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Fractals and Related Hierarchical Models in Polymer Science.

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Volume 29 . Number 2
February 1990
Pages 113 - 222
International Edition in English
Fractals and Related Hierarchical Models in Polymer Science
By Alexander Blumen" and Horst Schnorer
Polymer science, an interdisciplinary science well-rooted in organic chemistry and in materials
science, encompasses an inordinate number and diversity of substance classes and thus has
far-reaching applications. Interestingly, polymers also represent a great challenge to the theoreticians, since their theoretical treatment often necessitates appropriate extensions of the
classical methods from solid state physics and from statistical physics. Thus, new concepts
often have to be invoked when considering the special properties of polymers. In this review
we concentrate on one of the modern concepts in the theory of polymers, namely on scaling.
Scaling is closely associated with new developments in the field of fractals and of hierarchical
structures. Such concepts are invaluable for the modeling of complex geometries and for
describing dynamical processes in polymeric materials. Here, we focus on a presentation of
these ideas and we outline examples of recent research in which these concepts have been
successfully applied.
1. Introduction
One of the most fascinating aspects of today's polymer
science lies in the highly interesting combination of new materials' properties on the one hand and of recent theoretical
developments on the other. That polymers are substances
with enormous potential in the applied sciences is self-evident; that the same polymers display features making them
one of the favorite testing-grounds for new theoretical concepts is not so well known. In the authors' opinion, this
underscores the fact that progress in natural sciences occurs
historically, by which we mean that formerly, apparently
unrelated fields are suddenly perceived as related, and thereafter develop in parallel. Thus, in polymer science theoretical
modeling and chemical synthesis are, in fact, much more
intimately linked than what one might naively expect. In
fact, as we shall now demonstrate, most of the mathematical
apparatus required for modeling macromolecules as objects
Prof. Dr. A. Blumen, Dipl. Phys. H. Schnorer
Phystkalisches Institut der Universltit
and Bayreuther Institut fur Makromolekiilforschung (BIMF)
Postfach 101251. D-8580 Bayreuth (FRG)
Ange>i. ('hem. I n ( . Ed. EngI. 29 (1990) 113-125
sui generis was already available for quite some time; the
need to use such mathematical techniques, however, increased dramatically with the advent of highly sophisticated
devices for monitoring substances usually regarded as complex and irregular (i.e. not fitting simple model concepts of
the day).
The basic idea underlying the new theoretical models is
that of scaling, which, in its geometrical form is exemplified
by the concepts of percolation['] and of fractals.I2- '1 In Sections 2 and 3 we shall remind our readers of these ideas,
which, in view of the large number of recent publications,[2-141may by now be quite familiar (a fact which for
some time made us hesitate to write this article). Geometric
scaling (self-similarity) is, however, only one part of the story-scaling involves also the dynamical properties of polymers, an aspect often ignored in most of the recent reviews.
We will therefore also address questions related to the dynamics of excitation and charge carrier transport in polymeric materials,[1o, - '1 topics which require modeling
through time
and through hierarchical energy
distributions.[221 First, however, let us discuss the basic
X> VCH krlu,ggesellschufr m h H , 0-6940 Weinheim, 1990
3 02.5010
2. Polymers and Scaling Concepts
In 1979 de G e n n e published
~ ~ ~ ~ ~one of the seminal works
in the field, underscoring the importance of scaling ideas in
deriving the properties of numerous diverse materials. In this
brilliant work de Gennes demonstrates how scaling, combined with a feeling for the problems at hand, leads to a
seemingly effortless derivation of many polymeric features.
Of course, many of the foundations underlying de Gennes’
book were laid in the classical works of Flory. Rouse, Zimm,
and S t o ~ k m a y e r . [ ~ ~ - ~ ’ 1
As perhaps simplest example for scaling let us consider a
polymer chain and focus on its mean squared end-to-end
distance < R2 > . For a chain consisting of N monomers and
in the absence of any hindering interactions it is a simple
matter to show that < R2> is proportional to N [Eq. (a)],
where L is a (model-dependent) constant.
< R 2 > = NL2
The result (a) is very general, since it can be traced back to
one of the basic theorems of probability theory, the centrallimit theorem. In fact many t e ~ t b o o k s [ ~ ~use
, ~models
based on random walk (Fig. 1 a) ideas (the random flight
chain, the freely rotating chain, the Gaussian chain) and
show that Equation (a) holds true in all these (special) cases.
Interestingly, one also learns from these derivations that local hindrances (such as fixed bond lengths, limitations of the
rotations around the bonds, or short-ranged correlations in
orientation) d o not change the character of Equation (a).
Thus, different models only lead to differences in the constant L ( L is practically Kuhn’s persistence lengthc3’]).
It should also be pointed out, of course, that Equation
(a)-as distinct from many of the expressions which we will
encounter in the following-does not depend on the dimension dof the Euclidean space in which the random walk takes
place; Equation (a) is truly very general.
Fig. 1 . Random walk (a) and self-avoiding walk (b) with 100 steps each on a
square lattice, using the same sequence of random numbers. Starting points and
end points are marked with filled circles o r triangles, respectively. In case (b) the
walker cannot visit occupied sites again C‘excluded volume”), which leads t o an
increase of the spatial extension of the walk.
It is thus very remarkable that the introduction of an excluded
i.e. of steric hindrances between the
monomers (this results in so-called self-avoiding walks,
SAW) leads to a vastly different behavior for < R2> . One
has the relationship (b), where v (the F l ~ r y - p a r a m e t e r I ~is~ l )
in general a non-integral, dimension-dependent constant.
The physical reason for steric hindrances is the short-range
repulsion between different monomers, as occurs, for instance, in good solvents. Now we already know that interactions between neighboring monomers along the chain d o not
cause departures from Equation (a). What happens here is
that some monomers, which along the chain are quite distant, may come into close contact due to a loop in the chain.
