- Email: [email protected]
Dynamics of binary liquids near the glass transition: a mode-coupling theory
cQ% .-Klg
ELSEVIER
Journal of Non-Crystalline Solids 205-207 (1996) 472-475
Dynamics of binary liquids near the glass transition: a mode-coupling theory Y. Kaneko a1* , J. Bose b ’ Department ofApplied Mathematics and Physics, Faculty of Engineering, Kyoto Uniuersity, Kyoto 606, Japan b Instituffiir Theoretische Physik, Freie Ukxrsitiit Berlin, Arnimnllee 14, I4195 Berlin, Germmsy
Abstract The dynamicsof a disparate-size binary liquid nearthe glasstransitionareinvestigatedwithin a mode-coupling theory. The density-relaxation functionsandthe susceptibilityspectraof the largeparticlesare similarto thoseof a one-component system, the latter consisting of a large a-peak, a P-peak with low intensity and a microscopic peak. In the susceptibility
spectraof the smallparticles,a P-peakwhichis higherin its maximumthan the a-peakwasfound. The appearance of the large P-peakreflectsthe motionof the smallparticlesin an almoststatic randompotentialproducedby the largeparticles.
1. Introduction Ever since the work of Leutheufier [l] and Bengtzelius et al. [2], the glass transition of simple liquids has been extensively studied within a modecoupling theory. Many predictions have been made by Gijtze and co-workers [3] concerning the relaxation dynamics in simple supercooledliquids based on a one-componenttheory. Recently, the theory has been extended to two-component systems, which enabled us to study the localization-delocalization transition of small particles in disordered media within the same framework of the mode-coupling theory [4-61. In a previous paper [7] we studied the diffusion of small particles in a disparate-sizebinary hard-spheremixture with the aim of looking at the phenomenonof delocalization of small particles in a
glassy matrix from a dynamical point of view. Here we extend this analysis by presenting the cohewzt density-relaxation for the samemixture. In the following, we investigate the density-relaxation functions and the generalizedsusceptibility with a special attention to the difference in the relaxation dynamics of the small and large particles in the supercooled state.
2. Theory A brief summary of our theory is as follows. The temporal evolution of the density-relaxation functions of a two-component liquid (s, s’ = 1, 2) %(%
4
= & B
’ Corresponding author. Tel.: +81-75 753 5515; fax: +81-75 761 2437; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO22-3093(96)00262-l
,$ z-1
$ (exp[ -iq.
(+I(
f) -+“(O))])
j-1
(1)
Y. Kaneko,
is described tion &(q,
Journal
by the matrix oscillator-equation
t) +,n2(q)@(q,
c&q,
J. Bone/
t) +i’dt’K(q,
of Non-Crystalline
of mo-
t-t’)
t’) =o,
(2) where 0’(q) is a microscopic frequency matrix. The matrix of the relaxation kernel reads in modecoupling approximation [5-71: K,,(q, qy(
f> =W(q,
f> +G%>
0,
(3)
q> t)
x~:,~(kr-kl,
t) +k:(q-k&(k)
Xc,~,~(I~-~l)@,,,:(~~
~)@sl’(14-kl,
(I}>
(4)
with kr = k 1 q/q and C,,,(q) is the OrnsteinZemike direct correlation function. The regular part Kreg corresponds to the decay on the microscopic time scale defined by a2(q). The slow relaxation processes near the glass transition originate from the feed-back contribution KMC to the relaxation kernel. Eq. (2) is formally solved by the Fourier-Laplace transformation as @(q,
z) = -{zI-(zz+K(q, xqq,
z))-1i22(q)}-1
Solids 205-207
concentration of small particles. We solved the set of closed Eqs. (3)-(5) numerically. For the hard-sphere mixture the static structure factor, which serves as an input in solving Eqs. (5) with (3), is given by the solution of the Percus-Yevick equations [8]. It was found that the following three phases appear when 6 becomes as small as 0.2 [5,7]. (i) For r] < 0.52 (3 in), all particles are delocalized (a liquid phase) (ii) For rlB < q < 0.53 (= Q), the large particles are localized while the small particles retain a finite mobility (a delocalized phase) (iii) For r] > r]rl, both particle species are localized (a glassy phase). Note that the transition at 77= Q is the same as the ergodic-non-ergodic transition found for a one-component system (type B transition), while the transition at q = qA is the delocalization transition, a type A transition [3] (see fig. 1 in Ref. [7]). In the following we examine the density relaxation of the mixture in the liquid phase (‘I < r]n) assuming 6 = 0.2 and c1 = N,/N= 0.5.
