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Translation from Russian:

КОКОРЕВИЧ А. Г., ГРАВИТИС Я. А., ОЗОЛЬ-КАЛНИН В. Г.

РАЗВИТИЕ СКЕйЛИНГОВОГО ПОДХОДА ПРИ ИССЛЕДОВАНИИ НАДМОЛЕКУЛЯРНОЙ СТРУКТУРЫ ЛИГНИНА.

ЛИГНИН КАК ФРАКТАЛЬНЫЙ ОБЪЕКТ: ОБЗОР //

Химия древесины (ISSN 0201-7474). - 1989. - № 1. - с. 3-24.

UDC 634.0.813.11:54-126

Kokorevics A., Gravitis J., Ozol'-Kalnin V.

Institute of Wood Chemistry, Latvian SSR Academy of Sciences

THE DEVELOPMENT OF THE INVESTIGATION OF THE SUPERMOLECULAR STRUCTURE OF

LIGNIN FROM THE SCALING VIEWPOINT. LIGNIN AS A FRACTAL (REVIEW)

Khimiya Drevesiny (Wood Chemistry). - 1989. - N 1. - p. 3-24 (ISSN 0201-7474).

"...it is not the style, but standards of the style, that are the measure

of the investigator... As science progresses new styles, adapted to new

problems, are inevitable, and it is towards these new developments

that a tolerant and receptive attitude most becomes the thoughtful

scientist."

D.E.Green, R.F.Golberger "Molecular Insights into the Living Process",

Academic Press, NY, London, 1967, 418 p.

The works by Freudenberg [1] according to which lignin is a complex three-dimensional polymer became the

first step in understanding the structural organization of lignins. (It should be noted that almost any

macromolecule is three-dimensional). The next step was made by the Canadian physical chemist Goring [2],

who attributed lignin to the class of cross-linked polymers. The third step was a hypothesis proposed at the

Institute of Wood Chemistry, Latvian Academy of Sciences [3], according to which the lignin network has

definitely inhomogeneous cross-linking structure. Thus, if we move from a more cross-linked area to a less

cross-linked area, at certain scales of lengths, it is possible to decrease the density of packing of lignin

phenylpropane units. Let's consider the experimental data corroborating this proposition. Goring showed that

microgel particles of lignin fragments in the solution have a more cross-linked core and less dense periphery

[4]. In his later works, Lindström arrived at the same conclusion [5]. Heterogeneity of lignin fragments density

in the solution is corroborated by the works of Lindberg and coauthors [6] who performed the analysis of the

liability of nitroxil label. According to the data obtained by Hatakeyama and Nakano [7], the NMR wide line

spectra of isolated lignin preparations have wide and narrow components. Such spectra are characteristic for

semicrystallic polymers: the narrow and the wide part of the spectrum correspond to amorphous and

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crystalline regions, respectively. Lignin is a pure amorphous polymer. Consequently, the observed effect is a

result of different degrees of cross-linking leading to varying the mobility of chains. Data on lignin glass-

transition temperatures obtained by the American scientist Glasser and coauthors [8] also suggest different

degrees of cross-linking. The same was observed for lignin in situ during its destruction by the action of

metallic sodium in liquid ammonia [9] (method suggested by Shorigin [10]). The idea of inhomogeneous

density of lignin network is also corroborated by the results obtained recently by the French researcher

Monties [11].

It should be mentioned that the fact of inhomogeneity of lignin network cross-linking is still at the level of

ascertaining, i.e. this phenomenon will be described mathematically. It is clear that the Flory-Stockmayer

theory is unsuitable to describe the inhomogeneously cross-linked structure. Until the other approaches

became known, this theory was widely used by many authors. However, they tried to fit their experimental

data to this theory. In fact, Flory-Stockmayer model became only one among so called universality classes

about which we will talk later.

At the present level of scientific knowledge, it became possible to apply a number of new approaches and

methods for the description of supermolecular structure of lignins [12-14]. This review deals with the analysis

of the scaling approach, the concept of universality classes of physical phenomena, the concept of the fractals

geometry, and computerized simulation of physical processes [15-23].

FRACTALS GEOMETRY

The concept of the fractals geometry became well-known among researchers thanks the work by Mandelbrot

[24, 25]. In this review, only the basics of the fractals geometry will be discussed. For more detailed

description of this problem, we refer you to [26], where the history of the development of the concept of

fractals in mathematics, is also described. At first let's consider regular fractals, for instance, Koch's curve and

Serpinski's carpet (Serpinski's set) (Fig. 1). The Koch's curve is constructed in the following way: a fragment of

straight line is divided in three equal parts, the middle of which is replaced by two identical segments. Thus,

the size of each of the four segments is decreased by a factor of three. This procedure is repeated with each of

the obtained segments.

When constructing Serpinski's carpet, the initial square is divided in nine squares, the size of the latter is being

three times less than the size of the initial square. Then the middle square is removed. Figure 1 shows a thrice-

repeated reiteration of these procedures.

The fractal dimension df serving as characteristics of the above mentioned systems is determined for regular

fractals as:

N nrd

f or df = log N / log nr ,

where N is the number of the system parts obtained after the procedure;

nr is the number of times the scale is changed.

For Koch's curve df = 1.262, i.e. is more than the dimension of the initial construction - one dimensional line,

and less than the dimension of the two-dimensional space in which it is located. For Serpinski's carpet df =

1.893, i. e. it's less than for the initial object - the plane. A number of the procedures - the algorithms of the

formation of similar structures with different df values can be also elaborated. From the requirement of the

object's connectedness it follow that df ≥ 1, and from the assumption according to which the object should be

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enclosed in circumscribed space it follow that df ≤ d, where d is dimension of circumscribed space (in the case

when df = d, fractal dimension is trivial). Such an object - the fractal is characterized by fractional dimension df,

which is less than the dimension of real space.

Fig. 1. Procedures of construction of the regular fractals:

a - Koch's curve (df=log 4/log 3=1.262); b - Serpinski's carpet (df=log 8/log 3=1.893).

rmin and rmax minimum and maximum scales at which system is a fractal.

In physical sense, the fractal can be formed by connection of individual particles ( an atom, a molecule, a

colloid particle, a bacterium in culture, a star or a galaxy in the system of stars or galaxies, respectively, etc.

are taken to mean "a particle"). The particles combined in the integer are called clusters.