The repulsion between such monomers creates (mathematically speaking) long-range correlations. As forcefully
stressed by Flory, this leads to swelling of the polymer in
good solvents (see Fig. 1 b). A straightforward approximate
Alexander Blumen, born 1948 in BucharestlRumania,studiedphysics at the University of Munich
1966- 71. He gained his PhD in theoretical chemistry in 1976 from the Technical University
Munich under the guidance of G. L. Hofacker. After a post-doc period in the chemistry department of the Massachusetts Institute of Technology with R. Silbey and J: Ross he obtained his
Habilitation in 1982 at the T.U. Munich, where he was appointed Privatdozent in 1983. A
Heisenberg stipend was awarded to him by the Deutsche Forschungsgemeinschaft in 1984. He
spent a year, starting in 1985, at the Max-Planck-Institute of Polymer Science in Mainz with
E. W Fischer. Since 1986 he has been a member of the Physical Institute of the University of
Bayreuth, where he has a C3-Fiebiger projessorship. His special interests are the theory of disordered media (glasses, polymers), the dynamics of complex systems, and stochastic processes.
Horst Schnorer, born 1962 in Nankendorf (Bavaria), studied physics at the University of
Bayreuth from 1982 to 1987. After an experimental d‘iploma thesis deaiing with photoconductivity
in polymers, he now aims at his PhD in theoretical physics under the guidance of Alexander
Blumen. Central topics ofhis research are the theory oftransport processes on irregular structures
and the kinetics of diffusion-limited reactions.
Angew. Chrm. Int. Ed. EngI. 29 (1990) 113-125
calculation of the free energy for the polymer chains leads to
Equation (c).[~']The values obtained from Equation (c) are
the very recent theories of Scher, Shlesinger, Weiss and
Bendler give E = 10/3 as exact solution to their models.r34,3 5 1
in remarkable agreement with today's best estimates for
S A W S ' ~( v~ =
] 0.588 f o r d = 3 and v = 0.750 f o r d = 2). Furthermore, Equation (c) displays two limiting behaviors: In
d = 1 a SAW is trivial (being a straight segment of length
LN) so that < R 2 > = L 2 N 2 , i.e. v = 1. Moreover, in d 2 4
v = 1/2, i.e. the SAW behaves as a simple random walk; for
this reason the dimension d = 4 is marginal.
The expression (b) provides a classical example for a scaling law. First, it relates the mean end-to-end SAW distance
R = ,,/=
to the number of monomers N in a highly
nontrivial way (d). The exponent I / v in (d) is dimension-
3. Theories of Scaling
dependent [Eq. (c)], and we thus have qualitatively different
behaviors for different d, while the details of the local interactions are unimportant: the problem displays distinct universality classes. Secondly, introduction of the innocuouslylooking but fundamental condition of monomer-monomer
repulsion causes departures from the central-limit theorem.
In later sections we will show that the first point [Eq. (d)] is
directly related to (geometric) fractals, whereas the ideas
behind the second aspect are echoed in non-Gaussian (Levy)
stable laws and in time fractals.
The linear chain in a good solvent is only one of the many
examples for scaling, and we refer the reader to the books by
de Genne~,['~'
by Doi and Edwards[291
and by des Cloizeaux
and J a n t ~ i n k 'for
~ ~ an
] extensive list of scaling relations. To
present just a few spectacular examples, we consider the
semidilute regime, in which there is a relatively small concentration of chains in a good solvent; the concentration c is,
however, high enough, so that chains can mutually interpenetrate. In this regime the osmotic pressure 7~ behaves as in
Equation (e)123.291
(using v = 3/5), whereas the averaged
end-to-end distance R follows the law
the chains
A fundamental aspect of scaling-laws such as (b) or (d)(g) is their self-similarity. Take for instance Equation (d),
which relates a mesoscopic, i.e. in general fairly large (ca.
1000 A) length R to the (microscopically determined) degree
of polymerization N (average number of monomers per
chain). Equation (d) also holds if we group our monomers in
pairs (dimers) and relate R to the number ( N / 2 ) of dimers.
The same is, of course, true if we group 3. monomers together, and view R as a function of (N/A);as long as 3, stays much
less than N we have R (N/A)'/".On the mesoscopic level we
are unable to distinguish the microscopic details and a scale
change (within reasonable bounds) leads to the same physicochemical situation. Pictorially speaking, we cannot differentiate between the whole system and a (fairly substantial)
rescaled part of it-on the mesoscopic scale both look the
This idea is the basis of the renormalization group (RG)
theory, a very powerful method developed in conjunction
with the analysis of phase transition^.[^^-^^] Here the main
observation is that near a critical point the correlation length
of the order parameter diverges. We may exemplify the situation by considering a percolation cluster-probably the
best means for visualizing a phase transition geometrically.
In (site-)percolation, sites on a regular lattice are occupied at
random by conducting and insulating elements. Now an
electrical current can pass a bond only if both its ends are
occupied by conducting elements. The question is: when
does a current cross the entire sample? It is evident that for
a small concentration of conducting elements no current will
be transferred, whereas in the opposite case a current will
flow. Interestingly now, such a transition is not very gradual.
A detailed analysis shows that (for an infinite sample) below
a critical concentration pc no current will flow at all: the
resistivity is infinite. Above pc there is always a macroscopic
current Z,whose magnitude scales aroundPC.One has for the
conductance CJ above p,:
shrink therefore with increasing concentration, a result confirmed by neutron scattering.r331
A much-discussed scaling relation concerns the viscosity q.
