Untangling Complex Systems, page 58
Biological invasions, such as the spreading of insects, animals, seeds, and diseases into new ter-
ritories, and favored by intercontinental journeys, economic relationships and markets, and even the
circulation of ideas and fears among human communities fed by telecommunication technology, all
of those phenomena can be modelled by the theory of chemical waves and the Fisher-Kolmogorov
equation. Such theory is reasonable as long as the scale of the individual’s movement is small com-
pared with the size of the observations. There are several cases of monitored biological invasions.
A classic example is the spread of muskrat ( Ondatra zibethica) in Central Europe in the first half
of the twentieth century (Skellam 1951). The muskrat was brought to Europe from North America
for fur-breeding. In 1905, a few muskrats escaped from a farm near Prague. This small group
started multiplying, and in a few decades, the muskrats spread over the whole continental Europe
with a rate of ~3 km/year (Table 9.2).
Another example is the invasion of the gypsy moth ( Lymantria dispar) in North America. This insect
is thought to have been brought from France to a place near Boston by an amateur entomologist. When
it escaped around 1870, it spread at an estimated speed of ~10 km/year. By 1990, the whole Northwest
of the United States became heavily infested with resulting damages to the American agriculture.
Many more examples of biological invasions can be found in the book by Shigesada and Kawasaki
(1997), and in that by Petrovskii and Li (2006).
TRY EXERCISES 9.17 AND 9.18
9.8 LIESEGANG PATTERNS
Other exciting phenomena wherein chemical reactions coupled with diffusion originate stun-
ning patterns are the periodic precipitations discovered by the German chemist Raphael Eduard
Liesegang (1896). Some pictures of this impressive phenomenon are shown in Figure 9.27.
In the usual experimental setup, two strong electrolytes are needed. One is dissolved in a gel matrix
(the so-called inner electrolyte), whereas the other is poured onto the gel (outer electrolyte). The
concentration of the outer electrolyte is much larger than that of the inner one. The ions of the outer
electrolyte diffuse into the gel and react with the inner ions. An exchange reaction occurs and two new
salts form within the gel. One of the new two salts precipitates and forms a beautiful pattern. The
features of the pattern depend on the materials involved (included the gel) and the geometry of the
system. If the precipitation occurs in a test tube, we observe a sequence of solid bands (Figure 9.27); if it occurs in a Petri dish, a pattern of concentric rings emerges (see picture e of Figure 9.1).
286
Untangling Complex Systems
FIGURE 9.27 Periodic precipitation of Ag CrO in gelatin, within a test tube.
2
4
The shapes of the patterns can be described quantitatively by specifying the formation times
( tn) of the bands, their positions ( xn, measured from the interphase separating the solution and the
gel), and their width ( wn). The values of the variables tn, xn, and wn, can be predicted by using three phenomenological rules (Rácz 1999; Antal et al. 1998). The first is the time law:
xn ~ tn
[9.60]
The time law is a consequence of the diffusive motion of the reaction front within the gel (in fact,
x ~ Dt ). The positions of the bands form a geometric series according to the spacing law:
n
xn ~ Q(1+ p)
[9.61]
where p > 0 (typically included between 0.05 and 0.4) is the spacing coefficient, and Q represents the amplitude of the spacing. The spacing law can also be written in the following form known as
Jablczynski law:
xn ~1+ p
[9.62]
xn−1
The spacing coefficient is not a universal quantity. For example, it depends on the concentrations of
the outer ([ Ou] )
(
)
0 and inner [
]
In 0 electrolytes according to the Matalon-Packter law:
G ([ In]0 )
p = F ([ In]
[9.63]
0 ) +
[ Ou]0
where F ([ In] )
(
)
0 and G [ In]0 are decreasing functions of their argument [
]
In 0. Finally, there is the
width law stating that the width wn of the n-th band is an increasing function of n:
w
α
n ~ xn
[9.64]
with α > 0. Since the uncertainty in the measurements of wn is usually rather large, the width of
the bands has been ignored many times in the quantitative determination of the Liesegang patterns.