By using the dimension df, it is possible to characterize the dependence of the number of particles N in a

cluster on the distance r, where they are being counted:

N r d

f (1).

The number of the particles N is proportional to their mass M, and using the dimension df, it is also possible to

describe the distribution of the cluster mass, depending on its size:

M r d

f (2).

Only the particles which are located on the surface of the object can be considered. In this case we have the

right to speak about the fractal dimension of the surface dsurf. For Koch's curve dsurf = df, and for Serpinski's

carpet dsurf = 1.262, i.e. dsurf < df.

With the help of df it is possible to describe the change in the density of particles in space, depending on the

size under consideration:

(r) = N / rd r

df-d

(3).

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The (r) value is a decreasing function (with the exception of trivial fractal for which the (r) value is constant),

i.e. the bigger fractal or its part is considered the more porous it is.

The characteristic property of fractals is the property of self-similarity. The essence of the idea of self-

similarity is as follows. If the fractal is considered at one scale (or under the same magnification of a

microscope), and then at the different scale (magnification), the difference between the images is not evident.

It is connected with the procedure (the law) of fractal construction which doesn't change with the change of

scale. Self-similar description of the system is valid only at certain scales of rmin and rmax, at which the fractal is

constructed.

Let's examine Fig. 2. The Curve 1 correlates to Serpinski's carpet shown in Fig. 1. At the scales less than rmin,

the object is trivial and at the scale exceeding rmax, the final size of the object becomes apparent. If the

construction procedure is repeated infinite number of times for the object of infinity size, the obtained object

will be self-similar at any scale (Curve 2). When studying physical systems and their models we come across

the statistical fractals, i.e. the clusters for which the dependence (1) is done statistically on a certain segment

of scales.

Fig. 2. Examples of the fractal description of the system.

The graph is based on Fig. 2 from [15]. See the legend from the text.

Besides the dependencies (1)-(3), for determining df it is possible to use the dependence

Rg N1/d

f (4).

where Rg is radius of inertia of a cluster with the N particles.

The auto-correlation function C(r) is also widely used:

C(r) = <(r’) × (r’+r)> rd

f-d

(5).

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where (r’) and (r’+r) are the probabilities that, in the points r’ and r’+r a composite particle of the cluster is

located (the probability equals 1 and 0 in the presence and in the absence of the particle, respectively). All

other methods of determining df, as a rule, can be reduced to dependencies (1), (4) and (5). The advantages

and disadvantages of each method exemplified by modeling are discussed in reference [27].

SCALING APPROACH. UNIVERSALITY

The concept of scaling (scale invariance) is quite different in the works by the researchers belonging to

different schools. We basically share the point of view of de Gennes [28] who thinks that as much as possible,

it is important to disengage oneself from the details of the system's structure under consideration and to

single out simple universality features which are characteristic for the wide class of systems. Scaling approach

can determine only some asymptotics, the application of which, taking into account the specifics of the system

should be analyzed for each case. De Gennes outlined a new level in studies of polymers, just as how it was

done before by Flory [29]. Scaling characteristics (scaling indices) can be exemplified by fractal dimension

which, as shown for regular fractals, depends only on the object formation mechanism and determines the

global structure of a system - distribution of mass, depending on the scale.

The hypothesis of scaling is closely connected with the hypothesis of universality, although, in more strict

sense, they should be considered independent assumptions [30]. The essence of the hypothesis of universality

is as follows. If the same limiting conditions (interaction of the parts of the system) are characteristic for the

mechanism of various systems formation, these systems fall in the same universality class of physical

phenomena. Such a class is characterized by a certain set of scaling constants (indexes). The fact that

universality classes include not only physical systems (sometimes, such of them which can be referred to

various fields of natural sciences) but also theoretical models, provides an opportunity of wide application of

computer simulation (so called computer experiments). It should be noted that various universality classes can

have a particular form of scaling laws - various types of equations describing the system, the equation

connecting the scaling constants.

When analyzing more complex systems, one can note that under one scales, one mechanism of the formation

or interaction of the parts of a system is valid, while under other scales - the other mechanism. In this case,

when the scale is changed, there is a cross-over of a system from one universality class into another. Curves 1

and 2 in Fig. 2 illustrate the systems during the formation of which only one formation mechanism exists.

Curve 3 reflects the system in which under the scale rc transition from one class of universality into another

takes place. If the scale region of such a transition is wide (the universality class is not constant in any interval)

as shown on Curve 4, then the description of the system by the index df is not effective.

Let's consider a number of universality classes which, in our view, bear a direct relationship to the description

of the structure and processes in which colloids and polymers (including wood components) take part.

Basically, the theoretical models will be analyzed, because in this case, the limiting conditions of the

mechanism of the formation and interaction of particles are clearly evident, and their physical analogues will

be considered. When speaking of simulation, we will be aware only of computer experiments. When choosing

the material to illustrate it, the preference will be given to two-dimensional variants of models, because the

objects with df≥2, to which, in general, three-dimensional representations belong, give dense images when

projected on the plane. Depending on the size of the space where simulation or physical experiments take

place, scaling indices are designated as follows: df is marked as df(2) and df(3) for d=2 and d=3, respectively.

For more profound studies on scaling approach in the area of polymers, the reader may refer to [15,16,31 and

32].

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CLUSTER MODELS

In 1981, Witten and Sander developed a model of diffusion-limited aggregation of particles on the cluster (DLA

P-Cl) [33,34]. Let's assume that a particle is in space. Far from it, another particle appears performing

Brownian movement. When it meets the first particle, it aggregates to it, the aggregation being irreversible.

Then, in the distance, the next particle appears which also diffuses and when touching with the first two

particles, aggregates to them. This process is repeated many times. Some particles during wandering don't

touch the growing cluster and diffuse infinitely far from it. These particles are not take into account in the

future.

Such a cluster is shown in Fig. 3. In order to obtain this cluster, the Monte Carlo method realized by computer

was used. With this method, the whole set of events was imitated in details (Brownian movement is given by a

generator of random numbers). In [35] it has been found that df(2)=1.71±0.05, df(3)=2.51±0.06, and a

hypothesis has been put forward that df =5d/6. This evaluation, after more detailed measurements, can be

accepted only as approximate [17]. Why do such structures appear in the model DLA P-Cl ? The particle which

makes Brownian movement, most probably, at first touches the branches of a cluster increasing its size, the

probability of its penetrating inside the cluster which would cause the formation of a compact structure with

df=d being infinitely small. Thus, the branches of the cluster screen its inner structure as if freezing it. It should

be noted that it is convenient to consider the growth of such a cluster also as a phase transition between the

finite and infinite clusters. Such a formalization allows to compare DLA P-Cl with other models of formation of

infinite systems (polymer gels) which will be analyzed further in this article. Let's consider several other scaling

characteristics which are necessary and convenient for performing the analysis.