At a constant content of polymeric material in concentrated
solutions and in melts the viscosity scales with the degree of
polymerization N in accord with Equation (g), where the
- N"
exponent E is about 3.4. This behavior is quite universal,
independent of temperature and molecular species, and
requires only linear, flexible polymers in dense, entangled
fluids.[z91Remarkably, the reptation theory by de Gennes
predicts E = 3.[231In recent years, however, several improvements have been proposed for the determination of &; thus
A n g e n . Cliem lnl. Ed Engl. 29 (1990) 113-125
(p - p,>'
where t i s a universal constant (t-1.3 in d = 2 and Fz 2.0
in d = 3 for lattice percolation[']). This behavior is thus very
reminiscent of a second-order phase transition ; here the conductance plays the role of an order parameter. These are in
fact detailed works which stress the close correspondence
between typical features encountered in percolation and the
behavior of the thermodynamic parameters in second-order
phase transitions.r39.401
Furthermore, atp, the incipient infinite cluster (that set of
connected conducting elements which spans the system from
one side to the other) scales. Figure 2 illustrates the realization of such a cluster on a square lattice, for which
p, z 0.593. Note the appearance of voids of all sizes and that
one cannot distinguish the random pattern of this cluster
from that of a similar enlarged section of another incipient
we will discuss in the following section. The historical development followed, however, another course.
3.1. Fractals
Fig. 2. Percolation cluster o n a two-dimensional square lattice at the critical
concentration p. = 0.593.
The notion of fractals was introduced by Mandelbrot in a
book in 1975;1461he stressed that many structures in Nature
are self-similar.[2.461 Starting point for a mathematical selfsimilar description were the works of Cantor, Sierpinski, Julia, and Fatou, who provided us with classes of patterns
which display scale-inva~iance!~’]
In this section we present several ways of obtaining geometrical patterns which scale with distance. Common to all
objects is their underlying self-similarity, which is most easily
visualized in terms of dilation. We start thus by discussing
the fundamental features in terms of Sierpinski gaskets and
continue by describing the general stochastic and deterministic means of creating fractal shapes.
Perhaps the simplest fractal is the Sierpinski gasket[’, 481
embedded in the two-dimensional space, as displayed in Figure 3. The structure can be generated by a procedure which
infinite cluster. That such a cluster (barely) conducts current
is due to the divergence of its connectivity a t p c , i.e. to a
singularity in the correlation length.
Now the renormalization group theory makes use of this
observation through judicious use of scaling: Imagine that
we were to repeatedly enlarge (renormalize) our scale (or,
equivalently, shrink our sample) by a fixed factor. Obviously, finite clusters will then become smaller and smaller at
each step, whereas a self-similar infinite cluster would practically stay the same. Hence, the iterated renormalization procedure allows us to get rid of the uninteresting features (finite
clusters), while keeping that part of the problem which stays
invariant under the transformation. In mathematical terms,
the critical variables at the transition show up as fixed-points
of the corresponding hamiltonian function under the
An early implementation of the R G was suggested by
KadanoJf3’. 411 it consisted in systematically deleting (decimating) lattice points from a hamiltonian formulated in the
site representation and in recalculating the effective interactions between the remaining sites (renormalization of the
interactions). For regular lattices (but not for some simple
fractals!) this real-space renormalization (RSR) scheme runs
into difficulties,[421because it often alters the structure of the
hamiltonian (e.g. it may lead to long-range interactions).
Hence, a more successful way to implement the R G procedure is to renormalize in momentum space,[431also splendidly demonstrated by Wilson, who was awarded the Nobel
Prize for Physics in 1982 for his work in this field.
The real-space renormalization (RSR) could have been
the starting point for fractals. Indeed, one of Wilson’s students, the Indian scientist Dhar, proceeded along these lines
and asked himself what simple lattice-type structures stay
invariant under RSR. In this way he created a struct ~ r e [451~ which
~ * is similar to the Sierpinski-gaskets, which
Fig 3 . Sierpinski gasket in d = 2 at the sixth stage of iteration
renders clear the underlying symmetry. One starts with a
triangle of side length 2, which includes three smaller, upwards-pointing triangles of side length 1. In the nomenclature introduced by Mandelbrot[21this basic pattern is called
a generator. A dilation by a factor of 2 from the upper corner
transforms the upper small triangle into the large one, and
creates two additional, larger triangles. One may also view
the operation as attaching two copies (left and right) below
the original pattern. The procedure is then iterated n times,
and leads to the structure a t its nth stage. As an example, the
portion of the Sierpinski gasket depicted in Figure 3 corresponds to the 6th stage of iteration.
Sierpinski gaskets can be generated in embedding spaces
of arbitrary dimension d, by starting with the corresponding
Angew. Chew$.I n l . Ed. Engl. 29 (1990) 113-125
Fig. 4. Sierpinski gasket in d = 3 at the second (a) and the fifth (b) stage of iteration.
hyper tetrahedron^.[^'* Figure 4 shows a Sierpinski gasket
in three dimensions at the 2nd and 5th stage of iteration.
Also more general patterns can be constructed along the
same lines of reasoning.
