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The effect of visco-elastic behaviour of simple liquids on the observation of long-time tails
Volume 80A, number 5,6
PHYSICS LETTERS
22 December 1980
THE EFFECT OF VISCO-ELASTIC BEHAVIOUR OF SIMPLE LIQUIDS ON THE OBSERVATION OF LONG-TIME TAILS P.E. MASON and T. GASKELL Department of Physics, The University, Sheffield S3 7RH, UK Received 7 October 1980
312decay of the velocity autocorrelation function means that the frequency spectrum has a cusp at the origin. The t~ Its significance is shown to depend crucially on the visco-elastic character ofthe liquid. Our results suggest that its observation will be particularly difficult in a liquid metal.
The predicted asymptotic behaviour of the velocity autocorrelation function in a liquid, namely t,Vt = ~12n~~ 1ir1v + D~t’—3/2 -+ °°~
k
)
\
/
I
‘
~
‘
‘
‘
means that the frequency spectrum, ~i(w), has an infinitely negative slope, and hence a cusp, at the origin [1]. In a dense liquid this produces a dip in the spectrum, before it rises with increasing frequency to the characteristic broad peak. More precisely, it follows from eq. (1) that =
~“
-~ ~
dt ~Vt~ ~ cos
~
‘2~
~
‘ /
0
is such that i~-~0)=(Dm/irkfl[l—(w/w
1/2
0) +...], (3) 3(irnmD~kT)2.In the above where w0i,li(t)72(v +D)u(t))/(u (0)),u(t) being the analysis = (u(0) velocity of an atom in the liquid, D is the seif-diffusion coefficient, v the kinematic viscosity and n the number density. The physical reason for the slow loss of correlation is that the atom’s velocity is coupled to the transverse components of the fluctuations in the velocity field, setting up shear wave excitations. The transfer of momentum by this means, at sufficiently long times, is essentially a diffusion process. It is easy to show, at least qualitatively, that this leads to the t~312behaviour of the correlation function. This contrasts with the contribution to momentum transfer via -
sound waves, which produce an exponential decay of velocity correlation. The so-called long-time tail was discovered in molecular dynamics studies of a rigid-sphere fluid [2]. The frequency spectrum of a real liquid may be obtained from the incoherent contribution to the neutron inelastic scattering cross section, in the zero wavenumber limit. A study of the low frequency behaviour could, in principle, confirm the effect. Caneiro recently analyzed some neutron scattering data on liquid argon [3], with this in mind. The results show a low frequency dip, with a minimum at approximately 0.08 X 1013 s_I. It has to be remembered, however, that the smallest wavenumber,q, available in the experiments was I A1. It is far from clear that the data can be extrapolated accurately to q = 0. A recent investigation of the low frequency has behaviour, for models of liquid argon and rubidium, been made using mode-coupling theory [4]. Estimates of the location of the minimum were somewhat smaller than is suggested by Caneiro’s work. In fact a frequency close to 1011 ~-I was obtamed for both liquids, which, if correct, will mean that its experimental observation will be difficult. The visco-elastic character of the liquid was suggested as the major reason for this suppression of the onset of the long-time tail. The purpose of this note is to report some work on the same problem using a velocity field approach. It has been shown to give a very good representation of the gross characteristic features of the computed frequency spectrum in liquid rubidium 395
Volume 80A, number 5,6
PHYSICS LETTERS
[5],but the time scale was not sufficiently long to even see the anomalous low frequency behaviour. Here, we carry through a calculation of the frequency spec312tail carefully intrum with the effect of the t~ cluded. We have repeated the calculation for argon, and also for a rigid-sphere fluid, where the viscoelastic properties will be somewhat different. The derivation of the expression for Li(t) is thoroughly discussed in the latter reference. Within the ensemble average the velocity field,u(r, t), is represented by -
.
-
-
-
v(r,
t)
22 December 1980
M(q, t) [5]. In other words the memory function has an exponential time dependence, with a wavenumberdependent relaxation time r(q). The Fourier—Laplace transform of C~(q,t) is -
‘~-.