The chemical dimension dl: the distance between two points of a cluster is represented not as a geometric r

but as so called "chemical distance" l (the shortest distance over the particles of a cluster between these two

points). For calculating a number of particles of the cluster N, which are located within the chemical distance l,

it is possible to use a relation similar to equation (1):

N l d

l .

The geometrical and chemical distances are connected by so-called minimal dimension dmin:

l r d

min ; dmin df / dl .

From the statement about the l ≥ r, dl ≤ df and dmin ≥ 1. From the statement about the connectedness of

cluster it follows that dl ≥ 1 and dmin ≤ df. Dimensions dl and dmin are introduced in the studies of clusters by the

authors of works [36,37]. Sometimes dl is called spreading dimension or intrinsic dimension. The latter

emphasizes that dl reflects the intrinsic connection of the particles in clusters unlike df which shows how a

given cluster is located in certain space. The dmin value can be considered as a degree of self-similarity when

embeding a cluster in space. As has been shown in [13], dmin is characterized also by the ability of clusters to

self-similar swelling when r tends to l, and df to dl. The closer is the dmin value to one, the less is the ability of a

cluster for self-similar swelling. For the model DLA P-Cl, it has been found [38] that dl = df and dmin = 1, where

d=2 and d=3. Therefore, the DLA P-Cl clusters are not capable of self-similarity when they are embeded in

space.

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Fig. 3. Model of a cluster (10,000 particles) formed in the process of

diffusion-limited particle-cluster aggregation (DLA P-Cl) [35].

The spectral dimension ds: suppose there is some point (scout) on the cluster under consideration, which

performs some random wondering over this cluster moving from one neighboring particle to another. This

wandering is characterized by fractal dimension dw which is different from df of the cluster. In [39] it has been

suggested to regard the relation ds = 2df / dw, where ds - is so-called spectral or fracton dimension which

reflects the topology and inner binding of a cluster. For DLA P-Cl it has been found [40] that ds = 1.2...1.4,

where d=2 and d=3, but as the authors note, one can suggest that ds(2) < ds(3). More accurate calculation [41]

give ds(2)=1.205±0.018.

The conditions of cluster growth in the model DLA P-Cl are isotropic because all the directions of diffusion and

joining are equally probably. However, when regarding the general form of clusters of the finite size, one can

find asymmetrical bodies [42,43], differing from the circle (when d=2) or from the sphere (d=3) , which can be

conveniently approximated by an ellipse (d=2) or ellipsoid (d=3). This asymmetry is changed, depending on the

size of a cluster.

Indices for the description of the asymmetry of cluster form and its changes were suggested independently by

Ozol'-Kalnin and coauthors [44,45] and Family with coauthors [46]. Asymmetry is described by the following

scaling procedure:

A - AN N- ,

where A and AN - degree of asymmetry of the finite cluster (with N particles) and infinite cluster, respectively.

The value A is relation between the components of the square of radius of inertia along main axes of rotation.

The index shows the rate of convergence AN to the extreme value A. In [45], it was suggested DN also

describe the changes of dispersion of asymmetry by ensemble of clusters with help of the index ϰ:

DN N-ϰ

.

The clusters of the model DLA P-Cl become symmetrical as they growth (A(2)= A(3)=1) [42,43], and as has

been established in (2)=0.32±0.05.

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In 1961, Eden suggested a very simple model of simulating the growth of tumor cells colony [47]. The

attachment of each of the following particles in any place on the surface of a growing cluster is equally

possible. Then, with each step of simulation, any point of the surface can be chosen, and to this point the

following particle is attached, the history of its movement in space being ignored. Various types of Eden's

model have been regarded in [48]. With such a process, each point of the surface will be occupied, sooner or

later, which will lead to the formation of a dense cluster with df=d. The dimension of the surface of such a

cluster is also trivial [49] (dsurf=d-1), but the description of the structure of this surface requires a more

complex scaling device [45,48]. The cluster growing in isotropic conditions, similar to the cluster DLA P-Cl, also

tends to a symmetrical form, as the infinity is approached (A=1) [44,45]. The finite clusters are asymmetrical,

and, as it has been found by the authors of [45], =2/3d and ϰ=2. The value of for the Eden's model

coincides with the value of for the model DLA P-Cl, where d=2, which is quite unexpected, since for Eden's

model, the value of is connected with the scaling characteristics of the surface structure [45]. The fact that

ϰ>0, suggests that the dispersion of asymmetry is decreased, as the clusters growth.

The clusters with df =d are also formed if in the model DLA P-Cl the Brownian trajectory is replaced with a

straight-line trajectory. This is so called Vold-Sutherland model (this is discussed more comprehensively in

[27,50]).

The Vold-Sutherland and Eden model clusters are compact formations, and the wandering in such a medium,

irrespective of the d value, are characterized by the equality dw=2. Then ds=2df/dw=df=d. Self-similar embeding

of such clusters in space is impossible (dmin=1, and dl=df=d).

When interpreting the physical processes, it is of interest to estimate the affect of various factors on the

clusters formation mechanism. How are the length scales corresponding to various universality classes

changed in this case? Thus, for the model DLA P-Cl, it is possible to introduce joining probability coefficient p,

which determines a possibility of attaching the wondering particle to the cluster when they touch each other.

With the probability 1-p, a particle jumps off and continues wondering. With the help of coefficient p, it is

possible to describe chemical activity and other similar properties of particles. As it was established in [34,35],

where p<1 at small scales (r<rc), a dense structure is formed with df=d which transforms into loose structure of

the model DLA P-Cl when the scale is increased (r>rc). A cross-over between the universality classes of Eden's

model and the DLA P-Cl model takes place. The scale of this cross-over is inversely proportional to the joining

probability: rc 1/p [34]. At the infinitely small p value, the cluster with df=d is formed for all the scales of

lengths. In [50], a situation has been simulated, when directed external field influences the Brownian

wandering of particles. This cause a cross-over from the Brownian to directed movement. Then the growing

cluster at small scales corresponds to the model DLA P-Cl, and at large scales to the Vold-Sutherland model.