The Emergence of Order in Space
287
Although the Liesegang structures have been known for more than 100 years and despite an
intensive theoretical work devoted to them, there remain some experimental observations that have
not been explained.20 Moreover, there are still many theories to interpret periodic precipitations (Henisch 1988). The theories that have been proposed differ in the details treating the nucleation
and the kinetics of the precipitate growth (Antal et al. 1998; Henisch 1988). The simplest theory is
based on the supersaturation of ion-product: when the local product of the concentrations of the
outer electrolyte (A) and the inner electrolyte (B), [A][B], reaches a critical value, nucleation of
the precipitate starts. The nucleated particles grow and deplete A and B in their surroundings. The
local value of [A][B] drops and no new nucleation takes place. The reaction zone moves far from
the precipitation zone, and the critical value of the product [A][B] is reached again. The repetition
of these processes leads to the formation of the bands. In the theory of nucleation and droplet
growth, A and B react to produce a new species C, which also diffuses in the gel. When the local
concentration of C reaches some threshold value, nucleation occurs and the nucleated particles,
D, act as aggregation seeds. When the concentration of C around D is larger than a critical value,
the species C binds to D, and the droplet grows in dimension. This theory considers two threshold
values: one for nucleation and the other for droplet growth. A third theory considers the formation
of the bands as an induced sol-coagulation process. The electrolyte A and B bind to form a sol C.
The sol C coagulates if [C] exceeds a supersaturation threshold and if the local concentration of the
outer electrolyte is above a threshold. The band formation is a consequence of the coagulation of the
sol and the motion of the front defined by the critical concentration of the outer electrolyte. A fourth
theory interprets the phenomenon of periodic precipitation as the propagation of an autocatalytic
precipitation reaction. Such theory involves three chemical species and two chemical steps. The
species are the outer (A) and inner (B) electrolytes that diffuse within the gel and produce nuclei S
through an energetically unfavorable nucleation step:21
A + B → S
[9.65]
The nuclei S favor further associations between A and B:
A + B + ( n − )
1 S → n S
[9.66]
The transformation in equation [9.66] is called deposition step, and it represents the energetically
favorable autocatalytic growth of the nuclei S that gives rise to a precipitate (Lebedeva 2004). When
a deposition occurs, it automatically inhibits further precipitation in its neighborhood because it
depletes the concentration of A, B, and S. This model based on autocatalysis is corroborated by the
features the periodic precipitation patterns share with the transient pattern observed in exercise 9.1,
generated by the BZ reaction. After all, it is now accepted that a growing crystal is an example of
an excitable medium (Cartwright et al. 2012; Tinsley et al. 2013). It can give rise to target or spiral
patterns. The growth of spirals may be responsible for the helical structures we sometimes detect
in Liesegang structures.
TRY EXERCISE 9.19
20 Examples of observations that are tough to be explained are listed here. (I) Patterns form only under certain conditions.
(II) Some patterns are chiral and show helical structures. (III) A few substances can give rise to periodic precipitations where the interspacing of successive rings decreases with their ordinal numbers. These periodic precipitations are called revert Liesegang patterning. (IV) The coexistence of the so-called primary and secondary patterns wherein two types
of patterns are superimposed with different frequencies (Tóth et al. 2016). Many conundrums that we may find in the
literature are due to the use of non-standardized gels, reagents, and methods (Henisch, 1988).
21 The nucleation step is energetically unfavorable because when the nuclei are still small, molecules are relegated on the energetically unpleasant superficial positions. When the nuclei become larger, the ratio between the number of superficial molecules over those within the volume decreases and the growth of the nuclei becomes energetically favorable.
288
Untangling Complex Systems
9.9 LIESEGANG PHENOMENA IN NATURE
The wonderful phenomenon of periodic precipitations is around us in nature.
