Suppression of diffusion in the Lorentz gas

Journal of Non-CrystaUlne Solids 156-158 (1993) 205-209 North-Holland

IOURNA~ Or ~L_L~IE ~

Suppression of diffusion in the Lorentz gas R. B o n g r a t z a n d Ch. M o r k e l Phystk-Department, Techmsche Unwersttat Mtinchen, 8046 Garchmg, Germany

In an incoherent quasielastlc neutron scattering experiment on the binary system H 2-Ar, the diffusive dynamics of a light tagged particle m a matrLx of heavy scatterers have been studied The hydrogen-argon mixture well represents the Lorentz model of molecular dynamics. This simple model, although established long ago, has become tractable theoretically only recently owing to the development of mode-couphng techniques in the theory of dense many-particle systems According to recent theoretical results on the Lorentz gas for non-overlapping hard spheres, the &ffusion of the light species ~s found to be suppressed significantly below the Enskog value, DE, at increased "0, the packing fractton of the scatterers A density scan of the reduced diffusion coefficient D / D E measured at constant temperature indicates a diffusion locahzatlon transition in the system. The critical density is extrapolated to r/c = 0.30 in agreement with mode-coupling results for the H 2 - A r Lorentz model.

1. Introduction

In the study of transport phenomena in disordered systems, the Lorentz gas (LG) is a simple and rather instructive model system of statistical mechanics. Although established nearly a century ago by Lorentz [1], a satisfactory description of the L G dynamics - a tagged light particle diffusing in an amorphous matrix of fixed large scatterers - has only recently been achieved. Because of the application of mode-coupling techniques in the theory of the L G [2,3], the diffusion constant, D, is found to be renormalized by socalled correlated collisions, which a tagged particle experiences in a dense array of scatterers. These scattering events, occurring with increasing probability at higher packing fraction, induce a densitydependent feedback mechanism on the particle's dynamics. This has not been taken into consideration in the Enskog derivation of the diffusion coefficient, which makes the assumption of molecular chaos being justified only at low den-

Correspondence to Dr Ch Morkel, Fakultat fur Physlk der Technischen Unlversltht Munchen, James-Frank-strasse, W8046 Garching, Germany. Tel: 49-89 3209 2475. Telefax: + 4989 3209 2112.

sity. The well known Enskog result, DE, for the diffusion constant of a particle labeled 1 interacting via a hard-sphere potential with an array of scatterers labeled 2 is usually taken as a reference [4], so the quantity of interest here is the reduced diffusion constant, D / D E . In the basic theory of the L G [3], the quantity D I D E is predicted to decrease linearly with increasing density, given here in reduced units fi = nor3~

D/D E = 1 - (~/~c),

(~c = 9/4"n').

(1)

At some critical density, hc, the feedback mechanism described above leads to a blocking of the diffusion and the system undergoes a diffusion localization transition (DLT). The above result is valid only for the so-called overlapping LG, which deals with a point particle scattered from randomly distributed hard spheres, which are allowed to overlap. In a real experiment, neither the size of the small particle ~r~ nor the static structure of the scatterers tr 2, which is far from random, can be abstracted. Hence, up to now, a close comparison between theory and a real experiment suffered from idealizations of the theory. The above-mentioned features of realistic systems have been taken into account in the

0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B V All rights reserved

206

R Bongratz, Ch Morkel /Suppresston of dtffuslon m the Lorentz gas

subsequent theory of the non-overlapping L G [5]. Hence a quantitative comparison between theory and experiment is now an interesting task. A large amount of work in this direction was done earlier in Alder et al.'s extensive MD study of the mass and size dependence of a tagged particle's diffusion at different solvent densities [6]. Those results, fortunately reported in tabulated form, appear now to agree closely to the recent mode-coupling theory [5]. Whereas Alder used a simple hard-sphere interaction potential, Joslin and Egelstaff [7] as well as Sharma et al. [8] used a realistic Lennard-Jones potential in their MD studies of the L G model H 2-Ar. On the actual experimental side, the binary mixture of hydrogen in argon - shown to be a good model system for the L G [9] - has been studied persistently during the past three decades [9-11], concentrating mainly on the Q dependence of the peak shape of Ss( Q, to) - the selfdynamic structure factor as measured by incoherent neutron scattering. In this contribution, we restrict ourselves to the low-Q region in order to study the tagged particle's diffusive dynamics at various scatterer densities. Such a density scan of the reduced diffusion constant, D / D E , at constant temperature has not been conducted before, in a real experiment and offers the possibility of extrapolating the critical density for the D L T in a Lorentz gas.