The scale of the last cross-over is proportional to the scale of the cross-over of particles movement

mechanism.

In the initial model DLA P-Cl, the concentration of wandering particles isn't considered. However, it is

interesting to know whether the process of aggregation depends on the concentration of these particles c (the

initial concentration is c0) in same reactor, i.e. in finite space. In this case, it is assumed that only one of the

particles can serve as a nucleus of the growing immovable cluster. If during the growth of the cluster its

density is decreased down to the density of the surrounding medium (c in the given moment), then a cross-

over between the DLA P-Cl and Eden's universality classes takes places, and further the cluster grows as a

uniformly dense object. The scale of such a cross-over depends on the initial concentrations of the particles

rc c0-d/(d-df)

, where df corresponds to the model DLA P-Cl [51]. Simultaneous growth of several clusters in the

reactor was also simulated with the initial concentration of particles c0 [52]. The cluster during their growth

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use all the particles occurring in the reactor, after which their growth is stoped but the intergrowing of clusters

together doesn't take place. As it has been mentioned in [52], the mechanism responsible for the formation of

the DLA P-Cl structure provides also the screening from penetrating the wandering particles in the areas

where clusters intergrowing is possible. In order to obtain infinite systems, the rotation or diffusion of the

growing clusters themselves is necessary. If the initial concentration of particles is large enough, then, as in

the forementioned instance [51], the cross-over from the DLA P-Cl model to Eden's model takes place during

the growth of these clusters.

A surface or an object, for instance thread [53,54], can appear in the role of a nucleus or embryo in the model

DLA P-Cl. In such a process, a more complex structure - "the forest" is formed (Fig. 4), the "trees" of which

remind of the clusters parts in the model DLA P-CL. In the process of the forest growth, only the biggest trees

survive which screen smaller trees from wandering particles. Such trees have the same fractal dimensions as

the clusters of the model DLA P-CL with the nucleus as one particle [54], i.e. they belong to the same

universality class.

Fig. 4. Clusters of diffusion limited sedimentation of particles on the boundary of dimensional space [53].

A whole range of the objects were found for which the structure corresponding to the universality class DLA P-

Cl was determined experimentally [17,21,22,55,56]. One of such structures is shown in Fig. 5.

Recently, the attention of researchers has been attached by an experimentally discovered phenomena - so-

called "viscous fingers" (see: reviews [21,22,57]). Finger-like clusters obtained under certain conditions also

belongs to the universality class DLA P-Cl. Unlike other types of clusters, fingers-like clusters grows not due to

"obtaining" the material from the surrounding medium, but due to the redistribution of the material which is

supplied under pressure into the center of the cluster. The DLA P-Cl type structure can be found also when the

break-down of dielectrics takes place [21,22], in the case when as if there is no diffusion. The common thread

between the above processes is the fact that the clusters growth process is described by the Laplas's diffusion

equation [21,22], i.e. they enter a more general universality class, where the mechanisms described are the

expression of a more universal law.

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Fig. 5. Polypyrrole clusters (df(2)=1.74±0.03; ds(2)=1.26±0.04; the determined on the scales from 0.3 to 6 mm),

obtained by electrooxidation of neutral monomers in two dimensional space [56].

As has been mentioned in [20], the model DLA P-Cl doesn't allow to explain the formation of colloids and

aerosols. Their growth is explained by the cluster-cluster aggregation model. According to this model, the

number of clusters in the systems, due to aggregation, is decreasing, and their dimensions are increasing. For

the first time, the cluster-cluster aggregation was modeled by Sutherland [58]. He discovered the formation of

loose, branch-like aggregates similar to the soles which were observed in electron microscope. This looseness

could not be characterized quantitatively by the author because he didn't use the scaling approach. Modeling

of the cluster-cluster aggregation was performed independently also by Meakin [59] and by a group of

authors: Kolb, Botet and Jullien [60]. In two dimensional space, where c0 =0.03…0.15, both particles and the

growing clusters were moving according to Brownian trajectories, i.e. diffusion- limited cluster-cluster

aggregation (DLA Cl-Cl) took place. In the end the only one cluster was formed which absorbed all the particles

(Fig. 6). In various studies and publications, somewhat differing values of df have been obtained which is

accounted for by the finity of the sizes of simulating systems. We find the values df(2)=1.40...1.45 and

df(3)=1.75...1.80 (for more detailed information see: review [17]) to be the most substantiated ones. The

cluster of the model DLA Cl-Cl are more porous than the clusters of the model DLA P-Cl which allows them to

screen their cores to a greater extent.

Of interest is the problem how the structure of the model DLA Cl-Cl cluster is influenced by the correlation

between diffusion coefficient and the mass of clusters. If diffusion coefficient is constant or decreases as the

mass increases, the value df doesn't change [59,60], if the coefficient increases with the increase of the mass,

a cross-over to the model DLA P-Cl [59] takes place. The last- mentioned variant has no physical interpretation

that can be applied to colloids and polymers. In [61], a situation is described when the joining point between

two clusters serves in the future as if a joint around which the parts of a newly formed cluster are rotating. As

a result, the density of clusters is increase only under small scales, and minor increase of the df value under

large scales doesn't not permit us to indicate a new universality class different from DLA Cl-Cl. A variant of this

model was simulated [62], according to which the wandering particles and clusters, when touching the surface

of the reaction vessel deposit on it. Hereafter, all these immovable clusters are capable of joining the

wandering particles and clusters. The df values of the deposit clusters remain the same as in the usual process

DLA Cl-Cl.

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Fig. 6. The stages of the diffusion-limited cluster-cluster aggregation process (c0 =0.06):

a - 86 clusters; b - 8 clusters, c - 1 cluster [60].

