Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
Подождите немного. Документ загружается.
13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems 549
Fig. 13.29. Comparison of mea-
sured viscosities and viscosities
calculated from the neutron spin-
echo results for polyethylene
melts at 509 K as a function of
molecular weight (•: experimen-
tal result; ◦ viscosities calculated
on the basis of mode analysis).
It is evident that relaxation modes show deviations from Rouse behaviour
when their extension is greater than the entanglement distance in the long-
chain melt. We thus come to an interpretation of the mode analysis results.
Obviously, topological interactions block, or at least very much reduce, the
relaxation rate for those modes whose characteristic length becomes greater
than the entanglement distance formed in long-chain melts.
We are now going to compare the result of the mode analysis with mea-
surements of the viscosity on polyethylene melts. With the aid of (13.19),
which links the viscosity with the relaxation times, we can predict the vis-
cosity using the results of spin-echo measurements and compare it with the
measurement [54]. This is done in Fig. 13.29, where the viscosity is shown as
a function of molecular weight. The open circles represent the predictions of
the NSE result, whereas the filled circles represent the viscosity measurement.
Both data sets are in excellent agreement and demonstrate the consistency
of evaluation.
Let us draw a fifth conclusion: The Q-dependence of the mode contribu-
tions to the dynamic structure factor opens direct access to the individual
relaxation modes of a chain. This follows from the fact that neutrons permit
motions to be observed on their natural length and time scales. We find that
large-scale modes with characteristic lengths greater than the entanglement
distance or with a mode index p<N/N
e
are significantly slowed down. A
quantitative agreement with viscosity measurements is achieved with the aid
of the extracted relaxation rates.
550 Dieter Richter
13.8 Conclusion
The viscoelasticity of polymer melts is determined by the relaxations of the
individual chains. Their space-time analysis is possible with the aid of neutron
spin-echo spectroscopy. The experiments described in this chapter allow one
to draw the following conclusions:
(i) On the scale of molecular bonds the local relaxations depend on the spe-
cific local potentials. In (1–4) PB the two prevailing relaxation processes,
the α-andβ-relaxation, can be understood as interchain diffusion and
intrachain motion, whereby the β-process relates to rotational motions
of the chain building blocks. Above the merging temperature of the α-
and β-relaxation, both processes persist independently from each other.
(ii) For intermediate time and length scales the segment dynamics of poly-
mer melts follows the entropy-determined dynamics of the Rouse model.
The predictions of this model were confirmed in all facets. Detailed com-
parison with simulation show deviations in detail, which relate to remain-
ing interchain interactions – they are neglected in the Rouse model.
(iii) For long polymer chains the entanglement distance in polymer melts was
directly observed using NSE. Moreover, the NSE experiments led to a
quantitative validation of the dynamic structure factor for tube-confined
reptation. The predictions of all other models put forward so far could
be falsified.
(iv) The length- and time-scale regime in between universal Rouse relaxation
and local relaxation is dominated by intrachain dissipation effects. NSE
experiments on polymers with high (PIB) and low rotational barriers
(PDMS) but similar stiffness unambiguously identified the presence of
intrachain dissipation. In solution its activation energy coincides with
the height of the rotation barrier.
(v) If the chain length in a polymer melt is prolonged beyond the entan-
glement distance, the relaxation modes of this melt are slowed down,
provided that the characteristic length of the mode concerned is larger
than the entanglement arc length. This result was obtained by a mode
analysis of the dynamic structure factor; the microscopic mode relax-
ation rates lead to a quantitative agreement with macroscopic viscosity
measurements.