Properties of fractal objects related to physical quantities
(such as mass distribution, density of vibrational states, conductivity and elasticity) are describable by several, not necessarily integral parameters. These parameters play roles similar to that of the spatial dimension. Here we focus on the
fractal (Hausdorff) dimension 2 which characterizes the distribution of mass. Denoting by N the number of lattice
points inside a sphere of radius R, one has the proportionality (i), where 2 is defined by d = lim lnN/lnR. Hence,
N - Rz
for a Sierpinski gasket embedded in the d-dimensional Euclidean space one obtains Equation ('j).[', 'I
A comparison of the expressions (i) and (d) shows that a
polymer chain in a good solvent has a fractal dimension 2 of
Another important parameter which plays a role in connection with the dynamical properties of or on fractals (such
as diffusion-controlled reactions and heat conduction) is the
spectral (fracton) dimension 2. Using a scaling argument,
Alexander and O r b a ~ h [ " have
found that Equation (k)
holds for Sierpinski gaskets in d dimensions.
l n ( d + 1)
In (d
+ 3)
As example, for the Sierpinski gasket of Figure 3, (where
2, = 1.365. In many cases the
d = 2) one has as= 1.584 and
Angeu. Chem. Int. Ed. Engl. 29 (1990) 113-125
relation 2 < d < d holds; furthermore, for Sierpinski gaskets 1 < 2%
< 2, as may be seen from Equation (k). It should
be emphasized that d and 2 are amenable to experimental
observation.'". 5 1 , For modeling purposes it is therefore
desirable to construct deterministic fractals with prescribed
d a n d 2 values.
Based on the systematic construction of the Sierpinski
gaskets one can consider modified Sierpinski-type structures. The spectral dimensions 2 for these structures lie
between that of a 2d-Sierpinski gasket, 2, = 1.36, and the
value 2. In References [49, SO] one took as generators d-dimensional tetrahedrons of side length b filled with N smaller
tetrahedrons of unit side length. For such generators the
fractal pattern follows by iteration, and one has the relationship (1).
d = In Njln b
On the other hand, 2 can be obtained from the long-time
behavior of the probability of being at the origin (see References [49, 501 for details) and lies between 1 and 2. Hence
such symmetric fractals represent a wide class of lattices
which generalize the Sierpinski gaskets.
Not all fractal systems display the strong symmetry of the
Sierpinski gaskets, which are 'deterministic' fractals, designated as such since for each point in space it is unambiguously clear whether it belongs to the structure or not. The distribution of points in a fractal pattern may be random, as
exemplified through the previously discussed percolation
clusters. A related model is that of photopolymerization of
multifunctional monomers. Here the procedure is to pick at
random at each step a monomer adjacent to the boundary of
the growing cluster, and to attach it to the cluster. This
so-called Eden model is a special case of epidemic growth,
where, in a more general version, a fresh cell at the boundary
of an infected cluster either gets infected with probability p
(and hence joins the cluster) or becomes immune. The general version is identical to percolation:" for p smaller than the
critical concentration p c the growth of the cluster always
terminates, while for p 2 p , infinite clusters may be formed.
Models similar to epidemic growth have also been developed
in conjunction with the spread of forest fires.". 53*"1
A wide variety of fractal patterns is encountered in the
investigation of kinetic growth models. Here (see reviews [6]
and [12]) it is customary to distinguish the models according
to the mass transport and to accretion. Mass transport describes the way in which the reactants come in close contact:
in solvents the reactants diffuse towards each other, so that
one has Brownian motion; more rarely also ballistic aggregation may occur, in which case the reactants approach each
other along linear trajectories. Accretion may occur at first
encounter (the growth is then motion-limited) or several encounters of the reactants may happen before a bond is created (the growth is then reaction-limited). Furthermore, one
may allow only monomers to attach themselves to the already existing polymer cluster (monomer-cluster aggregation, MC) or the clusters may also combine (cluster-cluster
aggregation, CC). An example for a MC diffusion-limited
aggregate (DLA) is given in Figure 5. As a rule, CC leads to
Fig. 5. Cluster of monomers grown by diffusion-limited monomer-cluster
aggregation on a square lattice. The different colors show different stages of
more ramified clusters than MC, and the density of clusters
increases from diffusion-limited to ballistic to reaction-limited. Model
lead to the following fractal dimensions d for clusters grown in d = 3: a) MC: diffusionlimited d = 2.50, ballistic 2 = 3.00, reaction-limited (Eden
model) d = 3.00; b) CC: diffusion-limited: d = 1.80, ballistic
d = 1.95, reaction limited 2 = 2.09. All these patterns are
encountered in polymeric systems, and examples will be
provided in Section 4, where we will discuss the experimental techniques used to monitor the fractal dimensions.
First, however, let us turn to scaling aspects in dynamical
3.2. Time-Fractals and Ultrametric Spaces
A major field of application of scaling ideas in the time
domain is photocond~ctivity.~'~~
16] As pointed out in a recent review article" the use of polymeric photoconductors
in printing and copying techniques represents one of the
most sophisticated applications for organic materials. In fact
organic photoconductors have been improved in the last two
decades to the extent that they are nowadays at least equivalent (if not even superior) to their inorganic counterparts.
From the theoretical viewpoint, the issue at stake in amorphous photoconductors is the dispersive transport, as contrasted to the familiar diffusive, Gaussian behavior. Diffusive motion, in its many facets, ranging from the transfer of
heat (macroscopic) to Brownian motion (microscopic) is a
well-understood phenomenon and may be modeled using the
mathematical theory of Markov processes, in which the system keeps no memory of past events. Typical discrete examples are random walks on regular lattices-first introduced
by Bachelier at the turn of the century for monitoring the
fluctuations of the stock market[5s1-which nowadays comprise a vast mathematical literature.[s61 Simple random
walks on regular lattices for arbitrary dimensions lead to a
time-independent diffusion coefficient, or, expressed differently, in the absence of a biasing field, to a linear increase in
the mean squared displacement as a function of time.
In amorphous photoconductors the situation is different,
since one encounters dispersive transport: the motion of the
carriers through a sample becomes slower and slower with
the passage of time, so that no diffusion constant in the usual
sense can be defined. Here the diffusion coefficient D displays an algebraic dependence on time [(m) with y < I];
with a similar finding for the current Z(t) flowing in a biasing
field. Such behavior is incompatible with the usual diffusion
equation. Note, however, the nontrivial scaling behavior of
D with respect to t .
Progress in the advancement of theoretical models for dispersive transport was slow, due to the Gaussian character of
most of the probability distribution functions put forward.
