Orientation correlation of neopentane molecules in liquid state through available diffraction data

Chemical Physics 270 (2001) 197±203

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Orientation correlation of neopentane molecules in liquid state through available di€raction data S. Sarkar, P.P. Nath, R.N. Joarder * Department of Physics, Jadavpur University, Calcutta 700 032, India Received 4 October 2000; in ®nal form 20 March 2001

Abstract The orientation correlation model proposed by Misawa [J. Chem. Phys. 91 (1989) 5648] for liquid carbon tetrachloride reproduce remarkably well the intermolecular part of the available X-ray di€raction data of liquid neopentane [J. Chem. Phys. 70 (1979) 299]. This is far more superior than the simple Apollo model [Molec. Phys. 20 (1971) 881] which was used for nearest neighbor orientation correlations in simple tetrahedral molecular liquids and was applied by Rao and Joarder [J. Phys. C (Sol. State) 14 (1981) 4745] to liquid neopentane. The calculated radial distribution function too strongly supports the view. The partial structures and pair distribution functions (PDF) are calculated on Misawa's orientation correlation model. The PDF's are compared with those obtained through reference interaction site model and Monte Carlo simulation calculations [Discuss. Faraday Soc. 66 (1978) 39; Molec. Phys. 45 (1982) 1193] and results are discussed. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Molecular liquids are known to have strong intermolecular correlations and to describe the liquid structure, orientational correlations have been prescribed [1]. The orientation correlations modify the structure function enormously in intermediate scattering vector region and this was quite evident during the course of our recent crucial comparative study of the structure of molecular liquids with intermolecular H-bonding and those without H-bonding [2]. The orientation correlations a€ect, as well, the thermodynamic and transport properties of the liquid. In this communication, we report in detail the study of orienta-

*

Corresponding author. E-mail address: [email protected] (R.N. Joarder).

tion correlation and liquid structure of neopentane at room temperature through X-ray di€raction data [3]. With X-ray group scattering concept [3] neopentane molecules, CMe4 (CH3 BMe), have almost tetrahedral spherically symmetric structure and in the liquid state, the intermolecular structure and correlations are dominated by orientation correlation between neighboring molecules. Egelsta€ et al. [4] proposed the Apollo model as an elementary model for nearest neighbor correlations in simple tetrahedral molecular liquids (e.g. CCl4 ). Rao and Joarder [5] applied this model to the experimental data of Narten [3] on liquid neopentane with a fair amount of success. Misawa [6] introduced a much more detailed model, involving many parameters in order to describe the whole structure at all distances, and applied it successfully to liquid carbon tetrachloride. We test the

0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 5 1 - 2

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same model of molecular orientation for liquid neopentane through a more critical method of analysis and show that it gives much better result to ®t the X-ray data at room temperature than Apollo type orientation model prescribed earlier.

The model distinct structure function Hd …k† is then given by Hd …k† ˆ F2u …k†‰SC …k†

Apollo model:

As usual, the molecular structure function, H …k† for a molecular liquid consists of intra- and intermolecular parts, k being the scattering vector. In the study we assume that (i) at short distance of separation the distribution of molecules in liquid CMe4 is determined by preferred orientation of two neighboring molecules and (ii) beyond that short distance, there is uncorrelated orientation and the structure functions of molecular liquids are dominated by symmetric hard sphere contribution. The orientational correlations determine the asymmetrical part of the intermolecular or distinct structure function. The measure of the asymmetrical part of the intermolecular correlation Gas d …k† is given by [2]

Gas d …k† ˆ ‰F2C …k†

F2u …k†‰SC …k†

1Š

…1†

where Hd …k† is the distinct or intermolecular structure function, F2u …k†, uncorrelated form factor between neighboring molecules, SC …k†, the center structure factor, spherically symmetric i.e., for completely uncorrelated orientation, given by hard-sphere PY-model as [6] SC …k†

1 ˆ ‰Shs …k†

1Š exp… D2hs k 2 =2†

…2†

Dhs is introduced to account for the fact that damping of hard sphere structure with k is small compared to that for a realistic potential in a liquid. The expression for F2u …k† is given by F2u …k† ˆ M…k†‰fC …k† ‡ 4fMe …k† 2

 sin…krCMe †=…krCMe †Š with M…k† ˆ ‰fC …k† ‡ 4fMe …k†Š

…3†

2

fC …k† and fMe …k† being the atomic scattering factors of C and Me atoms respectively.