Notation
A
hop
elastic incoherent structure factor for local hopping
a
n
segment vector
B magnetic flux density
D, D
R
diffusion coefficients
d jump distance
13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems 551
d
t
tube diameter
E,∆E neutron energy, energy transfer
f
hop
quasielastic scattering contribution from two site jump pro-
cesses
G
S
self correlation function
gg-factor of the neutron
h Planck constant
= h/2π
H magnetic field
k, k neutron wave vector
k
B
Boltzmann constant
segment length
L chain contour length
m neutron mass
M polymer molecular weight
M
0
monomer weight
M
e
entanglement molecular weight
N degree of polymerization
N
A
Avogadro constant
N
e
number of monomers between entanglements
Prob probability function for chain conformation
P neutron polarization
Q momentum transfer
R
E
end-to-end vector
r monomer coordinates
S(Q, t), S(Q, ω) dynamic structure factor
S(Q) static structure factor
t time variable
T
m
,T
g
merging, glass transition temperature
W Rouse rate
x
s
,x
p
Rouse normal coordinates
α structural relaxation
β
diel
stretching exponents in the KWW function
β
slow/fast
secondary relaxations
γ magnetogyric ratio
ζ
0
monomeric friction coefficient
η viscosity
λ, δλ neutron wavelength, wavelength change
µ
N
nuclear magneton
ρ density
τ,τ
η
characteristic relaxation time; for viscosity relaxation
φ phase angle of neutron spin
ϕ
α
relaxation function for α relaxation
Ω characteristic relaxation rate
ω energy transfer during scattering
552 Dieter Richter
References
1. J.D. Ferry: Viscoelastic Properties of Polymers (Wiley, New York 1980)
2. P.E. Rouse: J. Chem. Phys. 21, 1273 (1953)
3. P.G. de Gennes: Scaling Concepts in Polymer Physics (Cornell University Press,
New York 1979)
4. D. Richter, B. Frick, B. Farago: Phys. Rev. Lett. 61, 2465 (1988)
5. D. Richter, B. Ewen, B. Farago, T. Wagner: Phys. Rev. Lett. 62, 2640 (1989)
6. B. Frick, B. Farago, D. Richter: Phys. Rev. Lett. 64, 2921 (1990)
7. D. Richter, B. Farago, L.J. Fetters, J.S. Huang, B. Ewen, C. Lartigue: Phys.
Rev. Lett. 64, 1389 (1990)
8.R.Butera,L.J.Fetters,J.S.Huang,D.Richter,W.Pyckhout-Hintzen,A.
Zirkel, B. Farago, B. Ewen: Phys. Rev. Lett. 66, 2088 (1991)
9. D. Richter, R. Zorn, B. Farago, B. Frick, L.J. Fetters: Phys. Rev. Lett. 68,71
(1992)
10. D. Richter, R. Butera, L.J. Fetters, J.S. Huang, B. Farago, B. Ewen: Macro-
molecules 25, 6156 (1992)
11. D. Richter, L. Willner, A. Zirkel, B. Farago, L.J. Fetters, J.S. Huang: Phys.
Rev. Lett. 71, 4158 (1993)
12. D. Richter, B. Farago, R. Butera, L.J. Fetters, J.S. Huang, B. Ewen: Macro-
molecules 26, 795 (1993)
13. D. Richter, L. Willner, A. Zirkel, B. Farago, L.J. Fetters, J.S. Huang: Macro-
molecules 27, 7437 (1994)
14. A. Arbe, U. Buchenau, L. Willner, D. Richter, B. Farago, J. Colmenero: Phys.
Rev. Lett. 76, 1872 (1996)
15. A. Arbe, D. Richter, J. Colmenero, B. Farago: Phys. Rev. E 54, 3853 (1996)
16. P. Schleger, B. Farago, A. Kollmar, C. Lartigue, D. Richter: Phys. Rev. Lett.
81, 124 (1998)
17. W. Paul, G.D. Smith, D.Y. Yoon, B. Farago, S. Rathgeber, A. Zirkel, L. Willner,
D. Richter: Phys. Rev. Lett. 80, 2346 (1998)
18. A.Arbe,J.Colmenero,M.Monkenbusch,D.Richter:Phys.Rev.Lett.81, 590
(1998)
19. A. Heuer, H.W. Spiess: Phys. Rev. Lett. 82, 1335 (1999)
20. A.Arbe,J.Colmenero,M.Monkenbusch,D.Richter:Phys.Rev.Lett82, 1336
(1999)
21. D. Richter, M. Monkenbusch, J. Allgaier, A. Arbe, J. Colmenero, B. Farago,
Y. Cheol, R. Faust: J. Chem. Phys. 111, 6107 (1999)
22. A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal, R.
Faust: Macromolecules 34, 1281 (2001)