3. Results Fig. 1 shows the wavenumber-dependence normalized function h
(5) where I denotes a 2 X 2 unit matrix. The closed set of non-linear Eqs. (5) and (3) can be solved iteratively with the initial conditions
of the
@3?,44, 0
f,A47
t=O),
473
119961 472-47.5
t)=
t=O)a$,(q,
[cp,,(q,
(*)
t=0)]“2
at tmO = 10’ for r = 0.516 and 0.518, where w;’ = is the unit of time in our numeri-
[k,T/(m, qv’2 1 0.9
Q&(4>
l=O)
=~(n,~J”‘s,,(q)
0.8
(6)
-B
and &(q,
t=O)
=o,
(7) is the partial structure
where n, = N,/V and S,,(q) factor. The above theory has been applied to a disparatesize binary hard-sphere mixture to examine the glass transition as well as the localization-delocalization transition within a glassy matrix [4,5,7]. This mixture is characterized by three parameters, that is, the total packing fraction r] = (9~/6>n,~:(l + c,S3/(1 c,)), the size ratio 6 = o,/g2 and cr = N,/N, the
=2
0.7
"0
0.6
il
0.5
2
0.4
0.3 0.2 0.1 0 2
4
6
8
10
12
14
16
18
20
q-2 Fig. 1. Wavenumber dependence of density-relaxation functions for a 1:l mixture of small (A-= 1) and large (s= 2) particles (diameter ratio 15). Full lines: x&q, f) at two = lo3 for supercooled liquid states (q = 0.516, 0.518). Dashed lines: DWF fsS(q) for glassy state (?I = 0.54).
474
Y. Kaneko,
J. Bosse / Journal
of Non-Crystalline
Solicls 205-207
(19963 472-475
cal calculations. For comparison, we plot for 7 = 0.54 > rlB the long time limit (non-ergodicity parameter) fd(4)
= ;;y%
t>3
(9)
which represents the Debye-Waller factor (DWF) of the glassy solid (dashed lines). We note that the q-dependence of As(q, t) in the supercooled-liquid state is - for sufficiently large t - quite similar to that of the DWF. Also, fll(q, t) decays more rapidly than &(q, t) as q increases. The main peaks of x1(4, t) and & at qa, G 7.05 are inherited from the first peak of &(q), implying that the spatial correlations of the small particles are strongly influenced by the static structure of the large particles. In Fig. 2 we plot f;l(q, t) as a function of time at qa2 = 7.05 for the same values of q as in Fig. 1. Three time regimes are observed: the short time decay (tw, < 1) arising from the microscopic vibrations (which are overdamped in the presented model), the P-relaxation at two = l-lo3 and the slow cr-relaxation at to, > lo”, which will disappear in a glassy state. In the a-relaxation regime both &(q, t) and &(q, t) are presented by the Kohlrausch law A exp[ - (t/7) @] quite well as expected from a onecomponent theory. The exponents are j3 = 0.626 and 0.841 for the small and large particles, respectively. The exponent for the large particles is close to the value found for a one-component Lennard-Jones system ( /3 = 0.88 at the position of the maximum of S(q) [9]>. It is interesting to note that for the small particles p is smaller than 0.88. This is due to the
1 0.8
~%li? w/w0
Fig.
3. The susceptibility fss(q, t) in Fig. 2.
spectra
,&(q,
w) corresponding
to
low plateau value of f^1[(q, t) in the p-relaxation regime, which originates from the rapid decrease of fil(q) as a function of q shown in Fig. 1. Fig. 3 shows the generalized susceptibility x;,Jq,
@> = @.f;&,
w),
W) where j$(q, w), is the imaginary part of the Fourier-Laplace transform of &(q, t)/2. In ,&(q, w), we observe a large a-peak at 10m7 < w/w0 < 1o-5 and a small microscopic peak at w/w0 N 10. For v = 0.518 we find a P-peak of low intensity at w/w0 N 10m2, which merges with the high-frequency wing of the a-peak for T= 0.516. Such a feature of &(q, w) is similar to that found for a one-component system. The susceptibility of the small particles ,y;,(q, w) consists of a small peak at w/w0 N 10V6 and a large peak at w/w0 slightly less than unity. The former is an a-peak which appears at the same frequency as that in x&(q, w). The large peak at w/,wO N 1 corresponds to the p-relaxation observed in fil(q, t) at to, = l103. The high-frequency shoulder of this peak is the remainder of the microscopic peak. Note that the intensity of the P-peak is much higher than that of the a-peak. The appearance of such a large P-peak suggests that the relaxation of the small particles is different from what has been argued for a simple one-component liquid.