In [63], it is pointed out that in many physical processes the most real is a situation when only rare collisions of

clusters lead to their connecting. Such a situation can occur when a chemical bound is formed. Hence, the

name of this process chemically-limited cluster-cluster aggregation (CLA Cl-Cl) (in other works and studies

reactive-limited aggregation). Under infinitely small probability that clusters, when colliding, become

connected, all the variants of the connection of two clusters are equally probable. When simulating in the

process CLA Cl-Cl a physically real situation when the clusters which become connected can be of various sizes,

the values are obtained which are df(2)=1.59±0.01 and df(3)=2.11±0.03 [64]. It was noted [63], that under CLA

Cl-Cl, the probability of interpenetration of clusters is higher than under DLA Cl-Cl, and this probability

increases under various sizes of clusters [64]. If the probability of connection has some finite value (but less

than one), then with the growth of the cluster, a cross-over from CLA Cl-Cl to DLA Cl-Cl takes places [65], and

the scale of this transition is increased with the decrease of the value . Under the finite probability of

connection, large clusters as if cannot "check" all the possible places of connecting with each other.

In many cases, the process of aggregation competes with the process of splitting when the clusters is

reconstructed during its growth. For simulating this effect [66], the cluster- cluster aggregation model was

supplemented with a condition that with some probability, any bond in clusters can be break-down. At first,

the process of clusters aggregation is prevails but after some time, dynamical equilibrium between

aggregation and dissociation sets in the system. At certain point in time, the number of the formed bonds

equals the number of broken ones, and a group of interacting clusters exists in the system. It appears that

irrespective of the presence or the absence of cycles in the clusters and also the fact whether the DLA Cl-Cl or

CLA Cl-Cl model was modified, df(2)=1.57±0.06 and df(3)=2.03±0.05. Another variant of clusters reorganization

is presented in [67]. A particle is split off from the surface of a clusters, but it doesn't result in cluster cleavage.

The particle performs Brownian movement and when touching the cluster, it connects to the latter in another

place. After reaching the dynamic equilibrium df(2)=1.54±0.08, irrespective of the type of initial cluster, the

role of which being played by a compact cluster (df(2)=2), a DLA P-Cl with (df(2)=1.70) and a line of particles

(df(2)=1).

Under the cluster-cluster aggregation as if two extreme cases are revealed: one, when the limiting factor is

diffusion (DLA Cl-Cl), and the other, when clusters have df(2)=1.55...1.60 and df(3)=2.0...2.1. The ways of

forming of such clusters can be different. The question whether these of clusters belong to one universality

class is not settled yet and requires further research [68]. So far, their formation can be represented as a

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limiting case CLA Cl-Cl. Such a combination is supported by experimental data [69,70,71] (Fig. 7). During the

experiments, reorganization of clusters was also observed [70].

Fig. 7. Transmission electron microphotographies of colloidal gold aggregates [69]:

a - under fast aggregation (DLA Cl-Cl, df(3)=1.77±0.05); b - under slow aggregation (CLA Cl-Cl, df(3)=2.05±0.05).

The value df(3) is determined by light scattering method.

During simulation, the cluster-cluster aggregation model requires large computer resources. Therefore, for the

time being, there are not enough data on other properties of these clusters besides the value df. In this

connection, several important works should be mentioned. For the model DLA Cl-Cl, df(2)=1.22±0.02 and

df(3)=1.42±0.02 (respectively dmin(2)=1.15±0.04 and dmin(3)=1.25±0.05) have been mentioned [38]. Since

dmin1, it can be said that the parts of this cluster are self-similarly built in the space unlike the parts of the

cluster DLA P-Cl. Since the mechanisms of cluster-cluster aggregation to various types are similar, dmin1 can

be also expected for the CLA Cl-Cl. Based on the assumption that the cycles don't influence the structure

classes DLA Cl-Cl, for such clusters it was estimated: ds(2)≤1.2 and ds(3)≤1.26 [72]. In [43] it was shown that

asymmetry of the general form of clusters both for DLA Cl-CL and CLA Cl-Cl, doesn't depend on their size and is

constant (AN = A < 1), therefore = 0 and ϰ = 0.

The effect of the initial concentration of particles c0 on DLA Cl-Cl was evaluated [73,74]. The process of

aggregation goes through three stages. At the first stage, the common process DLA Cl-Cl takes places, and the

clusters density is decreased. When = c0, the next stage begins, at which a gel is formed with df=d. If a

system has finite size, then after a time, a cluster is formed which is located throughout the whole space. At

the third stage, the finite clusters remaining in the system join the immovable cluster. In the infinite system, as

the author recognize [73,74], gel time also becomes infinite.

In the conclusion of the discussion of cluster-cluster aggregation, it should be noted that in many works, the

equation of Smoluhovski is used for description of this process. The volume of this paper doesn't permit us to

consider this problem and the problem related to it, such as interpretation of time, and the clusters

distribution by size. The reader is advised to refer to the review [17].

13

Fractal clusters, as it was noted by Sander [19], grow under the conditions which are far from equilibrium. The

time necessary for achieving equilibrium is "impermissible" luxury. The value df reflects the mechanism of

structures formation. The models discussed are called kinetic. A number of so-called equilibrium models exist

in which certain order of interaction of particles of the system under equilibrium conditions is expected. In

these case, the prehistory of the system formation is neglected. The equilibrium models are studied much

better than the kinetic ones.

In the studies on the formation and structure of polymer gels, much hope is pinned on percolation theory

[15,16,28,31,75], introduced in 1957 [76]. For instance, let's consider the percolation by sites on the square

lattice (Fig. 8.). Let's assume that, with the probability p, a site on the lattice is occupied by a particle, and

particles in the adjacent sites are interconnected. A group of such interconnected particles form a cluster.

Then, some minimal, critical value of this probability pc exists at which a cluster appears connecting the

opposite sides of the lattice. For this particular case pc=0.59. If me imagine the cluster is like a system of

canals, then at pc the liquid can percolate through the lattice (hence, the name of the theory). At pc, the initial

infinite cluster (IIC) is formed on the 1infinite lattice. The phase transition sol-gel takes place (between the set

of finite clusters and IIC). At p>pc, the remaining clusters join to the infinite cluster, but the latter now cannot

be consider to be an IIC. It is possible also to suggest the other models of percolation, for instance, along the

sides of the lattice which can be interpreted as the bonds between monomers. Similar picture takes place

though the value pc will be different. More thorough discussion of percolation has been presented

[15,16,18,31,77].

Fig. 8. Percolation by sites on square lattice (2020), pc=0.59.

Dark dots - particles of the percolation clusters; white dots - particles of finite clusters.

The scaling approach permits to study, firstly the changes of the properties of a system when it approach pc,

i.e. the phase transition, and secondly, the structure of IIC. The latter is characterized by the following scaling

constants: df(2)=91/48 1.89, df(3) 2.50 [15,18]; dl(2)=1.64±0.02, dl(3)=1.83±0.02 [36]; dmin(2)=1.16±0.02,

dmin(3)=1.37±0.02; ds(2) ds(3) 1.33 [18].