23. A. Wischnewski, M. Monkenbusch, L. Willner, D. Richer, A. Likthman, T.
McLeish:Phys.Rev.Lett.88, 058301 (2002)
24. W. Marshall, S.W. Lovesey: Theory of Thermal Neutron Scattering (Clarendon,
Oxford 1971)
25. Neutron Spin Echo, Lecture Notes in Physics, vol 128, ed by F. Mezei (Springer,
Berlin Heidelberg New York 1980)
26. Neutron Spin Echo Spectroscopy: Basics, Trends and Applications,Lecture
Notes in Physics, vol 601, ed by F. Mezei (Springer, Berlin Heidelberg New
York 2003)
13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems 553
27. R. Zorn, F.I. Mopsik, G.B. McKenna, L. Willner, D. Richter: J. Chem. Phys.
107, 3645 (1997)
28. R. Zorn, G.B. McKenna, L. Willner, D. Richter: Macromolecules 28, 8552
(1995)
29. B. Frick, D. Richter, W. Petry, U. Buchenau: Z. Phys. B70, 73 (1988)
30. R. Zorn, A. Arbe, J. Colmenero, B. Frick, D. Richter, U. Buchenau: Phys. Rev.
E 52, 782 (1995)
31. A.Arbe,J.Colmenero,B.Frick,M.Monkenbusch,D.Richter:Macromolecules
31, (1998) 4926
32. G.B. Berry, T.G. Fox: Adv. Polym. Sci. 5, 261 (1968)
33. J. Colmenero, A. Arbe, F. Alvarez, A. Narros, M. Monkenbusch, D. Richter:
Europhys. Lett., in press (2005)
34. G. Vineyard: Phys. Rev. 110, 999 (1959)
35. B. Farago, A. Arbe, J. Colmenero, R. Faust, U. Buchenau, D. Richter: Phys.
Rev. E 65, 051803 (2002)
36. A. Arbe, J. Colmenero, F. Alvarez, M. Monkenbusch, D. Richter, B. Farago,
B. Frick: Phys. Rev. Lett. 89, 245701 (2002)
37. A. Arbe, J. Colmenero, M. Monkenbusch, D. Richter: Phys. Rev. Lett. 82, 1336
(1999)
38. W.P.Slichter:J.Polym.Sci.PartC14, 33 (1966)
39. K. Karatasos, J.-P. Ryckaert: Macromolecules 34, 7323 (2001)
40. P.G. de Gennes: Physics (Long Island City, N.Y.) 3, 37 (1967)
41. A. Wischneswki, M. Monkenbusch, L. Willner, D. Richter, B. Farago, G. Kali:
Phys.Rev.Lett.90, 058302 (2003)
42. G. Smith, W. Paul, M. Monkenbusch, D. Richter: J. Chem. Phys. 261,61
(2000)
43. P.G. de Gennes: J. Chem. Phys. 55, 572 (1971)
44. M. Doi, S.F. Edwards: J. Chem. Soc. Faraday. Trans. 2, 74, 1789 (1978); 74,
1802 (1978); 75, 38 (1978)
45. G.J. Ronca: J. Chem. Phys. 79, 79 (1988)
46. W. Hess: Macromolecules 21, 2620 (1988)
47. P.G. de Gennes: J. Phys. (Paris) 42, 735 (1981)
48. J. des Cloizeaux: Macromolecules 23, 2753 (1990)
49. G. Allegra, F. Ganazzoli Macromolecules 14, 1110 (1981)
50. G. Allegra, F. Ganazzoli: Advances in Chemical Physics 75, 265 (1989)
51.L.Harnau,R.G.Winkler,P.J.Reinicker:J.Chem.Phys.102, 7750 (1995);
104, 6355 (1996)
52. R. Dejean de la Batie, F. Lauprˆetre, L. Monnerie: Macromolecules 22, 2617
(1989); A.A. Jones, R.P. Lubianez, M.A. Hanson, S.L.J. Shostak: Polym. Sci.,
Polym. Phys. Ed. 16, 1685 (1978); Y. Inoue, A. Nishioka, R. Chˆujˆo: J. Polym.