—
C~(q,z)mkT[—iz +M(q,z)] and 2
~
M(q z)~’qG(q)(nm)
—1
-
[1z+T
—1
(q)j
—1
G(q) represents the wavenuniber-dependent rigidity, and r(q) is such that r(q -÷ 0) = nmv/G(0), so that the hydrodynamic form is recovered in the small q and
~r~u 1(t)f(Ir
—
r,(t)I),
where u1(t) is an atomic velocity and the function f(r) is chosen so that the velocity field is essentially constantinacross an atomic diameter. order be correct the hydrodynamic limit it In must alsotosatisfy the condition ff(r)dr
n~.
(4)
large t limits. In fig. 1 we display the low frequency shape of the spectrum for a model of liquid rubidium at a temperature T 319 3Kand 0.01058 ~T(°-’)number is that density part of nit which, through X 1024 eq. (5), cm~ is a consequence of C~(q,t). The hydrodynamic contribution to ~T(’~)~ shown in the figure, refers to a result in which C~(q,t) is replaced by C~(q,1), thus neglecting the visco-elastic behaviour -
The result for the velocity autocorrelation function is ~J(t)3k~.
/
-_~fdq~q)[cL(q,t) t0~0.
—
+ 2C~(q,t)]F
5(q, -
-
(5)
t),
withf(q) and F~(q,t) the Fourier transforms off(r) and the self-correlation function, respectively. C~(q,t) and C~(q,t) represent the longitudinal and transverse momentum current density correlation functions respectively. In accordance with the earlier statements, we stress that it is the transverse current correlation function which is crucial in a discussion of the asymptotic behaviour. The result in eq. (1) is obtained from the small wavenumber contribution to the integral in eq. (5). This means that C~(q,t), F5(q, t), andf(q) may be replaced by their hydrodynamic forms, namely
Hydr-odynamio contn~but ion
—
.
—.-.
w
-
to
/ 0.36
/
0.35
2
~(q,t)~-mkTexp(—q yin), 2Djti), f(q)~n1. F~(q,t)~exp(~ To look at a wider range we have used an expression for C~(q,t)which is derived from a simple viscoelastic model for the associated memory function 396
~
~
~ 0
Fig. 1. Frequency spectrum, ~(c~), of liquid rubidium at T 319 K and number density n = 0.01058 X 10~cm3.
Volume 80A, number 5,6
PHYSICS LETTERS
of the liquid. Note, too, that ~~w) has been normalized so that it coincides with i~iT(W)at ci., = 0. The results in fig. 1 clearly show that the inclusion of viscoelastic effects produces the rapid frequency variation and the dip in the spectrum. The longitudinal current contribution does alter the location and depth to some extent. It is also responsible for a slight “kink” in the spectrum at w ~ 0.035 X 1013 ~1, although the reason for it remains unclear. Bosse et al. [4] quantify the relative size of the cusp by defining a parameter a, as U
=
(~(O)-_
22 December 1980
~P(a)) s l~0.42
—
Htdr-od>’nam tO
‘C
contr but
‘°~
~P
1c~)s
0.38
‘min))1~~(~).
1, and w Our result for wmth is 0024 X 1013 s 0 4.31 X 1013 s_i, which are not significantly different from those given in ref. [41.Both methods give 0.04. A similar calculation has been carried through for argon close to the triple point. The transverse current is again based on the visco-elastic approximation. More details of both current models are given by Gaskell and Miller [61.Fig. 2 contains the for Tof 3.Tresults he location = 86K and n = 0.0215 X 1024 cm~ the minimum at w 0.045 X 1013 ~I is substantially bigger than the value derived from mode-coupling theory, and more in line with that obtained from Caneiro’s analysis of the neutron scattering data. Our result for a is 0.096, much smaller than the 0.34 suggested by the scattering experiments, but considerably bigger than that obtained in ref. [4]. For this liquid the frequency w 0 = 1.91 X 1013 s_i. The longitudinal current plays a more significant role in the formation of the dip, than is the case in liquid rubidium. It should be mentioned that computer simulation studies of the transverse current in argon have shown that the association memory function is more appropriately described by two exponentials, one of which represents a long-range component [7]. Unfortunately, we do not have sufficiently detailed informaabout to theuse wavenumber dependence of the relaxation times this more sophisticated model. The work of Bosse et al. [4] suggests that it would tend to reduce the frequency minimum, Finally we consider the rigid-sphere fluid. The transverse current is again constructed from the memory function approach, but in this case only part of the memory is assumed to have an exponential decay. There is in addition a delta function, which has to be
0.34
\
0.30—_-----,_----—---,-——
0.15
0.20
0.25X10u
als
Fig. 2. Frequency spectrum, ~ of liquid argon at T = 86K = 0.0215 X 10~’cm3.