Here again the central-limit theorem of probability theory
plays an important role. One should recall that under very
general conditions, a sum of random variables will tend to
form a normal (Gaussian) distribution when the number of
elements in the sum increases. The position vector of a particle may be viewed as a sum of distance increments due to past
events, say collisions. If energies, directions etc., are randomized after such events, most distributions will lead to regular
diffusion. Hence, for a successful modeling of dispersive
transport in such a framework one has to look for the (relatively rare) distributions for which the central-limit theorem
does not hold.
The basic theoretical problem in modeling the transport of
light-generated charge carriers through an amorphous medium is to take care of the broad distribution of stepping times.
One models the motion of individual carriers which move
under the influence of an external electric field through hopping. Here it is important that the probability for performing
a step per unit time is not exponential, but decays much more
Angew. Chem. Inl. Ed. Engl. 29 (f990)113-125
slowly. Physically this may be pictured as arising from the
fact that an amorphous material has rather irregular lattice
spacings (geometrical disorder), which lead to large variations in stepping times (temporal disorder).
Mathematically, the most important parameter characterizing the transport is the distribution of waiting-times between subsequent carrier jumps. A theoretical breakthrough
occurred here through the works of Scher, Lax, and Montro", who showed that long tails in the waiting-time distribution $ ( t ) between the microscopic events can qualitatively
describe the dispersive transport,[57*
581 when one assumes
the relationship (n) with 0 < y < 1,
so that the zeroth but not the first moment of the distribution
$ ( t ) exists. Such distributions lead to Levy processes, a field
which was long considered to be a domain of mathematical
curiosities only.[2.591 The main technical point in investigating random walks in continuous time (CTRW) with such
distributions, was the realization that the basic CTRW
framework (as formulated ten years earlier by MontrolE and
Weiss, 1601 could readily incorporate these, less usual, $ (t)forms. However, although the theory developed in References 157, 581 was successful in providing a scheme for the
qualitative understanding of dispersive transport, one could
at the time only speculate on the nature of the microscopic
parameters leading to $ (2).
Today we know that this approach is only one of the
possible was of obtaining algebraic time dependences for
D ( t ) ,and that the key ingredient in the CTRW framework is
the idea of temporal scaling, or of time fractals, as formulated by Shlesinger."' -21*61]
As pointed out in Reference [21], scaling has a long history
in probability theory. As early as 1713 the work of N .
Bernoulli was reported, who analyzed a game of chance with
infinite average winning (the Petersburg paradox). Furthermore, scaling also shows up in a function introduced by
Weierstrass [Eq. (o)]. W(k) is everywhere continuous but
C a"cos(b"k)
W(k) = (1 -a)
(a < 1, b > a - ' )
nowhere differentiable. For our purposes, an expression related to Equation ( 0 ) provides an excellent means to obtain
the asymptotic behavior given by Equation (n).I1O-
It is a rather straightforward matter to verify that at large
values for t Equation (p) scales: From @) one has
ab$(bt) = $ ( t ) - (1 - a)b exp (- bt), so that for slowly decaying (slower than exponential) $ ( t ) one finds at longer
times: $ ( t ) z ab$(bt), i.e. scaling behavior. From the last
expression one obtains y of Equation (n) to be equal to
In a/ln b. Expression (p) is very valuable, since it allows one
to investigate the behavior of several dynamical quantities
near their marginal dimensions; notice that we can, through
Angew. Chem. Inr. Ed. EngI. 29
(19901 113-125
a judicious choice of a and b, get any preassigned value of y.
Some examples concerning photocurrents and excitation
depolarization in polymeric materials will be presented in
Sections 5 and 6.
One physical reason for such broad distributions of
waiting times is energy disorder. The motion of particles
through an amorphous solid as well as relaxation in glassy
materials can be described in terms of dynamical processes in
multidimensional random potentials with valleys and barriers of different
The microscopic motion over
the barriers is in many ways similar to Eying's ideas of the
activated complex in chemical
There are several models which are simplified variants of
the random potential, e.g. the random valley and random
barrier models[64] (see Fig. 6). Random valley models con-
Fig. 6. (a) Random potential, (b) random valley, and ( c ) random barrier mod-
sider a random distribution of valley depths with barriers at
equal top level. A similar model, in which charge carriers in
the conduction band get captured by and released from traps
of various depth (multiple trapping) was formulated by
The multiple trapping model was shown to
correspond to the CTRW-theory.'' 6 .
In random barrier models the valleys are at equal level.
These structures are amenable to hierarchical modeling
through so-called ultrametric spaces."',
In these models
sites are separated by energy barriers with random heights.
Thus, a particle positioned on a certain site needs thermal
energy to surmount the surrounding barriers. A given activation energy lets the particle visit only a subset (cluster) of
sites around the starting point. One may then classify the
sites through the energy required to reach them?'] To such
a classification corresponds an ultrametric space (UMS).
As examples, Figure 7 shows the regularly multifurcating
ultrametric spaces Z, and Z, . Note that only the points on
the baseline belong to the UMS and that the structure above
the baseline documents barrier heights and intersite connections. The height d ( x , y ) of the barrier between sites x and y
may be used as a generalized 'distance': It may be easily
verified that Equation (9) holds for all UMS sites. Equation
( 4 )is the 'strong' triangle inequality which leads to the name
'ultra'metric. Z , and Z , have branching ratios z = 2 and
adsorption techniques. It consists in determining the monolayer capacity N ( o ) of adsorbed molecules of cross section o
required to cover the surface to be investigated. For a fractal
surface one expects as a function of the cross section the
relationship ( u ) . ‘ ~ ~ ]
Fig. 7. The ultrametric spaces Z, and Z , .
z = 3, respectively, and, for simplicity, we have taken the
barrier heights to be hierarchically arranged, so that all consecutive energy levels differ by A . Now, it may be shown that
several dynamical parameters scale with 6 = kT Inz/d. We
refrain, however, from continuing with this topic and we
refer the interested reader to References [lo, 22, 681 and
works cited therein for further details.