Sci., Polym. Phys. Ed. 11
, 2237 (1973)
53. M. Doi, S.F. Edwards: The Theory of Polymer Dynamics (Clarendon, Oxford
UK 1986)
54. D.S. Pearsson, L.J. Fetters, W.W. Graessley, G. Verstrate, E.V. Merwall:
Macromolecules 27, 711 (1994)
14 The Molecular Description of Mutual
Diffusion Processes in Liquid Mixtures
Hermann Weing¨artner
14.1 Introduction
In a liquid, the constituent particles are in rapid translational motion due
to thermal agitation. The molecular displacements are characterized by the
self-diffusion coefficient D, which in three-dimensional space is related to the
mean square displacement after time t via the Einstein relation [1]
D = lim
t→∞
1
6t
|r
α
(t) − r
α
(0)|
2
. (14.1)
Here r
α
(t) describes the location of a particle numbered α at time t.The
brackets ··· denote the ensemble average. In a liquid mixture of r compo-
nents, self-diffusion coefficients D
a
can be assigned to any of the components
a =1, 2 ...r.
The limiting process in (14.1) implies that the displacements of molecules
are observed over a much longer time than the time scale set up by the un-
derlying molecular motions. This situation is often called normal diffusion.
In liquids of low or moderate viscosity the relevant molecular time scale is
typically of the order of some picoseconds. On the other hand, depending on
the experimental method, conventional diffusion experiments observe diffu-
sion over some milliseconds to hours or days. Clearly, the condition for normal
diffusion is fulfilled, e.g., in these cases. It may not be fulfilled for supercooled
liquids, glasses or polymer systems, where molecular relaxation times reach
the experimental observation time. Such cases of anomalous diffusion, dealt
with in Chaps. 10 and 13, will not be considered in the present chapter.
Following the general trend of looking for a molecular description of the
properties of matter, self-diffusion in liquids has become a key quantity for
interpreting and modelling transport in liquids, because among all transport
coefficients the self-diffusion coefficient is certainly the most simple to inter-
pret [1]. For example, the extraction of self-diffusion coefficients from mole-
cular dynamics simulations of liquids is straightforward by evaluation of the
mean square displacement from the simulated trajectories of the particles [2]
(see also Chap. 23).
If, as an external perturbation, a concentration inhomogeneity is set up
for one of the components, the liquid responds with another diffusion process,
because a concentration gradient of any component results in mass fluxes of
556 Hermann Weing¨artner
all other components. In this case the species diffuse cooperatively, so as to
smear out the concentration inhomogeneity. In work on fluid mixtures the
term mutual diffusion has become common for this process [1], in other fields
the terms interdiffusion, mass diffusion, chemical diffusion or transport diffu-
sion are also in use. The cooperativity of the mass flows distinguishes mutual
diffusion from the stochastic motion of individual molecules reflected by the
self-diffusion process. For example, the mass flows of the two components
in binary systems are coupled, and a single quantity, called mutual-diffusion
coefficient D
12
, is sufficient to characterize the diffusion process, regardless
of which of the mass flows is observed [1].
From the applications’ point of view, mutual diffusion is by far more
important than self-diffusion, and plays a major role in many physical and
chemical processes. Hence, the necessity of predicting mutual-diffusion co-
efficients often arises, and many efforts have been made to understand and
predict mutual-diffusion data [1]. Again, there is much interest in a micro-
scopic description of mutual diffusion processes, which is firmly based on
statistical mechanics and liquid-state theory. In particular, it is desirable to
put the description of mutual diffusion on a basis, which is analogous to that
used for interpreting self-diffusion data. Owing to its cooperative nature, a
theoretical description of mutual diffusion, however, is expected to be more
complex than that of self-diffusion. This chapter aims at discussing the ba-
sic foundations of such a molecular-based description of mutual diffusion in
relation to that of self-diffusion.
Actually, attempts to understand mutual diffusion processes at a micro-
scopic level are very closely related to the understanding of self-diffusion –
mutual-diffusion relationships. Many diffusion models in quite different fields
such as chemistry, metallurgy, chemical engineering or geochemistry give rela-
tions between self-diffusion and mutual-diffusion coefficients. As an example,
we mention the use of self-diffusion coefficients measured by NMR techniques
as input data in modelling multi-component mass transport in industrial
processes [3] or in oil recovery and oil technology [4,5].