and number density n
included because of the instantaneous character of the rigid sphere collisions [8]. Although this reflects a different visco-elastic character than either argon or rubidium it has nevertheless been demonstrated, in a computer study, that the rigid-spheres can support a propagating shear wave mode at the packing fraction (0.4628) used here [9]. The transverse current model we have derived incorporates this effect, and displays the physically correct features as the wavenumber is varied [8]. The results are given in fig. 3. The frequency unit is the Enskog collision frequency, which is m 2(irkT/m)1”2. In this equation a and = 4ng(a)a are the sphere’s diameter and mass respectively, and g(a) is the contact value of the radial distribution function. The most striking feature is the very different behaviour shown by ~/T(W) when compared to figs. 1 and 2. Presumably the formation of the shoulder, rather than the pronounced minimum shown in rubidium or argon, is a reflection of the less elastic nature of the fluid. In this case the contribution from 397
Volume 80A, number 5,6
PHYSICS LETTERS
22 December 1980
ly closer to the peak in the spectrum than it is in rubidium or argon. This is consistent with the fact that the long-time tail is easier to observe in molecular
0.5, o
dynamics _
“experiments” on a rigid 3rF), The is the Enskog result, 111(t) sphere-fluid. = exp( ~2t/ shown for comparison. The high frequency wing of the
spectrum of
spectrum quickly assumes the Enskog form, confirming that binary collisions are satisfactorily included in our
—
Ens ko
velocity field method.
9
show that the visco-elastic character of a dense liquid does tend to suppress the effect of the long-time tail. The more marked the visco-elasticity, the smaller the frequency at which the dip in the spectrum may be
0.2
0 1
~
H
/
—
seen. the The observation our result liquid of for the models rubidium long-time we have iswill tail, typical, considered by extremely means it means clearly of that cult neutron inIfathree scattering liquid metal. experiments, be diffi-
“..
References
-~
~/ 0.0 _~-~_—-i—-—--—--~——----—--———,—-———---—-i 0.0 0.5 1.0 1.5
2.0
Fig. 3. Solid line shows the frequency spectrum, ~(w)’ of a rigid-sphere fluid, packing fraction 0.4628. The open circles represent the spectrum of the Enskog result ~ (t) = exp(— 2t/ 3r~).The frequency unit is the Enskog collision frequency the spectrum is in TE.
the longitudinal current is responsible for the formation of the dip in the spectrum. The value of a is 0.098, but the formation of the minimum is significant-
398
[i] T. Gaskell and N.H. March, Phys. Lett. 33A (1970) 460. 121 B.J. Alder and T. Wainwright, Phys. Rev. Lett. 18 (1967) 988. [3] K. Caneiro, Phys. Rev. A14 (1976) 517. 141 J. Bosse, W. Gotze and M. Lücke, Phys. Rev. A20 (1979) 1603. 151 T. Gaskell and S. Miller, J. Phys. Cli (1978) 3749. 161 T. Gaskell and S. Miller, J. Phys. Cli (1978) 4839. 171 D. Levesque, L. Verlet and J. Kurkijarvi, Phys. Rev. A7 (1973) 1690. [8] P.E. Mason and T. Gaskell, Mol. Phys., to be published. 19] B.J. Alder, W.E. Alley and S. Yip, to be published.