4. Conclusion -2
0
2
4
6
8
log,,, two
Fig. 2. The density relaxation
functions
fSS(q,
t) at
qrz
= 7.05.
In this paper we have investigated the dynamics of a disparate-size binary liquid nea.r the glass transi-
Y. Kaneko,
J. Bosse / Journal
of Non-Crystalline
tion within the mode-coupling theory. The susceptibility spectrum of the large particles xz2(q, w) consists of a low-frequency a-peak, a P-peak and a high-frequency microscopic peak, as expected from a one-component theory. The intensity of the a-peak is much higher than that of the P-peak. For the small particles, however, the ratio of the peak-heights is reversed. We observed a large P-peak in xyl(q, w) with a high-frequency shoulder arising from a microscopic decay. As reported in our previous paper [7], the diffusion constant of the small particles near the transition at 7,~~ is about lo5 times larger than that of the large particles. This means that the large particles are almost frozen, producing a random potential which the small particles will experience when moving through the matrix. The P-peak of extremely high-intensity observed in x;r(q, w) is considered to arise from the relaxation of the small particles witlzin the almost-static random potential (or a ‘cage’ potential) of the large particles. On the other hand, the behavior of the large particles is almost the same as that in one-component liquids, and the density relaxation of the large particles gives rise to the relaxation of the
Solids 205-207
(1996)
472-475
475
random potential. Therefore the large or-peak observed in xi2(q, o) can be identified as the decay process of the random (cage) potential.
Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 337.
References [I] E. LeutheuBer, Phys. Rev. A29 (1984) 2765. [2] U. Bengtzelius, W. G&e and A. Sjijlander, J. Phys. Al7 (1984) 5915. [3] W. Giitze, in: Liquids, Freezing and the Glass Transition, ed. D. Levesque, J.P. Hansen and J. Zinn-Justin (North Holland, Amsterdam, 1990) p. 287. [4] J. Bosse and J.S. Thakur, Phys. Rev. Lett. 59 (1987) 998. [S] J.S. Thakur and J. Bosse, Phys. Rev. A43 (1991) 4378, 4388. [6] J. Bosse and M. Henel, Ber. Bunsenges. Phys. Chem. 95 (1991) 1007. [7] J. Bosse and Y. Kaneko, Phys. Rev. Lett. 74 (1995) 4023. [8] J.L. Lebowitz, Phys. Rev. Al33 (1964) 895. [9] U. Bengtzelius, Phys. Rev. A34 (1986) 5059.
ELSEVIER
Journal of Non-Crystalline Solids 205-207 (1996) 472-475
Dynamics of binary liquids near the glass transition: a mode-coupling theory Y. Kaneko a1* , J. Bose b ’ Department ofApplied Mathematics and Physics, Faculty of Engineering, Kyoto Uniuersity, Kyoto 606, Japan b Instituffiir Theoretische Physik, Freie Ukxrsitiit Berlin, Arnimnllee 14, I4195 Berlin, Germmsy
Abstract The dynamicsof a disparate-size binary liquid nearthe glasstransitionareinvestigatedwithin a mode-coupling theory. The density-relaxation functionsandthe susceptibilityspectraof the largeparticlesare similarto thoseof a one-component system, the latter consisting of a large a-peak, a P-peak with low intensity and a microscopic peak. In the susceptibility
spectraof the smallparticles,a P-peakwhichis higherin its maximumthan the a-peakwasfound. The appearance of the large P-peakreflectsthe motionof the smallparticlesin an almoststatic randompotentialproducedby the largeparticles.
1. Introduction Ever since the work of Leutheufier [l] and Bengtzelius et al. [2], the glass transition of simple liquids has been extensively studied within a modecoupling theory. Many predictions have been made by Gijtze and co-workers [3] concerning the relaxation dynamics in simple supercooledliquids based on a one-componenttheory. Recently, the theory has been extended to two-component systems, which enabled us to study the localization-delocalization transition of small particles in disordered media within the same framework of the mode-coupling theory [4-61. In a previous paper [7] we studied the diffusion of small particles in a disparate-sizebinary hard-spheremixture with the aim of looking at the phenomenonof delocalization of small particles in a
glassy matrix from a dynamical point of view. Here we extend this analysis by presenting the cohewzt density-relaxation for the samemixture. In the following, we investigate the density-relaxation functions and the generalizedsusceptibility with a special attention to the difference in the relaxation dynamics of the small and large particles in the supercooled state.