From the theoretical point of view, this problem is intriguing, because it
tackles our fundamental understanding of the interrelation between single-
particle and cooperative motions in liquids [6]. This problem is not only
present in translational dynamics, but is also a long-standing problem in the
interpretation of rotational dynamics in liquids, where some experiments such
as nuclear magnetic relaxation monitor single-particle motions, while other
techniques such as dielectric relaxation [7, 8] or depolarized light scattering
[9] monitor cooperative processes. Moreover, the electrical conductance in
molten salts and in electrolyte solutions is a cooperative process as well,
which is closely related to the diffusive motions considered here [6].
Finally, the use of nuclear magnetic resonance spin-echo experiments [10]
provides an elegant and convenient method for monitoring self-diffusion in liq-
uids [11,12]. Furthermore, by applying Fourier transform techniques in NMR
14 Mutual Diffusion in Mixtures 557
spectroscopy, self-diffusion coefficients can be determined simultaneously for
many components [10, 13]. From this experimental perspective, theoretical
relations, which interconnect mutual-diffusion coefficients with self-diffusion
coefficients, are of considerable interest as well.
In fact, the interrelation between mutual and self-diffusion in liquid mix-
tures has been under debate since a long time. The simplest self-diffusion –
mutual-diffusion relationship for binary systems is the Darken relation [14],
sometimes called Hartley-Crank relation [15]
D
12
= x
2
D
1
+ x
1
D
2
. (14.2)
Here, the x
i
are the mole fractions of the components. Equation (14.2) and
other closely related versions of this equation are well suited for estimat-
ing diffusion coefficients in thermodynamically near-ideal systems such as
mixtures of hydrocarbons [4, 16,17]. It will be shown below under which ap-
proximations this well-known relation evolves from rigorous theory.
On the other hand, (14.2) and many analogues reported in the literature
rapidly loose their validity in thermodynamically non-ideal systems [1,18,19],
because the effect of intermolecular interactions cannot easily be taken into
account theoretically. Thus, diffusion theory faces similar difficulties as the
calculation of equilibrium thermodynamic properties. Of course, the inter-
molecular interactions, which determine equilibrium properties, also govern
the transport processes. Therefore, it is often useful to take the thermody-
namic properties as a guide for interpreting diffusion data. On the other
hand, this sensitivity of diffusion coefficients to molecular aggregation pro-
vides useful information on molecular interactions in liquid mixtures [18,19].
The present chapter will discuss a conceptual framework for such interpreta-
tions.
The chapter is organized as follows: After a brief review of experimen-
tal aspects in Sect. 14.2, the basic quantities for describing diffusion on a
phenomenological level will be introduced in Sect. 14.3. In Sect. 14.4 we will
reformulate these relations within the framework of the thermodynamics of
irreversible processes. In Sect. 14.5 time correlation functions will be intro-
duced for describing transport processes on a microscopic level. In Sect. 14.6
this formalism is applied to mutual diffusion. As a result, the experimen-
tal diffusion coefficients can be rewritten in terms of generalized interaction
coefficients. These interaction coefficients, denoted as distinct-diffusion co-
efficients, describe the coupling of the diffusive motions of the particles in
the mixture. Their basic properties will be outlined in Sect. 14.7. Finally, in
Sect. 14.8 some predictive equations based on this approach are discussed
and illustrated in application by some examples for diffusion in liquid mix-
tures of non-electrolytes and electrolytes with specific interactions between
the components.
558 Hermann Weing¨artner
14.2 Experimental Background
Before considering theoretical aspects, it seems helpful to throw a glance at
the basic experimental situation. In the past, many sophisticated techniques
for measuring self-diffusion and mutual-diffusion coefficients have been de-
veloped [1, 20] . When assessing and comparing their benefits, the recent lit-
erature has, however, seen a considerable change in accentuation.
Mutual diffusion causes local changes in composition and can therefore
be studied by any physical and chemical method that is sufficiently sensitive
to the composition of the mixture. The most precise measurements in two-
and three-component liquid systems have been the Gouy and Rayleigh opti-
cal techniques [21]. In both methods fringe patterns are generated by passing
monochromatic light through a diffusion boundary. The basic principles of
these methods have been described almost five decades ago, and detailed re-
views can be found in the literature [1,21]. If equipped with suitable computer
control and electronic data recording, such techniques still today are methods
of choice, when requiring highly accurate mutual-diffusion data [22,23].