PHYSICS LETTERS
22 December 1980
THE EFFECT OF VISCO-ELASTIC BEHAVIOUR OF SIMPLE LIQUIDS ON THE OBSERVATION OF LONG-TIME TAILS P.E. MASON and T. GASKELL Department of Physics, The University, Sheffield S3 7RH, UK Received 7 October 1980
312decay of the velocity autocorrelation function means that the frequency spectrum has a cusp at the origin. The t~ Its significance is shown to depend crucially on the visco-elastic character ofthe liquid. Our results suggest that its observation will be particularly difficult in a liquid metal.
The predicted asymptotic behaviour of the velocity autocorrelation function in a liquid, namely t,Vt = ~12n~~ 1ir1v + D~t’—3/2 -+ °°~
k
)
\
/
I
‘
~
‘
‘
‘
means that the frequency spectrum, ~i(w), has an infinitely negative slope, and hence a cusp, at the origin [1]. In a dense liquid this produces a dip in the spectrum, before it rises with increasing frequency to the characteristic broad peak. More precisely, it follows from eq. (1) that =
~“
-~ ~
dt ~Vt~ ~ cos
~
‘2~
~
‘ /
0
is such that i~-~0)=(Dm/irkfl[l—(w/w
1/2
0) +...], (3) 3(irnmD~kT)2.In the above where w0i,li(t)72(v +D)u(t))/(u (0)),u(t) being the analysis = (u(0) velocity of an atom in the liquid, D is the seif-diffusion coefficient, v the kinematic viscosity and n the number density. The physical reason for the slow loss of correlation is that the atom’s velocity is coupled to the transverse components of the fluctuations in the velocity field, setting up shear wave excitations. The transfer of momentum by this means, at sufficiently long times, is essentially a diffusion process. It is easy to show, at least qualitatively, that this leads to the t~312behaviour of the correlation function. This contrasts with the contribution to momentum transfer via -
sound waves, which produce an exponential decay of velocity correlation. The so-called long-time tail was discovered in molecular dynamics studies of a rigid-sphere fluid [2]. The frequency spectrum of a real liquid may be obtained from the incoherent contribution to the neutron inelastic scattering cross section, in the zero wavenumber limit. A study of the low frequency behaviour could, in principle, confirm the effect. Caneiro recently analyzed some neutron scattering data on liquid argon [3], with this in mind. The results show a low frequency dip, with a minimum at approximately 0.08 X 1013 s_I. It has to be remembered, however, that the smallest wavenumber,q, available in the experiments was I A1. It is far from clear that the data can be extrapolated accurately to q = 0. A recent investigation of the low frequency has behaviour, for models of liquid argon and rubidium, been made using mode-coupling theory [4]. Estimates of the location of the minimum were somewhat smaller than is suggested by Caneiro’s work. In fact a frequency close to 1011 ~-I was obtamed for both liquids, which, if correct, will mean that its experimental observation will be difficult. The visco-elastic character of the liquid was suggested as the major reason for this suppression of the onset of the long-time tail. The purpose of this note is to report some work on the same problem using a velocity field approach. It has been shown to give a very good representation of the gross characteristic features of the computed frequency spectrum in liquid rubidium 395
Volume 80A, number 5,6
PHYSICS LETTERS
[5],but the time scale was not sufficiently long to even see the anomalous low frequency behaviour. Here, we carry through a calculation of the frequency spec312tail carefully intrum with the effect of the t~ cluded. We have repeated the calculation for argon, and also for a rigid-sphere fluid, where the viscoelastic properties will be somewhat different. The derivation of the expression for Li(t) is thoroughly discussed in the latter reference. Within the ensemble average the velocity field,u(r, t), is represented by -
.
-
-
-
v(r,
t)
22 December 1980
M(q, t) [5]. In other words the memory function has an exponential time dependence, with a wavenumberdependent relaxation time r(q). The Fourier—Laplace transform of C~(q,t) is -
‘~-.