…4†

In the model of orientation correlation the asymmetric part of the intermolecular correlation Gas d …k† is given by

2. Method of analysis

Gas d …k† ˆ Hd …k†

1Š ‡ Gas d …k†

F2u …k†Š‰SC …k†

1 ‡ …2=krc2 †

 …dSC …k†=dk†Š

…5†

where F2C …k† is the correlated form factor in the Apollo model [5] and rc is the correlation distance [4]. rc large means strongly correlated neighbors while small rc corresponds to weakly or uncorrelated neighbors [4] and the term containing rc represents a correction term to the anisotropic part in the Apollo model. The hard core diameter, r and correlation distance rc are suitably adjusted to ®t the model Hd …k† data to the experimental Hd …k† data [3]. The parameters are listed in Table 1. Misawa's model: " Gas d …k† ˆ nc M…k†

X

fa …k†fb …k†…sin krab =krab †

ab

 exp… D2ab k 2 =2†

F2u …k†

 …sin kRu =kRu † exp…

#

DR2u k 2 =2†

…6†

where a labels an atom in a molecule, b labels an atom in another neighboring molecule, summation

Table 1 Model parameters Apollo model  r ˆ 5:45 A  rc ˆ 5:0 A

Misawa model  Dhs ˆ 0:0744 A  r ˆ 5:469 A,  Ru ˆ 5:43 A  Rc ˆ 5:497 A,

nc ˆ 0:9498, dc ˆ 0:015976, du ˆ 0:097539 h ˆ 46:86°, / ˆ 49:02°, a ˆ 76:02°, b ˆ 38:14°, c ˆ 56:08°

S. Sarkar et al. / Chemical Physics 270 (2001) 197±203

being for all the atoms of both the correlated molecules, rab is the distance between a and b, Ru is a distance between centers of two molecules if they were uncorrelated and nc is a normalization constant. The ®rst term on the right-hand side of Eq. (6) is easily evaluated for a given orientation de®ned by six variables Rc , h, /, a, b, c as depicted in Ref. [6]. Rc is de®ned as rab ˆ jRc ‡ ra rb j. Rc , h, / are polar coordinates of the center of the neighboring molecule, taking carbon of one molecule as center. a, b, c are Euler angles of the neighboring molecule. Dab and DRu are introduced to accomodate the thermal ¯uctuations and these are assumed to be proportional to atomic spacing, namely Dab ˆ dc rab and DRu ˆ du Ru with constants dc and du . The ®ve angles h, /, a, b, c collectively determine the relative orientation of two neighboring molecules. The most probable orientation is estimated by ®tting experimental Hd …k† data [3] with model expression in Eq. (4) using Gas d …k† from Eq. (6) by means of v2 -method. The ®tted parameters are given in Table 1. The intraparameters used in the analysis are those from Refs. [3,5]. Calculated k-weighted Hd …k† for both the models are shown with experimental data in Fig. 1. It is quite evident that Misawa's model presents much better orientational correlation than that represented by Apollo model. The intermolecular radial distribution function (RDF), Gd …r† is ob-

Fig. 1. kHd …k†: (± ± ±) experimental (T ˆ 298 K, q ˆ  3 , X-ray data [3]), (


) Apollo model, (  ) MC 0:00488 A  3 ), (-
-
-) uncorrelated conresult (T ˆ 298 K, q ˆ 0:00488 A tribution.

