Chapter 4
Basic Physics of the Coupling Model: Direct Experimental Evidences Downloaded by STANFORD UNIV GREEN LIBR on October 5, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch004
K. L . Ngai and R. W. Rendell Naval Research Laboratory, Code 6807, 4555 Overlook Avenue, SW, Washington, DC 20375-5320
The fundamental results of the Coupling Model, i.e. crossover from independent relaxation at short-times to slowed down cooperative relaxation at a temperature independent time t , are represented by three equations. They are shown to be in accord with experimental data in molecular glass formers, glass-forming, glassy or crystalline ionic conductors, and concentrated colloidal suspensions. Emphasis is given in this work to the colloidal suspensions which have the advantage of the absence of vibrational contributions and the experimentally measured intermediate scattering function is attributed entirely to diffusion dynamics. Direct evidences of such crossover are demonstrated. These, together with the many successful applications of the predictions to several fields and many materials have led us to believe that the Coupling Model has captured the basic physics of relaxation in materials with many-body interactions. We mention in passing that the theoretical basis of the Coupling Model is closely related to chaos in Hamiltonian dynamical systems. c
The coupling model (CM) (1-3) is a general approach to dynamics of constrained or interacting systems, that has been shown to be applicable in depth to many problems of relaxation in different materials (4-6). Interaction between relaxing units implies cooperativity between them and vice versa. Thus, the effect of the many-body interactions on relaxation can be rephrased as cooperativity in the context of the C M . Several approaches to this problem have been proposed. Recent versions of the coupling theory are This chapter not subject to copyright. Published 1997 American Chemical Society
In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
45
46
SUPERCOOLED LIQUIDS
based on classical mechanics for systems that exhibit chaos caused by the anharmonic (rK>nlinear) nature of the interactions between the basic molecular units (2,3,7). For example, (i) intermolecular interaction between monomer units in polymers and small molecules in a glass-forming van der Waals liquid modeled by the Lennard-Jones potential; (ii) the entanglement interaction between polymer chains; (Hi) Coulomb interaction between ions in glass-forming electrolytes including the much studied 0.4Ca(NO3>0.6KNO (CKN); and (iv) hard-sphere like interaction between colloidal particles (8-10) are all highly anharmonic and they give rise to chaotic dynamics (6). Although a rigorous theory based on first principles is still at large, there are theoretical results supporting the basic elements of the C M (1-3). The common misconception that the C M is entirety phenomenological is not accurate. Future development in theory may best be in the hands of experts in the dynamics of nonlinear Hamiltonian systems (7) and not necessary by us who have no formal training in this field. Whatever the future course of development of the theory, we are quite confident that the results of the C M are essentially correct. This confidence is gained by: (a) the results being borne out by simple models (2,3) and the fact that manifestation of the effect of chaos is usually general, so that behavior found in simple systems carries over to more complicated ones; (b) the many successes of the C M in its applications to a variety of challenging problems in several fields (4-6) which represent the bulk of the effort of our group of coworkers, and (c) the direct verification of the fundamental physics of the C M by recent exr^riments. The latter include quasielastic neutron scattering exr^riments in several polymers including polyvinylchloride (11), polyisoprene and polybutadiene (12); very high frequency dielectric measurements for glassy and non-glassy ionic conductors (13,14) and glass-forming molten salts (14,15). D.c. conductivity measurements of glassy and non-glassy ionic conductors up to high temperatures where the conductivity relaxation time is of the order of a picosecond or less (16-20); and molecular dynamics simulations of small molecule liquids (21). The fundamental results of the coupling theory, which have remained unchanged since its inception fifteen years ago (1), are restated here as follows. There exists a temperature insensitive cross-over time, t before which (tt ) with a slowed-down nonexponential correlation function. A particularly convenient function which is compatible with both conputer simulations and experimental data for coupled systems is c
1
•(0 = a p H - xV ) - ]
(2)