9.9.1 in geology
The earth has provided and still provides chemical environments that mimic those employed by
Liesegang to obtain periodic precipitations. The Liesegang rings are obtained in closed systems, in
the presence of a gel that prevents convective motions of the fluid and allows only the diffusion of
the reactants, one of which is placed on the gel.22 Therefore, the optimal geological environment for observing Liesegang rings should be a homogeneous gelatinous substrate with pronounced boundaries and in physical contact with a chemical reagent. Many patterns resembling Liesegang rings are
found in quartz in its many natural forms. Beautiful examples are the agates (an agate is shown in
Figure 9.28). Some geologists justify the presence of distinct layers in agates, by assuming that all quartz on earth was at one time a silica hydrogel. This hypothesis was reinforced over one hundred
years ago when a vein of gelatinous silica was found in the course of deep excavations through the
Alps for the Simplon Tunnel, which is a railway tunnel that connects Italy and Switzerland (Henisch
1988). However, there are some geologists who believe that the formation of rings, like those in
agates, derive from the deposition of layers of silica filling voids in volcanic vesicles or other cavi-
ties. Each agate forms its own pattern based on the original cavity shape. The layers form in distinct
stages and may fill a cavity completely or partially. When the filling is not complete, a hollow void
can host crystalline quartz growth, and the agate becomes the outer lining of a geode (in the agate
shown in Figure 9.28, there is a small geode indicated by the black arrow).
Although natural specimens of silica hydrogel are rarely encountered, the same cannot be said
for quicksand that is a hydrogel made of fine sand, clay and salt water. When quicksand dehydrates,
it forms sandstone. Liesegang rings are frequently encountered in sandstone. They appear in con-
cretions that are discrete blocks having an ovoid or spherical shape formed by the precipitation
of mineral cement within the spaces between the sediment grains. One example is offered by the
Sydney sandstone that is composed of pure silica and small amounts of siderite (FeCO ), bound with
3
a clay matrix; it shows beautiful concentric yellow-brown rings.
FIGURE 9.28 An example of agate with a small geode.
22 It is worthwhile noticing that the Turing Reaction-Diffusion patterns require an open system to be maintained over time.
On the other hand, periodic precipitations do not need an open system as long as they have been formed.
The Emergence of Order in Space
289
The Liesegang phenomenon is sometimes invoked also to justify the crystal inclusions when a
material is trapped within a mineral. In the laboratory, it has been demonstrated that large crystals
can form after dissolving the microcrystalline material in, say, an acid and allowing its solution to
diffuse into a gel medium having a pH at which the solubility of the material is much lower (Henisch
1988). Most geologists do not interpret inclusions in minerals as examples of Liesegang phenomena.
In fact, they comply with the “Father of Modern Geology,” James Hutton, who formulated a law that
bears his name and states that “fragments included in a host rock are older than the host rock itself.”
However, Hutton’s Law is unlikely to be universally valid. For example, in the case of gold crystals
in quartz. Many gold deposits in quartz could originally have been deposited in natural silica gel
because it is reasonable to suppose that high temperatures and an ample supply of reducing agents
must have been present during eras of mineralization (Stong 1962).
9.9.2 in biology
The formation of patterns in biology similar to the Liesegang rings has been observed in bacterial
growth. When bacteria spread more or less circularly from a point, in search of food contained in
a gelatinous medium such as agar, they consume their nutrient within a ring of a certain thickness.
Within that ring, bacteria could not flourish due to the competition for food. Therefore, they move
further and form another ring (Henisch 1988). In the end, we observe a pattern of concentric rings
that look like a Liesegang ring structure.
The periodic precipitation phenomenon is also important in the context of human health. Cystic
and inflammatory lesions may give rise to concentric non-cellular lamellar structures, occurring
as a consequence of the accumulation of insoluble products in a colloidal matrix. The shape and
size of such Liesegang rings may vary significantly (for example in Tuur et al. 1987, the Liesegang
rings measured from 7 up to 800 μm). Some pathologists may mistake Liesegang rings for eggs and
larvae of parasites or tumoral lesions, but the use of polarized-light microscopy combined with the
right stains can be really useful for a correct diagnosis (Pegas et al., 2010).