More simple methods monitor diffusion through diaphragms or diffusion
out of capillaries by appropriate chemical or physical methods of concentra-
tion analysis [1, 20]. A more modern and faster technique uses dynamic light
scattering [24] which is described in Chaps. 15 and 16 of this book. Finally,
there is the Taylor dispersion method in which the diffusing component is
injected into the flowing solvent, and its dispersion in the flowing liquid is
subsequently measured at a detector [1, 20]. In recent years, Taylor diffusion
has become a convenient and economic method for collecting diffusion data
over wide ranges of state with fair accuracy, including, for example, the highly
fluid or supercritical regime [25].
In view of these powerful experimental techniques for studying mutual
diffusion, there rises the question, why to go a long way round by develop-
ing theories to estimate mutual-diffusion data from self-diffusion data. The
answer is that NMR spin-echo methods possess some major experimental
advantages [10] resulting, for example, from fast experiments, small sample
volumes, and easy applications over wide ranges of temperature and pressure.
Moreover, most techniques for determining mutual-diffusion coefficients ap-
ply non-equilibrium conditions. When working away from room temperature
and ambient pressure, it is often difficult to stabilize the resulting concen-
tration gradients against thermal or mechanical perturbations. In contrast,
self-diffusion measurements by NMR techniques are performed at equilib-
rium. (Other experimental methods for determining self-diffusion coefficients
based on isotopic tracer techniques, still extensively used in the 1970s and
1980s [1] have almost completely disappeared from the scene.)
The major advantage of NMR methods is however encountered, when
dealing with multi-component systems. In nature as well as in industrial
practice, systems are usually composed of three or more components. As dis-
cussed later in detail, a system composed of r components is characterized
14 Mutual Diffusion in Mixtures 559
by an r × r matrix of diffusion coefficients with r(r − 1)/2 independent el-
ements [1]. Except for the interferometric techniques, where the very high
accuracy of the data allows the determination of the matrix at least in the
ternary systems [1,21,22], the extraction of multi-component diffusion coeffi-
cients from measured concentration distributions is exceedingly difficult. The
possibility of measuring simultaneously the self-diffusion coefficients of many
constituents in multi-component mixtures by Fourier transform techniques is
one of the powerful features [10, 12] that make NMR experiments uniquely
suited for studying multi-component diffusion.
14.3 Phenomenological Description of Mutual Diffusion
At isothermal conditions mutual diffusion is described by Fick’s law. If, for
simplicity, we confine the discussion to binary mixtures, Fick’s law relates
the concentration gradients of the components a =1, 2 to their flux densities
j
a
. These flux densities depend on the reference frame in the experiment.
Usually, one observes diffusion in a laboratory-fixed frame, defined by some
plane in the diffusion cell perpendicular to the flux [1]. Then Fick’s law reads
for component 1
−j
1
= D
12
grad c
1
. (14.3)
If component 2 rather than component 1 is observed, one has
−j
2
= D
21
grad c
2
. (14.4)
Experimentally one finds the same value for D
12
and D
21
.
From the theoretical point of view, it is more adequate to define the
fluxes with respect to internal coordinate systems, for example, based on a
reference frame, which assumes the local center of volume to be at rest [1]. In
principle, flows in the laboratory-fixed and volume-fixed frames of reference
are not the same, because there may be diffusion-induced bulk volume flows.
In isothermal-isobaric liquid systems these flows can only result from volume
change on mixing. For small concentration gradients such volume changes
are negligible, so that in most experimental situations the equality of both
reference frames is assumed.
In passing we note that other reference frames can be chosen. For example,
when considering diffusion of a mobile phase into a comparatively immobile
phase, such as a gas into a solid, it is convenient to relate the flow of the mobile
phase to a coordinate system set up by the immobile phase, which is assumed
to be at rest. As an example, Chaps. 10 and 23 consider in detail the diffusion
properties of molecules in zeolites. The analogue in liquid systems is the so-
called solvent-fixed reference system, where the flux of one or more solutes
is considered relative to the solvent at rest [1]. In binary liquid mixtures
this reference frame is of comparatively little use. It may however be the