14

The general shape of IIC is asymmetrical and has A(2) 0.4, (2) 4.47 and ϰ(2) = 0 [46]. These

characteristics are universal and don't depend on details of model.

The structures in the percolation model can be characterized also at p different from pc. Such clusters, at the

scales less than correlation lengths , have a structure of IIC (Fig. 9.). If p > pc, then at the scale r = , a cross-

ver to the uniformly connected structure takes place (df = dl = ds = d ). At p < pc and r = a cross-over to the

set of clusters which are referred to the universality class of "lattice animals", takes place. The value is

determined by scaling law:

(p - pc)-

,

where (2)= 4/3 and (3)=0.88 [18]. In this case, the distance p - pc plays the role of scale variable.

Fig. 9. Changes of the percolation clusters structure, depending on the p value and the scale L [15].

A - "lattice animals" set; Б - dense cluster; B - IIC structure.

Gelation can be considered as a special type of percolation [31,75]. The classical theory of gelation was

proposed Flory and Stockmayer [29]. According to this theory, Bethe lattice, called also Caylee tree,

characterized by branching degree corresponding to particle functionality, plays the role of space (Fig. 10).

The formation of cycles is prohibited. IIC is formed when pc = 1 / ( -1) and has df =4 [18], dl =2, dmin =2 [36,37],

ds =4/3 [39]. This model coincides with the model of percolation if for later d ≥ 6 [18,28,31]. On the attempt to

transfer IIC from Bethe lattice into the space with d < 6, the condition of excluded volume is violated (the

particles of IIC will be spacely overlapped). Both theories of gelation have been discussed [31], where it has

been pointed out, that the results of the experimental studies on sol-gel phase transition are closer to those

predicted by the theory of percolation than to those the predicted by the classical theory. The discussion of

this problem is still in progress [28,51,75].

The so-called "lattice animals" [15,16], which are the ensemble of clusters where any configuration of a given

size is equally possible, can be the model of branched polymer in diluted solution. The scaling characteristics

are related only to the average clusters in the ensemble and not to the individual ones. When simulating, it

was found df(2)=1.55±0.05, df(3)=2.00±0.05; dl(2)=1.33±0.05, dl(3)=1.45±0.05; dmin(2)=1.17±10.05,

dmin(3)=1.38±0.05; ds(2)=1.12±0.05, ds(3)=1.19±0.05 [78]. The average form in the ensemble is characterized

15

by asymmetry with A(2) 0.29, (2)=1.01±0.04, ϰ(2)=0 [46]. There is also a point of view that the universality

class of lattice animals cannot be identified as general for all the types of branched polymers [79]. The kinetic

process of polymer formation affects the probability of the appearance of various topological structures, i.e. in

the equilibrium state all the configurations cannot be equally probable. The value df for "lattice animals" is

very close to the value df in the extreme instance of CLA Cl-Cl. The question of combining these models in the

one universality class (in the asymptotic limit of infinitely large clusters) is still open [68].

Fig. 10. Percolation on the Bethe lattice ( =3, pc =0.5 1).

Dark dots - particles of the percolation clusters, white dots - particles of finite clusters.

Nowadays, simulation of chain polymers is a separate branch of the theoretical physics of polymers, the

present state of this problem being discussed [80].

When studying various kinetic and equilibrium models, we didn't pay attention to such important detail of

polymer chemistry as functionality of particles (of monomers) . If ≥ 3, then the value doesn't affect the

mechanism of cluster formation (the order of clusters interaction), and then the universality class of a model

doesn't change. If =2, then chain clusters or cycle clusters are formed. In the intermediate case, when both

particles with =2 and ≥3, are presented in the system, the question of structure is vexed. An oligomer that is

large enough has ≥3, but the mechanism of formation (the order of interaction) of some models doesn't lead

to full realization of functionality. In such case, for a given model there is a certain concentration of particles

with ≥3, below which cross-over to another universality class takes place. It was noted [45] that this effect is

possible in Eden's model.

In conclusion, we can note that both simulation and experimental approaches in the area under study are

developing vigorously, and the number of papers and articles with this problem has increased like an

avalanche.

16

LIGNIN STRUCTURE ACCORDING TO THE FLORY-STOCKMAYER THEORY

It was natural that the first step in the description of lignin lattice (gel) was the use of the Flory-Stockmayer

theory [29]. When calculating according to Flory-Stockmayer, only the data on the functionality of monomer

(in the case of lignin, - the number of phenylpropane units connected with a monomer unit) and the degree

of conversion α (i.e. the fraction of the realized intermolecular bonds) are required. Szabo and Goring [81]

regarded lignin destruction during pulping processes as a reverse case of the Flory-Stockmayer theory of

gelation, if =3. It was assumed that at the zero degree of destruction (time t=0), sol-fraction in the cell wall is

absent, and two different gels coexist simultaneously, - the gel of the secondary layer S and the gel of the

complex middle lamella ML. When t=0, the degree of conversion αML = αS =1. It has been possible to explain

satisfactorily the topochemical effect by the destruction of two gels. It should be mentioned that Szabo and

Goring [81] used kinetic (t) conceptions of gelation, besides statistical ones (α). The kinetic conceptions are

emphasized when describing network formation in synthetic polymers [82]. Bolker and Brenner [83] used the

Flory-Stockmayer theory for studying lignin structure on the basis that the structure is cross-linked and consist

in the linear chains in which approximately, for each of the 18 phenylpropane units, there is one cross

benzylethere bond.

In mathematical description of polymer networks, it is very convenient to use the apparatus of producing

functions of probabilities (PFP) of the theory of branching processes [84]. For cross-linking, the cascade

structures are created - macromolecules which can be presented as a tree-like graph. When examing this

molecular forest, time sequence is replaced by the reaction completion sequence α [85].