2. Theory A brief summary of our theory is as follows. The temporal evolution of the density-relaxation functions of a two-component liquid (s, s’ = 1, 2) %(%
4
= & B
’ Corresponding author. Tel.: +81-75 753 5515; fax: +81-75 761 2437; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SOO22-3093(96)00262-l
,$ z-1
$ (exp[ -iq.
(+I(
f) -+“(O))])
j-1
(1)
Y. Kaneko,
is described tion &(q,
Journal
by the matrix oscillator-equation
t) +,n2(q)@(q,
c&q,
J. Bone/
t) +i’dt’K(q,
of Non-Crystalline
of mo-
t-t’)
t’) =o,
(2) where 0’(q) is a microscopic frequency matrix. The matrix of the relaxation kernel reads in modecoupling approximation [5-71: K,,(q, qy(
f> =W(q,
f> +G%>
0,
(3)
q> t)
x~:,~(kr-kl,
t) +k:(q-k&(k)
Xc,~,~(I~-~l)@,,,:(~~
~)@sl’(14-kl,
(I}>
(4)
with kr = k 1 q/q and C,,,(q) is the OrnsteinZemike direct correlation function. The regular part Kreg corresponds to the decay on the microscopic time scale defined by a2(q). The slow relaxation processes near the glass transition originate from the feed-back contribution KMC to the relaxation kernel. Eq. (2) is formally solved by the Fourier-Laplace transformation as @(q,
z) = -{zI-(zz+K(q, xqq,
z))-1i22(q)}-1
Solids 205-207
concentration of small particles. We solved the set of closed Eqs. (3)-(5) numerically. For the hard-sphere mixture the static structure factor, which serves as an input in solving Eqs. (5) with (3), is given by the solution of the Percus-Yevick equations [8]. It was found that the following three phases appear when 6 becomes as small as 0.2 [5,7]. (i) For r] < 0.52 (3 in), all particles are delocalized (a liquid phase) (ii) For rlB < q < 0.53 (= Q), the large particles are localized while the small particles retain a finite mobility (a delocalized phase) (iii) For r] > r]rl, both particle species are localized (a glassy phase). Note that the transition at 77= Q is the same as the ergodic-non-ergodic transition found for a one-component system (type B transition), while the transition at q = qA is the delocalization transition, a type A transition [3] (see fig. 1 in Ref. [7]). In the following we examine the density relaxation of the mixture in the liquid phase (‘I < r]n) assuming 6 = 0.2 and c1 = N,/N= 0.5.
3. Results Fig. 1 shows the wavenumber-dependence normalized function h
(5) where I denotes a 2 X 2 unit matrix. The closed set of non-linear Eqs. (5) and (3) can be solved iteratively with the initial conditions
of the
@3?,44, 0
f,A47
t=O),
473
119961 472-47.5
t)=
t=O)a$,(q,
[cp,,(q,
(*)
t=0)]“2
at tmO = 10’ for r = 0.516 and 0.518, where w;’ = is the unit of time in our numeri-
[k,T/(m, qv’2 1 0.9
Q&(4>
l=O)
=~(n,~J”‘s,,(q)
0.8
(6)
-B
and &(q,
t=O)
=o,
(7) is the partial structure
where n, = N,/V and S,,(q) factor. The above theory has been applied to a disparatesize binary hard-sphere mixture to examine the glass transition as well as the localization-delocalization transition within a glassy matrix [4,5,7]. This mixture is characterized by three parameters, that is, the total packing fraction r] = (9~/6>n,~:(l + c,S3/(1 c,)), the size ratio 6 = o,/g2 and cr = N,/N, the
=2
0.7
"0
0.6
il
0.5
2
0.4
0.3 0.2 0.1 0 2
4
6
8
10
12
14
16
18
20
q-2 Fig. 1. Wavenumber dependence of density-relaxation functions for a 1:l mixture of small (A-= 1) and large (s= 2) particles (diameter ratio 15). Full lines: x&q, f) at two = lo3 for supercooled liquid states (q = 0.516, 0.518). Dashed lines: DWF fsS(q) for glassy state (?I = 0.54).