4. Geometrical Aspects of Disordered Systems
In this section we focus on the determination of the geometric scaling aspect for aggregates. The basic quantity here
is the fractal dimension d, which was defined in Equation (1).
Since the total mass M of the aggregate is directly proportional to N , the number of monomers involved, the expression (i) may be replaced by (r).
Objects that obey Equation (r) are named “mass-fractals”.“ Distinct from these, one also encounters “surface
fractals”, i.e. aggregates which are uniformly dense (d = 3),
while having a rough surface S, for which the relationship (s)
holds, where is the surface fractal dimension.
The methods of choice for investigating fractal objects
include, in particular, scattering techniques. The intensity
profile I of a scattered beam (light, neutrons, X-rays) is in
such cases algebraically dependent upon the scattering wave
vector k in accordance with (t),”’] where -22 + is the
so-called Porod slope. Schaefer et al. have investigated whole
classes of substances and have demonstrated that, in general,
many structures can be distinguished by their Porod slopes;
they find that aggregates formed by polymer precursors give
rise to scattering curves whose slopes lie between -1 and
- 3, whereas rough colloids yield slopes between - 3 and - 4
and smooth colloids slopes of -4. The methods even allow
one to differentiate between substances grown by monomercluster (MC) and cluster-cluster (CC) aggregation; we refer
the reader to Reference 1121 for a summary of recent results.
Another method of choice in measuring the fractal dimension, advocated by Avnir et al.,[131makes extensive use of
Such behavior was reported by Avnir, Pfeifer et al.[69-721
in their study of crushed glass, carbon black, charcoal, and
of porous silica gels. The adsorbates that they use include
N,, alkanes and polycyclic arenes and alcohols.
This approach, however, has its problems: adsorption is
an intricate process, in which the local interactions of the
molecules with the surface are complex and may mask the
geometric details.[731It appears that both the measurements
of the monolayer capacity and the calculations of the molecular cross sections are quite delicate. Recent molecular-sizedependent experiments on porous silica
led to values
of d, even larger than 3, a clearly very problematic result.
Another method which provides morphological information (including the fractal dimension of self-similar objects)
centers on the direct transfer of electronic excitations. This
method, which monitors the fractal dimension and which
was proposed by us,[751is now of widespread use. Furthermore, the spectral dimension can be monitored through reaction-diffusion processes as propagated by us[751and by
K o p e l m a r ~ . For
newer developments on the spectral dimension see References [77, 781.
Let us now consider the direct energy transfer (ET), which
takes place from donor molecules to acceptor molecules, and
assume that donors and acceptors occupy substitutionally
sites on a regular lattice. In this classical problem[79.801
considers an excited donor molecule surrounded by acceptors which occupy some of the sites of the structure. The
transfer rates w (r) depend on the mutual distance r between
each donor-acceptor pair. An often encountered form for
w (r) which holds for isotropic multipolar interactions is
Equation (v). [’I The parameter s equals 6 for dipole-dipole,
8 for dipole-quadrupole and 10 for quadrupole-quadrupole
interactions. For regular, d-dimensional lattices one obtains
the decay function @ ( t )according to (w), where A is a con-
stant and B is the reciprocal intrinsic lifetime of the donor in
the absence of acceptors.
Let us now focus on optical properties related to fractals.
From Equation (w) we see that the Forster-type decays are
determined by the spatial dimension d of the region accessible to acceptors. Thus, the extension of Equation (w) to
fractal objects gives (x) (see Ref. [lo] for a detailed derivation).
Since d < s, the long-time decay is governed by intramolecular processes. This, on the other hand, also limits the
Angew. Chem. Int. Ed. Engl. 29 (1990) !!3-125
spatial range in which the ET method can probe the structure
of the underlying material. Typically, this range is of the
order of 100& which is also the region amenable to the
small angle X-ray scattering technique. Hence, in this region
these two methods give complementary information.
The ET method, as proposed in Reference [75], was first
used by Even et al.[81]for the investigation of the pore structure of Vycor glass. They studied the kinetics of the ET
between rhodamine Band malachite green in solution inside
the glass. Using Equation (x), Even et al. found a= 1.74.
This result has recently been questioned with regard to its
compatibility with light and X-ray scattering data.[82.831
Similar experiments have been conducted by Rojanski et
al.cE41on a porous silica gel; they obtained d, = 3 for the
silica gel surface, a result also corroborated by adsorption
and by X-ray scattering. These findings were later questioned by Levitz and Drake["] and by Levitz et a1.[861who
asserted that this type of silica gel is characterized by a
smooth surface on the molecular scale
= 2). As shown by
Levitz et al.[861it is problematic to claim that a structure is
fractal based only on results compatible with Equation (x).
Yang et al. performed computer simulations;[871they considered the ET among particles located on pores of spherical
or cylindrical shape, whose radii are of the order of the
typical transfer distance; the calculated kinetics show fractal-like decays. Similar geometrical restrictions for ET have
been studied by Baumann and Fayer[ssl and by our-
ET processes have been studied for polymeric systems.
Singlet ET mechanisms in anthracene-loaded copolymers of
vinylnaphthalene show complicated temporal behaviors,
which are thought to be partly due to fractal-like distributions of naphthalene and anthracene molecules.[g11Triplet
ET kinetics in doped polymer films has been investigated by
Lin, Nelson, and H a n ~ o n . [Their
~ ~ ] data fit their model better
when fractal structures are taken into account. Further discussions of polymeric systems in terms of fractals are given
in References [93 and 941.