—
C~(q,z)mkT[—iz +M(q,z)] and 2
~
M(q z)~’qG(q)(nm)
—1
-
[1z+T
—1
(q)j
—1
G(q) represents the wavenuniber-dependent rigidity, and r(q) is such that r(q -÷ 0) = nmv/G(0), so that the hydrodynamic form is recovered in the small q and
~r~u 1(t)f(Ir
—
r,(t)I),
where u1(t) is an atomic velocity and the function f(r) is chosen so that the velocity field is essentially constantinacross an atomic diameter. order be correct the hydrodynamic limit it In must alsotosatisfy the condition ff(r)dr
n~.
(4)
large t limits. In fig. 1 we display the low frequency shape of the spectrum for a model of liquid rubidium at a temperature T 319 3Kand 0.01058 ~T(°-’)number is that density part of nit which, through X 1024 eq. (5), cm~ is a consequence of C~(q,t). The hydrodynamic contribution to ~T(’~)~ shown in the figure, refers to a result in which C~(q,t) is replaced by C~(q,1), thus neglecting the visco-elastic behaviour -
The result for the velocity autocorrelation function is ~J(t)3k~.
/
-_~fdq~q)[cL(q,t) t0~0.
—
+ 2C~(q,t)]F
5(q, -
-
(5)
t),
withf(q) and F~(q,t) the Fourier transforms off(r) and the self-correlation function, respectively. C~(q,t) and C~(q,t) represent the longitudinal and transverse momentum current density correlation functions respectively. In accordance with the earlier statements, we stress that it is the transverse current correlation function which is crucial in a discussion of the asymptotic behaviour. The result in eq. (1) is obtained from the small wavenumber contribution to the integral in eq. (5). This means that C~(q,t), F5(q, t), andf(q) may be replaced by their hydrodynamic forms, namely
Hydr-odynamio contn~but ion
—
.
—.-.
w
-
to
/ 0.36
/
0.35
2
~(q,t)~-mkTexp(—q yin), 2Djti), f(q)~n1. F~(q,t)~exp(~ To look at a wider range we have used an expression for C~(q,t)which is derived from a simple viscoelastic model for the associated memory function 396
~
~
~ 0
Fig. 1. Frequency spectrum, ~(c~), of liquid rubidium at T 319 K and number density n = 0.01058 X 10~cm3.
Volume 80A, number 5,6
PHYSICS LETTERS
of the liquid. Note, too, that ~~w) has been normalized so that it coincides with i~iT(W)at ci., = 0. The results in fig. 1 clearly show that the inclusion of viscoelastic effects produces the rapid frequency variation and the dip in the spectrum. The longitudinal current contribution does alter the location and depth to some extent. It is also responsible for a slight “kink” in the spectrum at w ~ 0.035 X 1013 ~1, although the reason for it remains unclear. Bosse et al. [4] quantify the relative size of the cusp by defining a parameter a, as U
=
(~(O)-_
22 December 1980
~P(a)) s l~0.42
—
Htdr-od>’nam tO
‘C
contr but
‘°~
~P
1c~)s
0.38
‘min))1~~(~).
1, and w Our result for wmth is 0024 X 1013 s 0 4.31 X 1013 s_i, which are not significantly different from those given in ref. [41.Both methods give 0.04. A similar calculation has been carried through for argon close to the triple point. The transverse current is again based on the visco-elastic approximation. More details of both current models are given by Gaskell and Miller [61.Fig. 2 contains the for Tof 3.Tresults he location = 86K and n = 0.0215 X 1024 cm~ the minimum at w 0.045 X 1013 ~I is substantially bigger than the value derived from mode-coupling theory, and more in line with that obtained from Caneiro’s analysis of the neutron scattering data. Our result for a is 0.096, much smaller than the 0.34 suggested by the scattering experiments, but considerably bigger than that obtained in ref. [4]. For this liquid the frequency w 0 = 1.91 X 1013 s_i. The longitudinal current plays a more significant role in the formation of the dip, than is the case in liquid rubidium. It should be mentioned that computer simulation studies of the transverse current in argon have shown that the association memory function is more appropriately described by two exponentials, one of which represents a long-range component [7]. Unfortunately, we do not have sufficiently detailed informaabout to theuse wavenumber dependence of the relaxation times this more sophisticated model. The work of Bosse et al. [4] suggests that it would tend to reduce the frequency minimum, Finally we consider the rigid-sphere fluid. The transverse current is again constructed from the memory function approach, but in this case only part of the memory is assumed to have an exponential decay. There is in addition a delta function, which has to be
0.34
\
0.30—_-----,_----—---,-——
0.15
0.20
0.25X10u
als
Fig. 2. Frequency spectrum, ~ of liquid argon at T = 86K = 0.0215 X 10~’cm3.