474
Y. Kaneko,
J. Bosse / Journal
of Non-Crystalline
Solicls 205-207
(19963 472-475
cal calculations. For comparison, we plot for 7 = 0.54 > rlB the long time limit (non-ergodicity parameter) fd(4)
= ;;y%
t>3
(9)
which represents the Debye-Waller factor (DWF) of the glassy solid (dashed lines). We note that the q-dependence of As(q, t) in the supercooled-liquid state is - for sufficiently large t - quite similar to that of the DWF. Also, fll(q, t) decays more rapidly than &(q, t) as q increases. The main peaks of x1(4, t) and &
1 0.8
~%li? w/w0
Fig.
3. The susceptibility fss(q, t) in Fig. 2.
spectra
,&(q,
w) corresponding
to
low plateau value of f^1[(q, t) in the p-relaxation regime, which originates from the rapid decrease of fil(q) as a function of q shown in Fig. 1. Fig. 3 shows the generalized susceptibility x;,Jq,
@> = @.f;&,
w),
W) where j$(q, w), is the imaginary part of the Fourier-Laplace transform of &(q, t)/2. In ,&(q, w), we observe a large a-peak at 10m7 < w/w0 < 1o-5 and a small microscopic peak at w/w0 N 10. For v = 0.518 we find a P-peak of low intensity at w/w0 N 10m2, which merges with the high-frequency wing of the a-peak for T= 0.516. Such a feature of &(q, w) is similar to that found for a one-component system. The susceptibility of the small particles ,y;,(q, w) consists of a small peak at w/w0 N 10V6 and a large peak at w/w0 slightly less than unity. The former is an a-peak which appears at the same frequency as that in x&(q, w). The large peak at w/,wO N 1 corresponds to the p-relaxation observed in fil(q, t) at to, = l103. The high-frequency shoulder of this peak is the remainder of the microscopic peak. Note that the intensity of the P-peak is much higher than that of the a-peak. The appearance of such a large P-peak suggests that the relaxation of the small particles is different from what has been argued for a simple one-component liquid.
4. Conclusion -2
0
2
4
6
8
log,,, two
Fig. 2. The density relaxation
functions
fSS(q,
t) at
qrz
= 7.05.
In this paper we have investigated the dynamics of a disparate-size binary liquid nea.r the glass transi-
Y. Kaneko,
J. Bosse / Journal
of Non-Crystalline
tion within the mode-coupling theory. The susceptibility spectrum of the large particles xz2(q, w) consists of a low-frequency a-peak, a P-peak and a high-frequency microscopic peak, as expected from a one-component theory. The intensity of the a-peak is much higher than that of the P-peak. For the small particles, however, the ratio of the peak-heights is reversed. We observed a large P-peak in xyl(q, w) with a high-frequency shoulder arising from a microscopic decay. As reported in our previous paper [7], the diffusion constant of the small particles near the transition at 7,~~ is about lo5 times larger than that of the large particles. This means that the large particles are almost frozen, producing a random potential which the small particles will experience when moving through the matrix. The P-peak of extremely high-intensity observed in x;r(q, w) is considered to arise from the relaxation of the small particles witlzin the almost-static random potential (or a ‘cage’ potential) of the large particles. On the other hand, the behavior of the large particles is almost the same as that in one-component liquids, and the density relaxation of the large particles gives rise to the relaxation of the
Solids 205-207
(1996)
472-475
475
random potential. Therefore the large or-peak observed in xi2(q, o) can be identified as the decay process of the random (cage) potential.
Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 337.
References [I] E. LeutheuBer, Phys. Rev. A29 (1984) 2765. [2] U. Bengtzelius, W. G&e and A. Sjijlander, J. Phys. Al7 (1984) 5915. [3] W. Giitze, in: Liquids, Freezing and the Glass Transition, ed. D. Levesque, J.P. Hansen and J. Zinn-Justin (North Holland, Amsterdam, 1990) p. 287. [4] J. Bosse and J.S. Thakur, Phys. Rev. Lett. 59 (1987) 998. [S] J.S. Thakur and J. Bosse, Phys. Rev. A43 (1991) 4378, 4388. [6] J. Bosse and M. Henel, Ber. Bunsenges. Phys. Chem. 95 (1991) 1007. [7] J. Bosse and Y. Kaneko, Phys. Rev. Lett. 74 (1995) 4023. [8] J.L. Lebowitz, Phys. Rev. Al33 (1964) 895. [9] U. Bengtzelius, Phys. Rev. A34 (1986) 5059.