e l a t i o n functions
may be obtained over a l i m i t e d time range and details of the short time dynamics of o r i e n t a t i o n a l fluctuations un- ravelled. N M R studies cannot y i e l d such detailed informa- t i o n since i n the short c o r r e l a t i o n time l i m i t o n l y the area of the relevant c o r r e l a t i o n function is determined and not its shape. However, if spin-rotation interactions dominate,
Ber. Bunsenges. Phys. Chem. 91, 1025-1033 (t987) - © V C H Verlagsgesellschaft m b H , D-6940 Weinheim, 1987.
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