9.10 A FINAL NOTE: THE REACTION-DIFFUSION STRUCTURES
IN ART AND TECHNOLOGY
Despite many efforts, there is no a universal theory of pattern formation for system out-of-
equilibrium. This situation is in sharp contrast to what happens at equilibrium. For predicting the
structural properties of a system at equilibrium, we have the universal principle of minimization of
its free energy. Such principle is a local version of the Second Law of Thermodynamics because
it is focused on the system rather than on the entire Universe. Based on the principle of minimiza-
tion of system’s free energy, it is possible to build phase diagrams. A phase diagram allows us to
predict the most stable phase for a material, after specifying the contour conditions. An analogous
diagram, which we might call “morphology diagram,” for the prediction of pattern formation out-of-
equilibrium, does not exist, because we lack universal principles of dynamic evolution (Ben-Jacob
and Levine 2001). Nevertheless, the unpredictable phenomena of spontaneous pattern formation have
unthinkable aesthetic and technological possibilities that spur us to explore more deeply.
9.10.1 reacTion-diffusion Processes as arT
Originally, painters tried to represent the surrounding world in realistic manners with their works of
art. Correspondingly, the artists used professional skills and specific techniques to reproduce even the
most minute details on canvas. The advent of photography, in the first half of the nineteenth century,
contributed to change the role of painters. A picture can reproduce reality with extreme accuracy.
Therefore, a painter should not make efforts to describe the surrounding world with fidelity, but rather
he should just interpret it and communicate his impressions. And in fact, new artistic movements
290
Untangling Complex Systems
were born, shifting the content of the paintings from the figurative to the abstract. Correspondingly,
techniques evolved, tracing new artistic trends. They moved from the Impressionism of Claude Monet
(for instance, the oil in canvas titled “Impression, soleil levant,” painted in 1872), where the paint-
ing starts to be made of relatively small, thin, yet visible brush strokes, to the Pointillism of Georges
Seurat and Paul Signac in 1886 in which small, distinct dots or patches of color are applied in patterns
to form an image, to the unique style of Jackson Pollack’s drip paintings (created in the 1930s), where
the artist only marginally controls the structure of his composition.
Reaction-Diffusion processes performed by using micro-patterned hydrogel stamps to deliver a
solution of one or more reactants into a film of dry gels doped with chemicals that react with those
delivered from the stamp can be regarded as a new micro-scale painting technique (Grzybowski
2009). It is possible to control the initial conditions, but the images form on their own. Since fluctua-
ritories, and favored by intercontinental journeys, economic relationships and markets, and even the
circulation of ideas and fears among human communities fed by telecommunication technology, all
of those phenomena can be modelled by the theory of chemical waves and the Fisher-Kolmogorov
equation. Such theory is reasonable as long as the scale of the individual’s movement is small com-
pared with the size of the observations. There are several cases of monitored biological invasions.
A classic example is the spread of muskrat ( Ondatra zibethica) in Central Europe in the first half
of the twentieth century (Skellam 1951). The muskrat was brought to Europe from North America
for fur-breeding. In 1905, a few muskrats escaped from a farm near Prague. This small group
started multiplying, and in a few decades, the muskrats spread over the whole continental Europe
with a rate of ~3 km/year (Table 9.2).
Another example is the invasion of the gypsy moth ( Lymantria dispar) in North America. This insect
is thought to have been brought from France to a place near Boston by an amateur entomologist. When
it escaped around 1870, it spread at an estimated speed of ~10 km/year. By 1990, the whole Northwest
of the United States became heavily infested with resulting damages to the American agriculture.
Many more examples of biological invasions can be found in the book by Shigesada and Kawasaki
(1997), and in that by Petrovskii and Li (2006).
TRY EXERCISES 9.17 AND 9.18
9.8 LIESEGANG PATTERNS
Other exciting phenomena wherein chemical reactions coupled with diffusion originate stun-
ning patterns are the periodic precipitations discovered by the German chemist Raphael Eduard
Liesegang (1896). Some pictures of this impressive phenomenon are shown in Figure 9.27.
In the usual experimental setup, two strong electrolytes are needed. One is dissolved in a gel matrix
(the so-called inner electrolyte), whereas the other is poured onto the gel (outer electrolyte). The
concentration of the outer electrolyte is much larger than that of the inner one. The ions of the outer
electrolyte diffuse into the gel and react with the inner ions. An exchange reaction occurs and two new
salts form within the gel. One of the new two salts precipitates and forms a beautiful pattern. The
features of the pattern depend on the materials involved (included the gel) and the geometry of the
system. If the precipitation occurs in a test tube, we observe a sequence of solid bands (Figure 9.27); if it occurs in a Petri dish, a pattern of concentric rings emerges (see picture e of Figure 9.1).