199

 3, Fig. 2. Gd …r†: (± ± ±) experimental (T ˆ 298 K, q ˆ 0:00488 A X-ray data [3]), (±±) Misawa's model, (


) Apollo model, (-
-
-) uncorrelated contribution.

tained by Fourier inverse transform of the distinct structure function Hd …k†. Z …7† Gd …r† ˆ 1=…2p†3 kHd …k† exp…ik
r† dk In Fig. 2 we show Gd …r† for both the models with one obtained from experimental data [3]. It is again evident that Misawa's model of orientational correlation is remarkably better. The most probable orientation (approximate) of two neighboring neopentane molecules according to Misawa's model is shown in Fig. 3. One Me group of the molecule `A' lies symmetrically close to the two Me groups of the neighboring molecule `B' near its one hollow and the one other Me group of the molecule `B' lies a little bit apart from a hollow of the molecule `A' and the rest two Me groups of the two molecules lie almost farthest distance apart. The two molecules are somewhat locked but very di€erent from ideal closed packed locking. Though one Me group of one molecule is nestled near the hollow of the neighboring molecule it cannot freely rotate as in Apollo model. This is of course an average orientation of two neighboring molecules. 3. Partial structures and correlations Since Misawa's orientational correlation model describes the liquid structure remarkably well, we

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S. Sarkar et al. / Chemical Physics 270 (2001) 197±203

Again from Eq. (4) we have

h  Hd …k† ˆ M…k† fC2 …k† SC …k† 1 ‡ Gas d …k†jCC  ‡ 8fC …k†fMe …k† …SC …k† 1† sin krCMe =krCMe n 2 ‡ Gas 1† d …k†jCMe ‡ 16fMe …k† …SC …k† oi  …sin krCMe =krCMe †2 ‡ Gas d …k†jMeMe

…9†

We note that SCC …k†BSC …k† here and Eq. (9) can be written as sum of partial terms as in Eq. (8). Then comparing Eqs. (8) and (9) we obtain partial structure factors SCC …k†, SCMe …k† and SMeMe …k† given by SCC …k† ˆ 1 ‡ ‰Shs …k†

1Š exp… D2hs k 2 =2†

‡ nc fsin…kjRc j=kjRc j† exp… DR2c k 2 =2† sin…kRu =kRu † exp… DRu k 2 =2†g …10a† SCMe …k† ˆ 1 ‡ …sin krCMe =krCMe †…SC …k† ( X ‡ nc sin…krCMe =krCMe †

1†

C;Me

 exp… D2CMe k 2 =2† …sin kRu =kRu † )  exp… DR2u k 2 =2†

…10b† 2

Fig. 3. (a) Top view (from Z-direction). (b) Side view (along Ydirection). A, circles with bold lines; B, circles with thin lines.

SMeMe …k† ˆ 1 ‡ …sin krCMe =krCMe † …SC …k† 1† ( X sin…krMe;Me =krMe;Me † ‡ nc Me;Me

 exp… D2Me;Me k 2 =2† ) consider the calculations of partials on this model only. To ®nd the partial structure functions for the model, we write the distinct structure function for the model, Hd …k† as the sum of the partial structure factors. Thus [5] Hd …k† ˆ M…k†‰fC2 …k†…SCC …k†

1†

‡ 8fC …k†fMe …k†…SCMe …k† 2 ‡ 16fMe …k†…SMeMe …k†

1†Š

1† …8†

 exp… DR2u k 2 =2†

sin…kRu =kRu † …10c†

The intermolecular partial pair distribution functions (PDF) gCC …r† … gC …r††, gCMe …r† and gMeMe …r† are obtained respectively from the Fourier inversion of SCC …k†, SCMe …k† and SMeMe …k† given by Z 3 gab …r† ˆ 1 ‡ 1=…2p† q …Sab …k† 1† exp…ik
rab † dk …11†

S. Sarkar et al. / Chemical Physics 270 (2001) 197±203

201

 3 ), (  ) MC result (T ˆ Fig. 4. (a) gCC …r†: (±±) Misawa's model, (± ± ±) RISM result (T ˆ 298 K, q ˆ 0:00488 A  3 ), (  ) MC result  3 ). (b) gCMe …r†: (±±) Misawa's model, (± ± ±) RISM result (T ˆ 298 K, q ˆ 0:00488 A 298 K, q ˆ 0:00488 A  3 ). (c) gMeMe …r†: (±±) Misawa's model, (± ± ±) RISM result (T ˆ 298 K, q ˆ 0:00488 A  3 ), (  ) MC result (T ˆ 298 K, q ˆ 0:00488 A 3  (T ˆ 298 K, q ˆ 0:00488 A ).