2. Experimental The mixture H 2-Ar has been chosen as model system for a L G because of its precisely known and favourable parameters for such an experiment [9]. There is first the mass ratio m l / m 2 = 0.05, which is close to zero as required from theory. Besides the mass ratio, the second parameter of relevance for a L G model system is the size ratio, 6 = ~rl/o"z. This ratio is of great influence on the diffusion of the tagged particle in the matrix of scatterers for geometrical reasons and is taken into account with 6 = 0.81 here [12]. As far as the internal degrees of freedom of the hydrogen molecule are concerned, it has been shown elsewhere that, in the low-energy-transfer region

of the scattering law explored here, the contributions due to coherent scattering and the first rotational level of the H 2 molecule at 14.7 meV are negligibly small [9,11]. Hence the translational part of Ss(Q, to) is observed in the experiment, yielding the desired information about the diffusion constant in the hydrodynamic limit. Looking at the neutron scattering cross-sections for hydrogen and argon, it is obvious that the array of argon scatterers is practically invisible in the scattering intensity compared with the overwhelming cross-section of hydrogen [13]. The experiment was carried out on the statistical chopper time-of-flight spectrometer of the Forschungsreaktor Miinchen (FRM) [12,14] with an incident energy E 0 = 8.93 meV and an energy resolution A E / E o = 2.5 × 10 -2. Spectra were taken at momentum transfers, Q, ranging from 0.12 to 0.55 ~ - 1 . The cylindrical sample container (wall thickness between 0.7 and 1.8 mm according to pressure) was manufactured from an aluminum alloy of extraordinarily high tensile strength [12] and was pressurized at constant temperature T = 294 K between 300 and 1000 atm in order to achieve the required argon number densities. In table 1, the relevant thermophysical parameters of the experiment are noted.

3. Results Spectra of the H 2 - A r system for an intermediate packing fraction r / = 0.253 are shown in fig. 1 for two different momentum transfers, Q. The width of the normalized spectra is seen to increase insignificantly with Q and - not shown in this figure - decrease characteristically with

Table 1 Pressure, n u m b e r densities n 1 (H2) , n 2 (Ar) and packing fraction ~ = (~r/6)n2cr23 for the LG H 2 - A r at T = 294 K p (105 Pa)

nx ( 1022 c m - 3 )

n2 ( 1022 cm -3)

7/

325 511 528 745 982

0.05 0.05 0.05 0.05 0 05

0.678 1 008 1.022 1.228 1.382

0.140 0.207 0.210 0.253 0.284

R Bongratz, Ch. Morkel

/Suppresstonofdiffuston m the Lorentz gas

Lorentzian yielding the diffusion constant via the half-width oJ1/2(Q), which is known to vary for low Q as [3,5]

0.8

0=0.12 A4 0.6

o J 1 / 2 ( Q ) = DQ2(1 - a * Q 2 ) . e.,,

(2)

o.#

The coefficient a* is a small positive constant, which is known only in the low-density regime [15]. According to eq. (2) the diffusion constant, D, has been extrapolated from the Q --* 0 limit of a sequence of the measured half-widths for each density. The density scan of the reduced diffusion constant D / D E evaluated in this way is presented in fig. 2, the main result of the experiment. The values of this work (dots) cover only the intermediate to high density range, where the suppression with respect to the Enskog approximation is most serious. It is interesting to include the MD results of Alder for a hard-sphere mixture at three different densities [6]. This gives the three triangles in fig. 2, two of which lie far outside the density range accessed in our experiment. Since the present state of the theory for a LG is not yet at a level to take into account mass ratios different from zero, an extrapolation of the measured D / D E values to m l / m 2 = 0 is necessary and has been included in fig. 2. The ratio m l / m 2 = 0.05 for our system being close to zero, the correction factors were extrapolated in agreement with previous estimations [5] from the Alder tables [6]. Such an extrapolation was not neces-

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growing argon density. For lack of a closed theoretical prescription of the scattering law Ss(Q, to) for the LG, the spectra have been fitted with a

H2 - Ar T = 294 K

1,0

t~

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R Bongratz, Ch Morkel /Suppresston of dtffuslon m the Lorentz gas