286
Untangling Complex Systems
FIGURE 9.27 Periodic precipitation of Ag CrO in gelatin, within a test tube.
2
4
The shapes of the patterns can be described quantitatively by specifying the formation times
( tn) of the bands, their positions ( xn, measured from the interphase separating the solution and the
gel), and their width ( wn). The values of the variables tn, xn, and wn, can be predicted by using three phenomenological rules (Rácz 1999; Antal et al. 1998). The first is the time law:
xn ~ tn
[9.60]
The time law is a consequence of the diffusive motion of the reaction front within the gel (in fact,
x ~ Dt ). The positions of the bands form a geometric series according to the spacing law:
n
xn ~ Q(1+ p)
[9.61]
where p > 0 (typically included between 0.05 and 0.4) is the spacing coefficient, and Q represents the amplitude of the spacing. The spacing law can also be written in the following form known as
Jablczynski law:
xn ~1+ p
[9.62]
xn−1
The spacing coefficient is not a universal quantity. For example, it depends on the concentrations of
the outer ([ Ou] )
(
)
0 and inner [
]
In 0 electrolytes according to the Matalon-Packter law:
G ([ In]0 )
p = F ([ In]
[9.63]
0 ) +
[ Ou]0
where F ([ In] )
(
)
0 and G [ In]0 are decreasing functions of their argument [
]
In 0. Finally, there is the
width law stating that the width wn of the n-th band is an increasing function of n:
w
α
n ~ xn
[9.64]
with α > 0. Since the uncertainty in the measurements of wn is usually rather large, the width of
the bands has been ignored many times in the quantitative determination of the Liesegang patterns.
The Emergence of Order in Space
287
Although the Liesegang structures have been known for more than 100 years and despite an
intensive theoretical work devoted to them, there remain some experimental observations that have
not been explained.20 Moreover, there are still many theories to interpret periodic precipitations (Henisch 1988). The theories that have been proposed differ in the details treating the nucleation
and the kinetics of the precipitate growth (Antal et al. 1998; Henisch 1988). The simplest theory is
based on the supersaturation of ion-product: when the local product of the concentrations of the
outer electrolyte (A) and the inner electrolyte (B), [A][B], reaches a critical value, nucleation of
the precipitate starts. The nucleated particles grow and deplete A and B in their surroundings. The
local value of [A][B] drops and no new nucleation takes place. The reaction zone moves far from
the precipitation zone, and the critical value of the product [A][B] is reached again. The repetition
of these processes leads to the formation of the bands. In the theory of nucleation and droplet
growth, A and B react to produce a new species C, which also diffuses in the gel. When the local
concentration of C reaches some threshold value, nucleation occurs and the nucleated particles,
D, act as aggregation seeds. When the concentration of C around D is larger than a critical value,
the species C binds to D, and the droplet grows in dimension. This theory considers two threshold
values: one for nucleation and the other for droplet growth. A third theory considers the formation
of the bands as an induced sol-coagulation process. The electrolyte A and B bind to form a sol C.
The sol C coagulates if [C] exceeds a supersaturation threshold and if the local concentration of the
outer electrolyte is above a threshold. The band formation is a consequence of the coagulation of the
sol and the motion of the front defined by the critical concentration of the outer electrolyte. A fourth
theory interprets the phenomenon of periodic precipitation as the propagation of an autocatalytic
precipitation reaction. Such theory involves three chemical species and two chemical steps. The
species are the outer (A) and inner (B) electrolytes that diffuse within the gel and produce nuclei S
through an energetically unfavorable nucleation step:21
A + B → S
[9.65]
The nuclei S favor further associations between A and B:
A + B + ( n − )
1 S → n S
[9.66]
The transformation in equation [9.66] is called deposition step, and it represents the energetically
favorable autocatalytic growth of the nuclei S that gives rise to a precipitate (Lebedeva 2004). When
a deposition occurs, it automatically inhibits further precipitation in its neighborhood because it
depletes the concentration of A, B, and S. This model based on autocatalysis is corroborated by the
features the periodic precipitation patterns share with the transient pattern observed in exercise 9.1,
generated by the BZ reaction. After all, it is now accepted that a growing crystal is an example of
an excitable medium (Cartwright et al. 2012; Tinsley et al. 2013). It can give rise to target or spiral
patterns. The growth of spirals may be responsible for the helical structures we sometimes detect
in Liesegang structures.