and number density n
included because of the instantaneous character of the rigid sphere collisions [8]. Although this reflects a different visco-elastic character than either argon or rubidium it has nevertheless been demonstrated, in a computer study, that the rigid-spheres can support a propagating shear wave mode at the packing fraction (0.4628) used here [9]. The transverse current model we have derived incorporates this effect, and displays the physically correct features as the wavenumber is varied [8]. The results are given in fig. 3. The frequency unit is the Enskog collision frequency, which is m 2(irkT/m)1”2. In this equation a and = 4ng(a)a are the sphere’s diameter and mass respectively, and g(a) is the contact value of the radial distribution function. The most striking feature is the very different behaviour shown by ~/T(W) when compared to figs. 1 and 2. Presumably the formation of the shoulder, rather than the pronounced minimum shown in rubidium or argon, is a reflection of the less elastic nature of the fluid. In this case the contribution from 397
Volume 80A, number 5,6
PHYSICS LETTERS
22 December 1980
ly closer to the peak in the spectrum than it is in rubidium or argon. This is consistent with the fact that the long-time tail is easier to observe in molecular
0.5, o
dynamics _
“experiments” on a rigid 3rF), The is the Enskog result, 111(t) sphere-fluid. = exp( ~2t/ shown for comparison. The high frequency wing of the
spectrum of
spectrum quickly assumes the Enskog form, confirming that binary collisions are satisfactorily included in our
—
Ens ko
velocity field method.
9
show that the visco-elastic character of a dense liquid does tend to suppress the effect of the long-time tail. The more marked the visco-elasticity, the smaller the frequency at which the dip in the spectrum may be
0.2
0 1
~
H
/
—
seen. the The observation our result liquid of for the models rubidium long-time we have iswill tail, typical, considered by extremely means it means clearly of that cult neutron inIfathree scattering liquid metal. experiments, be diffi-
“..
References
-~
~/ 0.0 _~-~_—-i—-—--—--~——----—--———,—-———---—-i 0.0 0.5 1.0 1.5
2.0
Fig. 3. Solid line shows the frequency spectrum, ~(w)’ of a rigid-sphere fluid, packing fraction 0.4628. The open circles represent the spectrum of the Enskog result ~ (t) = exp(— 2t/ 3r~).The frequency unit is the Enskog collision frequency the spectrum is in TE.
the longitudinal current is responsible for the formation of the dip in the spectrum. The value of a is 0.098, but the formation of the minimum is significant-
398
[i] T. Gaskell and N.H. March, Phys. Lett. 33A (1970) 460. 121 B.J. Alder and T. Wainwright, Phys. Rev. Lett. 18 (1967) 988. [3] K. Caneiro, Phys. Rev. A14 (1976) 517. 141 J. Bosse, W. Gotze and M. Lücke, Phys. Rev. A20 (1979) 1603. 151 T. Gaskell and S. Miller, J. Phys. Cli (1978) 3749. 161 T. Gaskell and S. Miller, J. Phys. Cli (1978) 4839. 171 D. Levesque, L. Verlet and J. Kurkijarvi, Phys. Rev. A7 (1973) 1690. [8] P.E. Mason and T. Gaskell, Mol. Phys., to be published. 19] B.J. Alder, W.E. Alley and S. Yip, to be published.