The results for gCC …r†, gCMe …r† and gMeMe …r† are shown in Fig. 4. The partial PDF calculations based on reference interaction site model (RISM) and Monte Carlo (MC) simulation are available [7,8]. The orientation correlation model results are compared with RISM and MC results. There is considerable degree of agreement among the three models. 4. Discussion of the results It is evident from the results shown in Figs. 1 and 2 that Misawa's orientational correlation model is far better compared to Apollo model prescribed earlier. In the ®gures we have shown the uncorrelated contributions as well and evidently there is no feature in this curves except those for a spherically symmetric molecule. It is however to be

noted that though Apollo model is less successful in describing the orientation correlation of neighboring molecules in liquid neopentane, the model has only two parameters to adjust while Misawa's model involves 12 parameters. Although a v2 minimum method was used for the best ®tting, the relative magnitudes of various parameters are hardly unique and as such not much physical signi®cance could be set on the magnitudes of the parameters involved. The accuracy of the analysis also depends on the accuracy of the experimental data. The data used here is quite old but at the same time known to be quite accurate [3]. Regarding the predictive powers of the two models we note the following. The Apollo type orientation correlation is consistent with the structure of the molecule and takes into account the van der Waals distances and it is not seriously inconsistent with the known liquid density [4].

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S. Sarkar et al. / Chemical Physics 270 (2001) 197±203

However, the agreement in Ref. [5] shown for molecular structure factor was good except in the  1 where the orientaimportant region 1.75±3.5 A tion correlation plays major role. In RDF (Fig. 2)  where peaks occur the important region 3.5±5 A giving signature of speci®c orientation correlation, the agreement is poor. It is infact somewhat similar to uncorrelated model. In liquid carbon tetrachloride too, there are distinct peaks in this region showing speci®c form of orientation correlation di€erent from that in Apollo model [9,10]. Apollo  model does not yield Me±Me peak at about 4.1 A as in the case for similar liquid e.g. carbon tetra Also the chloride (Cl±Cl peak occurs at 3.95 A). present detailed tests (including one mentioned in Section 5) show that it fails miserably. Misawa's model, on the other hand, predicts the intermolecular structure, anisotropic part of the structure measuring orientation correlation. As a result the intermolecular atom±atom distribution functions derived in the present study would be acceptable with con®dence. The Me±Me peaks lie at expected distances (Fig. 4(c)). We now discuss in detail the intermolecular partial distribution functions, gCC …r†, gCMe …r† and gMeMe …r† shown in Fig. 4(a)±(c). It is gratifying to note that the orientation correlation model results agree reasonably with RISM and MC results. It is to be noted that RISM theory for fused hard sphere molecules is only an approximate representation of real liquids and reasonably accurate for hard molecules [7]. The MC simulation results are strongly model potential dependent and results

for exponential-6 potential are only more close to the experimental di€raction data (see Fig. 1 for kHd …k† data) [8] than other potential models and shown here for comparison. The present orientational model result for gCC …r† agrees more with the MC result for L-J potential than exponential-6 potential showing evidently that attractive forces play a major role for this liquid with tetrahedral molecules. The present model results for gCMe …r† and gMeMe …r† agree, however, more closely with exponential-6 potential MC results and less closely with RISM results. The present gCMe …r† curve does not exhibit sharp double peak like those in MC and RISM calculations while in gMeMe …r† curve there is an extra small peak (vide Fig. 4(b) and (c)). The results for gMeMe …r† however appear to support the view that molecular rotations are hindered in this liquid and attractive forces are important in deciding the intermolecular correlations. 5. General remarks In our previous communication [2] we have compared the asymmetrical part of the intermolecular structure from experimental data and that obtained from Misawa's orientational correlation model for two neighboring molecules in k space. We have also compared asymmetrical parts of the r-space correlations. This gives a very sensitive test to identify the very probable average intermolecular orientational correlation for liquid structure. The results are shown in Fig. 5(a) and (b) together