TRY EXERCISE 9.19
20 Examples of observations that are tough to be explained are listed here. (I) Patterns form only under certain conditions.
(II) Some patterns are chiral and show helical structures. (III) A few substances can give rise to periodic precipitations where the interspacing of successive rings decreases with their ordinal numbers. These periodic precipitations are called revert Liesegang patterning. (IV) The coexistence of the so-called primary and secondary patterns wherein two types
of patterns are superimposed with different frequencies (Tóth et al. 2016). Many conundrums that we may find in the
literature are due to the use of non-standardized gels, reagents, and methods (Henisch, 1988).
21 The nucleation step is energetically unfavorable because when the nuclei are still small, molecules are relegated on the energetically unpleasant superficial positions. When the nuclei become larger, the ratio between the number of superficial molecules over those within the volume decreases and the growth of the nuclei becomes energetically favorable.
288
Untangling Complex Systems
9.9 LIESEGANG PHENOMENA IN NATURE
The wonderful phenomenon of periodic precipitations is around us in nature.
9.9.1 in geology
The earth has provided and still provides chemical environments that mimic those employed by
Liesegang to obtain periodic precipitations. The Liesegang rings are obtained in closed systems, in
the presence of a gel that prevents convective motions of the fluid and allows only the diffusion of
the reactants, one of which is placed on the gel.22 Therefore, the optimal geological environment for observing Liesegang rings should be a homogeneous gelatinous substrate with pronounced boundaries and in physical contact with a chemical reagent. Many patterns resembling Liesegang rings are
found in quartz in its many natural forms. Beautiful examples are the agates (an agate is shown in
Figure 9.28). Some geologists justify the presence of distinct layers in agates, by assuming that all quartz on earth was at one time a silica hydrogel. This hypothesis was reinforced over one hundred
years ago when a vein of gelatinous silica was found in the course of deep excavations through the
Alps for the Simplon Tunnel, which is a railway tunnel that connects Italy and Switzerland (Henisch
1988). However, there are some geologists who believe that the formation of rings, like those in
agates, derive from the deposition of layers of silica filling voids in volcanic vesicles or other cavi-
ties. Each agate forms its own pattern based on the original cavity shape. The layers form in distinct
stages and may fill a cavity completely or partially. When the filling is not complete, a hollow void
can host crystalline quartz growth, and the agate becomes the outer lining of a geode (in the agate
shown in Figure 9.28, there is a small geode indicated by the black arrow).
Although natural specimens of silica hydrogel are rarely encountered, the same cannot be said
for quicksand that is a hydrogel made of fine sand, clay and salt water. When quicksand dehydrates,
it forms sandstone. Liesegang rings are frequently encountered in sandstone. They appear in con-
cretions that are discrete blocks having an ovoid or spherical shape formed by the precipitation
of mineral cement within the spaces between the sediment grains. One example is offered by the
Sydney sandstone that is composed of pure silica and small amounts of siderite (FeCO ), bound with
3
a clay matrix; it shows beautiful concentric yellow-brown rings.
FIGURE 9.28 An example of agate with a small geode.
22 It is worthwhile noticing that the Turing Reaction-Diffusion patterns require an open system to be maintained over time.
On the other hand, periodic precipitations do not need an open system as long as they have been formed.
The Emergence of Order in Space
289
The Liesegang phenomenon is sometimes invoked also to justify the crystal inclusions when a
material is trapped within a mineral. In the laboratory, it has been demonstrated that large crystals
can form after dissolving the microcrystalline material in, say, an acid and allowing its solution to
diffuse into a gel medium having a pH at which the solubility of the material is much lower (Henisch
1988). Most geologists do not interpret inclusions in minerals as examples of Liesegang phenomena.