We close this section by mentioning that the ET model has
also initiated new ideas; thus, fractals have been considered
in the discussion of the energy migration and trapping
among cationic porphyrins adsorbed on anionic vesicle surface~.[~']
Furthermore, a stretched exponential modulation
model has been proposed for simulating vibrational
dephasing in locally amorphous
Finally, hierarchical and fractal concepts have been used in the analysis of
time-resolved spectra of mixed crystals of dichlorobenzene
and dibromoben~ene.[~']
A flow through a porous system, as any motion through
a fractal space, may show dispersion, i.e. the diffusion is in
general governed by Equation (m). This field has recently
been well reviewed by Po-Zen Wong,[981so we shall contend
ourselves here with a summary of the results. The ionic conductivity of rocks arscales with their porosity cp, following
ar cp-", where rn is around 2. This result
corresponds to the 'shrinking tube' model, which is
analogous to percolation,['] see for instance Equation (h),
with p , = 0.[991
An extension of the model leads to the 'grain
consolidation' model, obtained from random sphere packing.['"' This model again leads to Archie's law over much of
the range of practical interest.["'1
Special topics of attention in flows in random systems
involve problems which arise when two (or more) liquids are
present at the same time, e.g. the question of how oil and
water displace each other. Here one encounters 'viscous fingering' aspects, and 'invasion percolation'.['''] The latter
model ensures that the invader flows along a continuous
path while the defender may form trapped clusters. For details on these models we refer the reader to the literature.[981
Another field of research in which temporal scaling and
the CTRW scheme play an important role is charge carrier
transport in amorphous materials, especially in polymeric
photoconductors. A technique which is often applied for
investigating the properties of amorphous photoconductors
is the time-of-flight method. Charge carriers, generated by a
short light pulse near the top surface of a thin film of the
photoconductive material, drift under the influence of an
external electric field through the sample and give rise to a
transient photocurrent in the external circuit. In many inorganic and organic amorphous materials the shapes of the
photocurrents Z ( t ) show dispersive behavior, i.e. the current
decreases monotonically without exhibiting a regime with
constant current, which one would expect if all charge carriers moved with the same mean veIocity through the sample.
As also noted in Section 4, Scher and M o n t r ~ l l [applied
the CTRW-theory to this problem and using the algebraic
waiting-time distribution of the form (n) derived expressions
for the asymptotic decay of the current. Two regimes have to
be distinguished. In the time regime before the so-called
transit time t, (the time when the fastest carriers passed
through the sample) the current decays according to (y),
whereas in the time range t > t, the current decays according
to (z).
5. Dynamical Aspects of Disordered Systems
We will now turn our attention to dynamical aspects for
which the ideas of scaling in time play an important role. The
dominant feature, distinct from the topics of the previous
section, is that here we focus on changes in dynamics
brought about by irregular objects, rather than on the geometrical aspects. Important problems in this field are
(macroscopically) liquid flows through porous systems and
(microscopically) charge and excitation transfer through
polymeric samples.
Angew. Chem. In[. Ed. Engl. 29 (1990) 113-125
The results (y) and (2) mean that here the temporal aspect
dominates the dynamics. As an example for which Equations
(y) and (z) provide the right description for experimental
data we present in Figure 8 the transient photocurrent in
polysiloxanes with pendant carbazole groups.['021Note that
both axes are logarithmic, leading to an algebraic decay of
the form t - @ appearing as a straight line with slope -p. The
experimental curve (1) and the theoretical curve (2), calculated with y = 0.58, coincide over a period of four decades in
I [A
Fig. 8. Transient photocurrent in polysiloxane (1) and theoretical curve, calculated from an algebraic $ ( I ) with y = 0.58 (2). double logarithmic plot.
To close this section let us consider the problem of electronic energy transfer in substitutionally disordered molecular and ionic crystals. Here the general interest has focused
on the diffusion coefficient, which is related to the meansquared distance traveled by the excitation, or on the decay
of the excitation, due to trapping and annihilation."'. 104-1061 In the case of dipolar interactions, a less
direct but experimentally very versatile method to test the
transport consists in performing depolarization measureFor randomly oriented molecules already a
single-step transfer depolarizes very efficiently, so that in the
long run the memory of the original polarization is lost,
except for those excitations which are (still or again) on the
originally excited sites. Thus depolarization measurements
probe Po(t),the probability that an originally excited site is
excited at time t. As we will show, Po(t)scales with time.
We have treated this problem in the framework of the
previously described CTRW theory!".
the local disorder in the waiting-time distribution $ (I), for
which we used the time fractal Weierstrass-form Equation
(p). Now for a simple random walk in d = 3 the relationship
(A) holds true, and for general dimension d the relationship
(B). Here the geometry of the lattice plays an important role.
time. Therefore, also in photoconductivity we obtain in
many instances scaling behavior (see Ref. [15]).
However, the behavior of the photocurrents in polymeric
systems is manifold. In the next section we will show some
limitations of the idea of temporal scaling and we will also
present crossover effects.
Let us now turn to percolative aspects in photoconductivity, an aspect more akin to geometric scaling. Percolation is
important, if the substrate consists of impurity centers
embedded in an inert matrix. In a recent investigation11031a
benzotriazole derivative embedded in a polycarbonate
matrix was used. In such a situation, especially at relatively
low impurity concentrations, the fluctuations in the intersite
distances become very drastic. As a result, the influence of
the geometric disorder becomes dominant, and percolation,
as described in Section 3, prevails. As discussed there, one
has a critical concentration p c , below which no current can
flow. Slightly above p , the current depends on the impurity
concentration p , according to the expression (h), where p , is
nonuniversal while t is a universal constant (which depends
mainly on the dimension). Figure 9 shows the carrier mobilities
found for the benzotriazole-polycarbonate system.