208

sary in the case of the MD work of Sharma et al. [8] (square in fig. 2). These authors simulated an H 2 - A r L G at T = 297 K, using a realistic Lennard-Jones interaction potential with the argon particles having infinite mass. The agreement between this point and theory is remarkably good. The mode-coupling theory for the non-overlapping L G [5] is plotted as full curve in fig. 2 for the H a - A t system. The theoretical curve for a point particle (orI = 0) is also shown (dashed line). The comparison of both curves indicates the seriously reduced mobility of a large tagged particle and emphasizes the importance of the size parameter 6 in theory and experiment. The renormalization of D - predicted in the mode-coupling description of the L G - leads to a significant suppression of D below the Enskog value, showing up in agreement with theory more and more pronounced at high density. The density scan of D / D E offers the possibility of extrapolating the critical density, r/c , from the data, although the density range of localization could not be reached in our experiment. A linear extrapolation from the last four points in fig. 2 yields a critical density ~/c = 0.30 + 0.03. In fig. 3 we compare this result (dot) with the mode-coupling theory for the H e - A r system (full curve), giving ~7c = 0.29. Obviously there is good agreement between several experiments [6,8,12] and theory [5]. In fig. 3, the critical density of a

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0,5

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0,5

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t

6

i

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point particle in the non-overlapping L G is indicated at ~/c = 0.75 (open circle), which is higher then the density of close packing r/p = ~ / - 2 / 6 = 0.740 (dashed line) [5]. Hence a point particle can diffuse even at close packing density in a threedimensional array, indicating once more the importance of the size parameters, 6. From fig. 3 one can also see the relevance of the mass ratio for the DLT. Whereas in the case m l / m 2 = 0 the critical density for a particle of equal size (6 = 1) is r/c = 0.22 (full curve in fig. 3), the critical density for a particle of equal mass and size is ~7c = 0.60 [16]. To avoid a zero mass correction of the experimental data, progress in the theoretical description of the L G including a variable mass ratio besides the size ratio 6 would be highly desirable.

4.

Conclusions

The L G system hydrogen in argon has been studied, allowing for a quantitative comparison of quasielastic neutron scattering data with recent predictions of mode-coupling theory. In a density scan of the diffusion coefficient of a light particle in an array of heavy scatterers, the mobility is found to be seriously suppressed below the Enskog value at increasing packing fraction. The measured slowing down of diffusion is in quantitative agreement with the mode-coupling theory for the non-overlapping LG. A diffusion localization transition is extrapolated from the data at a critical density r/c = 0.30. This density agrees well with theoretical calculations, taking the size ratio of the two species into account. The relevance of size and mass ratio is emphasized, while a full theoretical account of the mass ratio is still missing. Nevertheless, our experiment can serve as a testing ground and confirmation for a promising theoretical concept, which quantitatively describes the diffusion localization transition and, more generally, the liquid-glass transition in the dynamics of disordered media [16]. This work was supported by the German Bundesministerium fiir Forschung und Technologie.

R Bongratz, Ch. Morkel / Suppression of dtffuston m the Lorentz gas

References [1] H. Lorentz, Proc. Amsterdam Acad 7 (1905) 438. [2] H Ernst and A Weyland, Phys. Lett. 34A (1971) 39. [3] W Gotze, E Leutheusser and S Ylp, Phys. Rev A24 (1981) 1008 [4] S. Chapmann, T Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University, Cambridge, 1952). [5] E. Leutheusser, Phys Rev. A28 (1983) 2510 [6] B Alder, W. Alley and J Dymont, J Chem. Phys 61 (1974) 1415. [7] C. Joshn and P. Egelstaff, J Stat. Phys 56 (1989) 127 [8] K Sharma, S. Ranganathan, P. Egelstaff and A Soper, Phys. Rev A36 (1987) 809.

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[9] O. Eder, S Chen and P Egelstaff, Proc Phys Soc 89 (1966) 833 [10] R McPherson and P. Egelstaff, Can J Phys 58 (1980) 289. [11] P Egelstaff, O. Eder, W Glaser, J Polo B Renker and A Soper, Phys Rev. A41 (1990) 1936. [12] R Bongratz, PhD thesm, Tech. Unlv Mimchen (1990). [13] V. Sears, Thermal Neutron Scattering Cross Sections (AECL-8490, CNRL 1984) [14] U Freudenberg and W. Glaser, Nucl Instrum and Meth. A243 (1986) 429 [15] S. Hess, Z Naturforsch 32a (1972) 678 [16] W Gotze and L. Sjogren, Rep Prog Phys 55 (1992) 241