 3 Fig. 5. (a) Gas d …k†: (± ± ±) experimental (T ˆ 298 K, q ˆ 0:00488 A , X-ray data [3]), (±±) Misawa's model, (


) Apollo model. (b)  3 , X-ray data [3]), (±±) Misawa's model, (


) Apollo model. …r†: (± ± ±) experimental (T ˆ 298 K, q ˆ 0:00488 A Gas d

S. Sarkar et al. / Chemical Physics 270 (2001) 197±203

with results for Apollo model. Clearly, the Apollo model is untenable. In this analysis the model studied gives remarkably good agreement with experimental data at room temperature although some discrepancies exist among the partial PDFs obtained from the model and those obtained from the RISM and MC simulation calculations. In the study with Apollo type orientation correlation model in Ref. [5] the agreement was shown between the model and the experimental molecular structure factor i.e., Sm …k† data. In Sm …k† data the major contribution being the intramolecular structure part, the agreement between the model and the experimental contributions for intermolecular structure which measures the intermolecular correlation (mainly, orientation) could not be properly ascertained and it was not subject to sensitive test like one in the present analysis. Misawa's orientational correlation model, therefore, represents a far more superior model for liquid neopentane structure. Another possible representation of the orientation correlation is through the use of spherical harmonics expansion of total RDF ± the coecients giving magnitudes of orientation correlation [11]. The construction of the partial structure factors of molecular liquids with a very large number of coecients (e.g. as many as 170) is possible and a very good ®t to the experimental data is obtainable and thereby extract orientation correlation [12]. The method uses Max. Ent. like reconstruction of the spherical harmonics expansion with weighting functions determined largely by trial and error. For liquid neopentane the experimental partial structure factors are not available and so the method cannot be tested though in principle the method should be applicable. Again the experimental determination of the partial structure factors using the method of isotopic substitution in neutron di€raction itself is a tricky method and the accuracy of these partials is subject to several limitations and so the extraction of the orientation correlations. Infact, the spherical harmonics coecients if available from computer simulations could establish the degree of strengths

203

and weaknesses of the direct spherical harmonics expansion ®tting method. The present method, in comparison is very straightforward and with a relatively small number of orientational parameters gives ®ttings to experimental data (Figs. 1 and 2) extremely well almost similar to spherical harmonics expansion method [12]. The features in the anisotropic part are represented extremely well by the model (Fig. 5(a) and (b)) and descrepancies that occur can be attributed to our assumption of isotropic part being given by HS expression and the limitations of the experimental data. So the present method is very useful way for extracting orientation correlation in the liquid state. Further, the partials are obtainable and could be compared with experimental and other theoritical and simulation model data whichever available.

Acknowledgements The authors are grateful to DST (New Delhi) and IUC-DAEF (Mumbai Center) for ®nancial assistance. References [1] P.A. Egelsta€, An Introduction to the Liquid State, Oxford Science Publication, 1994. [2] S. Sarkar, P.P. Nath, R.N. Joarder, Phys. Lett. A 275 (2000) 138. [3] A.H. Narten, J. Chem. Phys. 70 (1979) 299. [4] P.A. Egelsta€, D.I. Page, J.G. Powels, Molec. Phys. 20 (1971) 881. [5] R.V.G. Rao, R.N. Joarder, J. Phys. C (Sol. State) 14 (1981) 4745. [6] M. Misawa, J. Chem. Phys. 91 (1989) 5648. [7] A.H. Narten, S.I. Sandler, T.A. Rensi, Discuss. Faraday Soc. 66 (1978) 39. [8] D.S.H. Wong, S.I. Sandler, Molec. Phys. 45 (1982) 1193. [9] R.W. Gruebel, G.T. Clayton, J. Chem. Phys. 46 (1967) 639. [10] A.H. Narten, M.D. Danford, H.A. Levy, J. Chem. Phys. 46 (1967) 4875. [11] C.G. Gray, K.E. Gubbins, Theory of Molecular Fluids I: Fundamentals, Vol. I, Clarendon Press, Oxford, 1984. [12] A.K. Soper, J. Chem. Phys. 101 (1994) 6888.