The experimental data appear to agree fairly well with the
percolation picture.
On the other hand, when we perform a continuous-time
we find that the
random walk with an algebraic $ ( t ) I long-time Po(t) expression is a direct reflection of the waiting-time distribution $ (t), as long as this distribution is
broad, while for sufficiently narrow waiting-time distributions the classical random-walk behavior [Eq. (B)] for P,(t)
is recovered. The decay behavior follows Equation (C)
(see also Fig. 10). Equation (C) is an example for the interplay between temporal and spatial aspects of excitation
transport, the temporal aspect being dominant for small y
(high dispersion, wide distribution of jump times).
' boo
-'1 1 0
* 200
P Fig. 9. Effective charge carrier mobilities pT of polycarbonate matrices contaming different concentrationsp of benzotriazole. The pT data were measured
at the U-values given in the inset (sample thickness ca. 10 pm) at two different
temperatures (293 K and 333 K). All curves are fitted to the percolation model
[(h), with p. = 0.1 and r= 2.51
Fig. 10. Decay behavior of P,,(I) for a bcc lattice under CTRW with
0.25 5 y I2, plotted on log-log scales. The y-values are as indicated. A is a
scaling factor with the dimension s - ' .
Angew. Chem. I n t . Ed. Engl. 29 (1990) 113-125
In summary, in this section we have shown that the ideas
of temporal and geometrical scaling (exemplified by the
CTRW and by the percolation scheme) are fruitful methods
for describing transport phenomena in disordered systems
like polymers. In the next section we will outline some limitations of these methods.
6. Concluding Remarks
In a real physical system one cannot expect to have an
infinite hierarchy of temporal or spatial scales. There always
exist maximal and minimal cut-offs, like the size of the system or the distances between atoms, where the hierarchy of
scales ends. Therefore, it depends on the range of the experimental parameters, whether the idea of scaling is fully applicable. Moreover, a more subtle limitation for scaling occurs
when one works in the marginal regime for the critical
parameters; in this regime one must be careful, since
crossover effects (which may lead to a transition to normal
behavior) may have to be taken into account. In such transition regions scaling is attainable only at exceedingly long
times, possibly far outside the experimentally interesting
In order to illustrate this comment let us consider a very
transparent model: we focus on a particle (charge carrier)
diffusing under the influence of an external bias (electric
field) on a linear chain and take the sites (traps) to have
different energy levels E with a distribution e ( E ) (Fig. 11).
For y > 1 there exists a mean waiting-time 7 = y/(y- 1)
and thus in the long-time limit the transport behavior is
nondispersive. As long as y is less than unity, the resulting
current decays asymptotically as I ( t ) t - ' + Y . These findings are illustrated in Figure 13. Now we have a crossover
from dispersive to nondispersive behavior depending on the
parameter y. Note, however, and this is the important point
here, that the time to reach the asymptotic regime diverges as
y approaches the marginal value 1. Thus, if y is around unity,
the asymptotic behavior may well be of no experimental
lg ( I / I oI
Fig. 11. Charge carrier hopping on a linear chain of equidistant sites with
different trap energies E , and transition rates ri. The ti are randomly chosen for
a distribution efz). A = absorber.
lg ( u t )
Assuming thermal activation, we readily calculate the transition rates and, after averaging over the distribution e (E), we
get the corresponding waiting-time distribution $ (t).I1lol
This leads, like in the CTRW theory, to the current Z ( t ) . The
analysis shows that for a finite mean waiting-time
T =
<t >
7 t $ ( t )d t, a crossover from dispersive behav0
ior for t < 7 to nondispersive behavior for t 7 occurs. This
is illustrated in Figure 12, which shows the current Z for a
Gaussian distribution of energy levels. Depending on the
length N of the chain, the regime of normal transport (constant I ) is more or less pronounced.
The situation is quite different for exponential energy distributions of the form (D). These lead to $ ( t ) t - l - Y , just
as considered in Section 3.
Angen. Chem. Inr. Ed. Engl. 29 (1990) 113-125
Fig. 13. Current curves for an infinite chain with an exponential distribution
The parameters y are as indicated.
e (E).
This example shows that the ideas of scaling are quite
useful if one is well inside the range of validity of the critical
parameters. However, in a marginal regime one has to be
aware that a simple asymptotic scaling analysis may not be
sufficient and that crossover effects have to be taken into
In conclusion, there is no doubt that ideas based on scaling, such as fractals and related hierarchical models have
entered into almost all scientific disciplines. Scaling concepts
are becoming increasingly standard for physicists and
chemists, and such concepts are ideally suited for the analysis
of polymers. Nevertheless, on a note of caution, it must be
emphasized that not all irregular structures are ultimately
fractals; furthermore, switching now to the time scale, it
must be pointed out that not all irregular time evolutions are
indicative of scaling. In applications one has to be aware of
crossover effects and of the fact that self-similarity is generally bounded by lower and upper cut-offs, a feature that renders the interpretation of experimental data delicate and
which requires care in order to attain a sound physicochemical understanding.
The support of this work by the Deutsche Forschungsgemeinschaft (SFB213), by the Fonds der Chemischen Industrie
(grant of an IRIS-workstation) and by grants of computer
time from the Hochschulrechenzentrum Bayreuth is gratefully
acknowledged. We have greatly benefited from discussions
with our friends and colleagues, Professors D. Haarer,
J. Friedrich, J. Klafter and G. Zumofen, who have callaborated
with us on many of the here-mentioned articles. Many thanks
are also due to Dr. G. Zumofenfor providing us with Figures 1
to 3 and 5 .
Received: May 3, 1989 [A 747 IE]
German version: Angew. Chem. 102 (1990) 158
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