In fact, they comply with the “Father of Modern Geology,” James Hutton, who formulated a law that
bears his name and states that “fragments included in a host rock are older than the host rock itself.”
However, Hutton’s Law is unlikely to be universally valid. For example, in the case of gold crystals
in quartz. Many gold deposits in quartz could originally have been deposited in natural silica gel
because it is reasonable to suppose that high temperatures and an ample supply of reducing agents
must have been present during eras of mineralization (Stong 1962).
9.9.2 in biology
The formation of patterns in biology similar to the Liesegang rings has been observed in bacterial
growth. When bacteria spread more or less circularly from a point, in search of food contained in
a gelatinous medium such as agar, they consume their nutrient within a ring of a certain thickness.
Within that ring, bacteria could not flourish due to the competition for food. Therefore, they move
further and form another ring (Henisch 1988). In the end, we observe a pattern of concentric rings
that look like a Liesegang ring structure.
The periodic precipitation phenomenon is also important in the context of human health. Cystic
and inflammatory lesions may give rise to concentric non-cellular lamellar structures, occurring
as a consequence of the accumulation of insoluble products in a colloidal matrix. The shape and
size of such Liesegang rings may vary significantly (for example in Tuur et al. 1987, the Liesegang
rings measured from 7 up to 800 μm). Some pathologists may mistake Liesegang rings for eggs and
larvae of parasites or tumoral lesions, but the use of polarized-light microscopy combined with the
right stains can be really useful for a correct diagnosis (Pegas et al., 2010).
9.10 A FINAL NOTE: THE REACTION-DIFFUSION STRUCTURES
IN ART AND TECHNOLOGY
Despite many efforts, there is no a universal theory of pattern formation for system out-of-
equilibrium. This situation is in sharp contrast to what happens at equilibrium. For predicting the
structural properties of a system at equilibrium, we have the universal principle of minimization of
its free energy. Such principle is a local version of the Second Law of Thermodynamics because
it is focused on the system rather than on the entire Universe. Based on the principle of minimiza-
tion of system’s free energy, it is possible to build phase diagrams. A phase diagram allows us to
predict the most stable phase for a material, after specifying the contour conditions. An analogous
diagram, which we might call “morphology diagram,” for the prediction of pattern formation out-of-
equilibrium, does not exist, because we lack universal principles of dynamic evolution (Ben-Jacob
and Levine 2001). Nevertheless, the unpredictable phenomena of spontaneous pattern formation have
unthinkable aesthetic and technological possibilities that spur us to explore more deeply.
9.10.1 reacTion-diffusion Processes as arT
Originally, painters tried to represent the surrounding world in realistic manners with their works of
art. Correspondingly, the artists used professional skills and specific techniques to reproduce even the
most minute details on canvas. The advent of photography, in the first half of the nineteenth century,
contributed to change the role of painters. A picture can reproduce reality with extreme accuracy.
Therefore, a painter should not make efforts to describe the surrounding world with fidelity, but rather
he should just interpret it and communicate his impressions. And in fact, new artistic movements
290
Untangling Complex Systems
were born, shifting the content of the paintings from the figurative to the abstract. Correspondingly,
techniques evolved, tracing new artistic trends. They moved from the Impressionism of Claude Monet
(for instance, the oil in canvas titled “Impression, soleil levant,” painted in 1872), where the paint-
ing starts to be made of relatively small, thin, yet visible brush strokes, to the Pointillism of Georges
Seurat and Paul Signac in 1886 in which small, distinct dots or patches of color are applied in patterns
to form an image, to the unique style of Jackson Pollack’s drip paintings (created in the 1930s), where
the artist only marginally controls the structure of his composition.
Reaction-Diffusion processes performed by using micro-patterned hydrogel stamps to deliver a
solution of one or more reactants into a film of dry gels doped with chemicals that react with those
delivered from the stamp can be regarded as a new micro-scale painting technique (Grzybowski
2009). It is possible to control the initial conditions, but the images form on their own. Since fluctua-