where ft is the coupling parameter whose value lies within the range 0Tg=335 K. C K N is definitely anharmonic and it is moot to check the C M using the harmonic phonon approximation. Since the criticism of the harmonic approximation comes mainly from supporters of the Mode Coupling Theory (MCT) (39), it is fair to examine the impact of the vibrational contribution on the interpretation of neutron and light scattering data by MCT. In the earlier comparison of M C T with incoherent neutron scattering data by Kiebel et al and coherent neutron scattering data by Bartsch et al (40), the harmonic phonon hypothesis was used in the procedure to isolate out 7^(Q,r). Nevertheless, good agreements of IidQf) with the M C T predictions were found (33,40). If the harmonic phonon hypothesis were not valid, then these good fits to illegitimately deduced relaxation data by MCT are fortuitous and suggests that the good fit to the susceptibility minimum data is neither necessary nor sufficient to conclude that MCT correctly captures the physics of the short-time dynamics. Most comparisons of experimental data with M C T were made without removing the vibrational contribution from the data. The fast P-process of the MCT fit was laid near the I(Q t) data which increases with decreasing for the susceptibility %"(©) data which increases with increasing Co. Without knowing the vibrational contribution in this time or frequency c
c
c
c
Q
t
In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
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SUPERCOOLED LIQUIDS
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region, it is hard to evaluate how good the fit really is. Recent inelastic coherent neutron scattering data (41) indicates the importance of the vibrational contribution in the region of the fast process in polybutadiene. Finally, if the comparison of the fast dynamics of the C M with experiment were carried out in the same manner as that involves M C T (Le. leaving the vibrational contribution being uncertain), then good fits to the data by the C M would be already guaranteed (36). Ionic Glass Formers C K N is a good example of this class of ion-containing materials which have the structural relaxation time or the shear relaxation time being nearly the same as the ionic conductivity relaxation time at least at high temperatures where these relaxation times become of the order of picoseconds. This condition ensures that one can study the fast dynamics of structural relaxation by making ionic conductivity relaxation measurements. If the C M holds and there is no contribution to the ax. conductivity a(co) other than the diffusing ions, then it predicts that a(co) consists of three stages (41- 44). At high frequencies, (Q>(*c) *> where ions execute independent motions with correlation function given by equation 1, a(co) is a frequency independent number which is denoted by a . At lower frequencies, co r ( ^ ^ ) r * = f (regime HI). (14) 1-n
= 6Dft
0
i
1-n
elf
2
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Defining the generalized diffusion coefficient by D* (t)= (l/6) t . In these systems, the constraints imposed on the relaxing unit remain permanent and thus the coupling parameter remains unchanged. This is not the case for the concentrated colloidal particles studied here. The diffusive motions of the colloidal particles become impeded and slowed down by its neighbors which form a cage surrounding each particle. This cage is expected to be well-defined until timescales are reached which are on the order of the mobility of the cage particles, which is approximately t = x*. For t > x*, the cage will begin disintegrating as these particle diffuse and the constraints of the cage on the c
0
L
S
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c
3
0
c
2
2
2
s
c
In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
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SUPERCOOLED LIQUIDS
-4.0
-3.5
-3.0
-2.5 -2.0 iogf
-1.5
-1.0
-0.5
Figure 4. Normalized time-dependent diffusion coefficient for colloidal suspensions with volume fraction $ = 0.465. Symbols are data replotted from Figure 2a of Ref.10. Solid (dashed) line is C M fit with (without) constraint mitigation. The parameters used are given in the text.
Figure 5. Scaled scattering function (^Ds^^MfiQt) I (?Ds(Q) for colloidal suspensions with volume fraction $ = 0.465. Symbols are data replotted from Figure 3b of Ref.10. Solid (dashed) line is C M fit with (without) constraint mitigation. The parameters used are given in the text.
In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.
4. NGAI & RENDELL
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Basic Physics of the Coupling Model
enclosed colloidal particle will be softened or mitigated. For situations in which such constraint mitigation occurs, we would expect a constant value of n for t < x*, followed by decreasing values of n for t > x*. This can be incorporated into the C M by introducing a time dependence for n and using this in a relaxation rate formulation of the C M . Note that the C M relations, eqs. 1-3, are solutions of the following timedependent rate equation: d$/dt = -W(t)ty(t)
(16)
W(t)=%l\tr (18) ,
c
c
This has the solution :
$ (0 = e x p ( - r / x cw
0
) , tt
(20)
c
where the subscript 'cm' is used to indicate the inclusion of constraint mitigation. We now parameterize n(t) to reflect constraint mitigation for t > t = x*: t
n(t) = n + (n - n )exp(-(* -1 ) IX „ ) , f
t
f
t>t
{
(21) •
{
where x„ is expected to be of order x*. The tcm of C M with constraint mitigation is described by eq.19 for t t\. Without constraint mitigation n(t) = n,=n/, f -**>, and §cm reduces to the usual C M equations 1-2. From 0^(0 we can calculate , lnf(Q t)/Q D (Q) and D(Q t) / D (Q) by the same procedure discussed before and, wherever applicable, replacing 4K0 hy fy^it) in the equations. To fit the calculated quantities to the experimental data, the parameters 4, Xo and nf have to be consistent with that required by the experimental facts and they can be considered to be predetermined. The two new parameters U and n» are adjusted to improve the fit to D(Q t)f D (Q) in fig.4 for