Well-known diagrams of lignin fragments were presented as a tree-like graph [86]. Calculations with the use of

PFP have shown that the branching of a graph can proceed infinitely, i.e. α in the diagrams reaches the critical

value αc, and a gel is formed. The calculated molecular mass of the sol fraction coincides with the molecular

mass of Braun's lignin which can be arbitrarily considered as sol fraction of lignin in situ. Calculations of various

characteristics of lignin, for instance, the number of elasticity active chains and their average lengths were

performed with the use PFP [87]. We have to remind that the point of a network is active if it has not less than

three network-forming bonds, by which the bonds coming from a certain point by non-intersecting pathways

and reaching the boundaries of the sample are implied. A segment of the chain between two active points

corresponds to the elastically active chain. It is precisely their number that governs swelling as well as

mechanical, relaxation, etc. properties of network polymers. Experimental data on these parameters of lignin

networks are not available.

The results obtained with the theory of branching processes coincide with the results obtained with Flory-

Stockmayer theory. Naturally, the assumptions and drawbacks of both approaches are also similar.

At present, Yan and coauthors [88,89] use Flory-Stockmayer theory combining it with the formal kinetics of

lignin degradation during various sulphate pulping processes. Sulphite pulping is discussed in the work by

Berry and Bolker [90]. According to Yan and coauthors, the process of delignification includes three kinetically

distinct stages (Fig. 11): gel degradation with the formation of sol, reverse condensation of sol and post-

hydrolysis of bonds with the formation of low-temperature fragments so-called "ω-lignins". Batch

delignifications includes all the stages, and the use of a flow-through reactor permits to perform pulping,

without the second stage complicated calculations. Bonds cleavage processes in lignin are described by the

equation

α = Σ αi e-k

it + α ,

17

where αi - the fraction of the i-type bonds in the initial moment,

ki - reaction rate constant of the i-type bonds cleavage;

α - fraction of non-cleavage bonds of lignin.

Fig. 11. Kinetic diagrams of lignin gel cleavage [89]: A - gel degradation with the formation of sol;

B - reverse condensation of sol; C - post-hydrolysis of bonds with the formation of ω-lignins.

Altogether, three types of bonds were taken into consideration by the authors. In the above mentioned works,

Flory-Stockmayer theory was modified in such a way that the results of calculations would coincide with

experimental data - the degree of delignification, depending on time of pulping and molecular mass

characteristics of the soluble fragments of lignin (sol-fractions) The calculated data [88] agree satisfactory with

the experimental data on sulphite pulping obtained by Kleinert (Fig. 12).

Fig. 12. The degree of delignification Wg depending on time [88] during the process at temperatures 170oC (1),

180oC (2), 185

oC (3). The curves- calculation; the points - the experimental data obtained by Kleinert.

18

Lignins characteristics were studied also on the basis of the hydrodynamic data on lignin solutions under θ-

conditions or close to them [91-93]. In general, the concept of g-factor, which is relations of the mean squares

of radius of inertia of branched and linear macromolecules with the same molecular mass was used. From the

latest data [93], the fragment of lignin is described better if the branched phenylpropane units are assumed to

have an average =4. In any case, all the results of the studies of solutions support the theory of branching of

lignins fragments.

LIGNIN AS A FRACTAL OBJECT

The comparison of limiting conditions of the construction of cluster models and mechanisms of the synthesis

in vitro of dehydropolymers (DHP) of high-molecular weight models of lignin allowed to suggest a hypothesis

that they correspond to definite universal classes [13]. With the many time repeated procedure of adding the

substrates [1], when the so-called end-wise structure of DHP is formed [94], the following conditions are

realized. The concentration of free phenoxyl radicals in buffer solution is low, polymerization proceeds by

adding the radicals to the already existing growing macromolecules, and not by forming new centers of

polymerization (hence, the term "end-wise" polymer).the reactivity of free phenylpropane radicals are high:

even by application of the EPR method it was impossible to determined half-lives of the radicals of guaiacyl

units in aqueous medium [95]. Consequently, diffusion is a limiting factor when end-wise structures are

formed, and the process of end-wise DHP growth correspond to the DLA P-Cl process: addition of separate

DHP particles to the cluster, Brownian trajectory of particles movement, and the finite probability of their

connection with the cluster when they join.

With single addition of substrates [1], the so-called bulk-structure of DHP is formed [94], which can be

explained by a relatively high concentration of free radicals. The growth of DHP macromolecules takes place at

once in many centers of polymerization, and different variants of the recombinations of intermediate products

are also possible (hence, the term "bulk-structure"). Therefore, the limiting conditions of cluster-cluster

aggregation correspond more to the formation of DHP bulk-structure. However, which of the limits DLA Cl-CL

(limiting conditions - diffusion of clusters) or CLA Cl-Cl (limiting conditions - formation of chemical bonds or

reversibility of aggregation) is preferable still is not clear. Obviously, a cross-over between these two limits

takes place with the increase of the scale. The established correlations allow to make a conclusion on the

relationship of the properties of end-wise and bulk DHP depending on the scale [13]. The end-wise-polymer in

the scaling understanding is more compact than the bulk-polymer ( dfew

> dfb

). Bulk-polymers is capable of

self-similar swelling ( dminb >1 ), while end-wise-polymer is incapable of such swelling (dmin

ew =1 ). According to

the latest data [43], one can conclude that the end-wise-polymer, during its growth, tends to the symmetry of

the form (θew

< 1 and Aew

=1), while with bulk polymerization, clusters are characterized by the constant

asymmetry in the process of growth ( θb = 1 and A

b1 ).

It is known that the lignin middle lamella ML corresponds more to the end-wise structure, while the lignin of

the secondary layer S - to the bulk-polymer structure [96]. It should be noted that a direct application of the

conceptions of end-wise- and bulk-structures of DHP formation in vitro onto lignin in situ is hardly appropriate.

Diffusion in cell wall is strongly hampered by the presence of cellulose and hemicelluloses. Under such

conditions, the diffusion of clusters in bulk-polymerization is hardly probable. The mechanism of end-wise-

polymerization both in ML and the layer S seems to be more realistic.

One should note the interesting work by Shumilin [97], in which the synthesis of end-wise- and bulk-structures

is examined from different positions. The drawback of this work, however, is the fact that the author uses the

19

concepts of end-wise and bulk structures in somehow different sense than it is accepted in the chemistry of

lignins.

At small scales (especially during the synthesis of an end-wise-structure) the appearance of dense clusters (in

the scaling sense) with df = dl = d, i.e. the clusters falling into universal class of the Eden model is possible [13].

In order to realize the Eden model, the reactivity of free radicals depending on their concentration and

stability, has to be low. The concentration of radicals in physiological conditions is low, and the reactivity of

syringyl units is also not very high. By application of the EPR method, half-lives of syringyl radicals have been

determined to be as high as 30 minutes [98], which indicates their low reactivity. Hence, the Eden model is

more suitable for describing hardwood lignins, because the specific weight of their syringyl units is higher than

that of soft-wood lignins. On the other hand, syringyl units yield less branching than guaiacyl units which

doesn't promote the realization of the Eden models.

In the number of works [99-102] gradual growth of the clusters consisting of 50 two- and three-functional

units was modeled by Monte-Carlo method. An increase in the share of two-functional units leads to a

decrease in the reactivity of the formed clusters, and to an increase in the asymmetry of their form and that of

porosity. The number of cycles is proportionate to the share of three-functional units. The distribution of

cycles, depending on their sizes has definitely decreasing nature. The cycles of small lengths connected with

each other, prevail, forming an interconnected system. With an increase in the share of two-functional units,

the share of longer cycles and inhomogeneity of clusters cross-linking, are also increased. These results

indicate that the correlation between the share of two- and three-functional units affect the properties of

clusters. Apparently, these tendencies can be characteristic both for small size and large size clusters.

We consider the experimental data indicating the presence of fractal non-trivial structure in lignin

preparations, to be especially important [14]. In order to approximate the form of macromolecules of isolated

fragments of lignin, the dependence of intrinsic viscosity [], diffusion coefficient D and sedimentation

constant s on their molecular mass M is used [4]:

[] Ma ; D M

-b ; s M

c .

In fact, these dependences are scaling by nature, and as shown by the authors of paper [14], the exponents a

and b can be converted to fractal dimension df (Table 1). The absence of the data on the accuracy of factors a

and b determination does not permit to draw simple conclusions about the correspondence between the

structures of these fragments and the structure of the objects of certain universal classes. One would note the

following: the value df for slightly decomposed lignin preparations does not contradict the hypothesis of its

correspondence with the universal class of DLA P-Cl. Low df values for lignosulphonates reflect a structure

which is formed during pulping as a result of degradation processes and reverse condensation rather than the

initial structure of lignins in vivo, therefore, in this case, the df value is close to cluster-cluster aggregation

models. This is in agreement with a conclusion drawn by Goring [4] that sulphonate compounds of lignin have

a more sparse network. However these lignin preparations (Table 1) were obtained under the conditions of

strong destruction of the lignin network and can be considered, to a certain degree, altered.

Experimental data on the properties of the lignin fractions obtained by soft destruction at 70oC in solution (10

ml of HCl per 1000 ml of dioxane) obtained by Pla and Yan [103] are of paramount importance to substantiate

the interpretation of lignin structure in vivo as the universal class DLA P-Cl. The values of average mass degree

of polymerization <Xw> and average mass molecular mass <Mw> of lignin fractions were compared with the

value g' = []b,θ / []l,θ , determined by the viscosimetry method for correlation between intrinsic viscosities of

20

lignin macromolecules under θ-conditions and linear chains under the same <Xw> and <Mw>. Having used the

relation from the contribution by Bohdanescy [104], Pla and Yan found the relationship

1/g’ = A + B <Mw>1/2

.

Table 1. Fractal dimension values df of isolated preparations of lignin*.

However, the interpretation relation of Bohdanescy for defining the df value does not fit. According to de

Gennes [28], intrinsic viscosity [] R3 / N ,where R - macromolecules radius, N - a number of monomer unit.

Hence, using general expressions for df, the authors [105] have deduced the expression

g’ = C <Mw> 3/d

f-3/2

.

The interpretation of the data by Pla and Yan (Table 2) within the framework of the concept of fractal

dimensions results: df =2.439±0.007 (significance level 0.95). Under the approximation, the fraction P-23 has

not been taken into account. We have to note, however, that interpretation, as it is known, depends on the

proposed theoretical model. And, to our point of view, it was precisely the interpretation within framework of

the theory of branching processes, i.e. within framework of the Flory theory, that prevented Pla and Yan, who

obtained excellent experimental data, to come to the conclusions presented by the authors of this paper.

Determination of df values [14,105] can be considered only as the first step in the studies on the fractal

structure of lignin. Ti is also necessary to find the other experimental methods, which would help to determine

scaling characteristics of lignin both in vitro and in vivo. A comparison of the scaling characteristics of the lignin

from the different cell layers would be also of great interest. This would permit to judge not only the

macromolecular structure but also diffusion rate of lignin oligomers, the effect of hemicelluloses on their

diffusion, and possible penetration of mediators into the cell where a macromolecule of lignin has been

formed [14]. One can regard lignin distribution in the cell wall as in an integral system. Then the scaling indices

will characterize the structure on a globular and not monomer level.

In our opinion, the scaling approach is the most promising method of qualitatively and quantitative

description of the structure of network polymers (which is also lignin), on a supermolecular level.

Lignin preparation Solvent

Hydrotropic lignin Dioxane lignin Alkaline lignin Alkaline lignin Lignosulphonate amine Lignosulphonate -"- -"- -"- -"-

Dioxane Pyridine Dioxane 0.1M buffer Methanol Methanol 2M NaCl 0.5M NaCl 0.1M NaCl 0.02M NaCl

* Data on correlation exponent between intrinsic viscosity a and diffusion coefficient b are taken from the work [4].

21

Table 2. Factor g' (determined viscosimetrically) and

average mass degree of polymerization <Xw> of dioxane lignin

The scaling approach can be, undoubtedly, used to study not only lignin structure but also the other cell wall

components, as well and to determined some parameters (for example, the distribution of micropores) of cell

wall as a whole. We should be mention the work [106] by scientist who studied small-angle x-ray scattering of

native cellulose and come to a conclusion that it is a cluster aggregate, consisting of microcrystals with df < d.

Recently, the scaling approach was used when studying another biopolymer - melanin [107,108]. The fractal

characteristics help to obtain new information about such classical objects as proteins [109]. Even in the case.

when polymer chain is collapsed into a globula (on the whole, having trivial dimensions df = d =3, the data

have been obtained [110] offering a real insight into the phenomena. The scaling approach is widely used to

study the properties and dynamics of polymers and biopolymers formation and it also is applicable to study

cell wall biopolymers of plants.

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Fraction Fraction

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Received 25.12.1987

Translation re-prepared for distribution 11.03.2014

(Thank you for translation Ms. Irina Morozova.)