3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast
The paper is devoted to the theoretical study of elementary permeable objects percolation and its application to real physical objects. Spheres and isotropic oriented capped sticks were chosen as elementary geometrical objects for percolation simulation, physically adequate for radiation defects b...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1199692017-06-11T03:04:29Z 3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast Zhydkov, V.O. The paper is devoted to the theoretical study of elementary permeable objects percolation and its application to real physical objects. Spheres and isotropic oriented capped sticks were chosen as elementary geometrical objects for percolation simulation, physically adequate for radiation defects behaviour description in brittle dielectrics, particularly in the so-called Lava-like Fuel Containing Materials (LFCM), where it effects their mechanical steadiness. LFCMs is high-radioactive glass, which was formed during active stage of well-known heavy nuclear accident, that occurred at Chornobyl nuclear facility in 1986. Physical processes taking place in the materials are of great practical interest. Furthermore, when applying percolation models to LFCM objects, an approximate behaviour forecast can be created. From the results of simulation, it appears that physical properties of the LFCM should drastically change within in the period of 2015/2045 calendar years, depending on variations in nuclear fuel content. Ця робота присвячена теоретичному вивченню перколяцiйних задач на елементарних проникаючих об’єктах та його застосуванню до реальних фiзичних об’єктiв. Сфери та iзотропно орiєнтованi скругленi палички були обранi як елементарнi геометричнi об’єкти для моделювання перколяцiї, фiзично адекватнi для опису поведiнки крихких матерiалiв, що мiстять радiацiйнi дефекти, зокрема до так званих лавоподiбних паливовмiсних матерiалiв (ЛПВМ), де цi дефекти суттєвим чином впливають на механiчну стiйкiсть. ЛПВМ являють собою високорадiоактивне скло, що сформувалося у перебiгу активної фази добре вiдомої важкої ядерної аварiї, що вiдбулася на Чорнобильськiй Атомнiй Електростанцiї у 1986 роцi. Фiзичнi процеси, що мають мiсце у цих матерiалах, становлять великий практичний iнтерес. Застосовуючи перколяцiйнi моделi до ЛПВМ, можливо створити приблизний прогноз їх поведiнки. З результатiв моделювання випливає, що фiзичнi властивостi ЛПВМ корiнним чином змiняться в перiод 2015 2045 року, в залежностi вiд вмiсту палива в них. 2009 Article 3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast / V.O. Zhydkov // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 193-203. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 64.60.Ak, 78.70.-g DOI:10.5488/CMP.12.2.193 http://dspace.nbuv.gov.ua/handle/123456789/119969 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The paper is devoted to the theoretical study of elementary permeable objects percolation and its application
to real physical objects. Spheres and isotropic oriented capped sticks were chosen as elementary geometrical
objects for percolation simulation, physically adequate for radiation defects behaviour description in brittle
dielectrics, particularly in the so-called Lava-like Fuel Containing Materials (LFCM), where it effects their
mechanical steadiness. LFCMs is high-radioactive glass, which was formed during active stage of well-known
heavy nuclear accident, that occurred at Chornobyl nuclear facility in 1986. Physical processes taking place in
the materials are of great practical interest. Furthermore, when applying percolation models to LFCM objects,
an approximate behaviour forecast can be created. From the results of simulation, it appears that physical
properties of the LFCM should drastically change within in the period of 2015/2045 calendar years, depending
on variations in nuclear fuel content. |
| format |
Article |
| author |
Zhydkov, V.O. |
| spellingShingle |
Zhydkov, V.O. 3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast Condensed Matter Physics |
| author_facet |
Zhydkov, V.O. |
| author_sort |
Zhydkov, V.O. |
| title |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| title_short |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| title_full |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| title_fullStr |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| title_full_unstemmed |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| title_sort |
3d continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119969 |
| citation_txt |
3D continuum percolation approach and its application to lava-like fuel-containing materials behaviour forecast / V.O. Zhydkov // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 193-203. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT zhydkovvo 3dcontinuumpercolationapproachanditsapplicationtolavalikefuelcontainingmaterialsbehaviourforecast |
| first_indexed |
2025-07-08T16:59:54Z |
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2025-07-08T16:59:54Z |
| _version_ |
1837098856600305664 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 193–203
3D continuum percolation approach and its application
to lava-like fuel-containing materials behaviour forecast
V.O.Zhydkov
Institute for NPP Safety Problems of the National Academy of Sciences of Ukraine
Received February 9, 2009, in final form May 25, 2009
The paper is devoted to the theoretical study of elementary permeable objects percolation and its application
to real physical objects. Spheres and isotropic oriented capped sticks were chosen as elementary geometrical
objects for percolation simulation, physically adequate for radiation defects behaviour description in brittle
dielectrics, particularly in the so-called Lava-like Fuel Containing Materials (LFCM), where it effects their
mechanical steadiness. LFCMs is high-radioactive glass, which was formed during active stage of well-known
heavy nuclear accident, that occurred at Chornobyl nuclear facility in 1986. Physical processes taking place in
the materials are of great practical interest. Furthermore, when applying percolation models to LFCM objects,
an approximate behaviour forecast can be created. From the results of simulation, it appears that physical
properties of the LFCM should drastically change within in the period of 2015÷2045 calendar years, depending
on variations in nuclear fuel content.
Key words: continuum percolation, irradiated nuclear fuel, radiation damages
PACS: 64.60.Ak, 78.70.-g
1. Introduction
The percolation approach is widely used in solid-state physics, particularly for its description
as a unified system, containing inclusions of another structure. To these may be referred compos-
ite materials, glass-ceramics, and complex structures with radiation damages in its volume, and
many others. Roughly, a generalized three-dimensional percolation problem can be formulated in
the following manner: there are some random non-homogeneous media, for example, stone with
randomly distributed pores, and there is some liquid (water) or gas from one side of the stone,
which may percolate inside and through the pores. And there arises a question, concerning the con-
ditions (concentration, size, form, positions of the pores) under which the substance will propagate
(percolate) from one side to the other. Such conditions are called as critical conditions or critical
points. Those sorts of tasks are known as percolation problems. Many static and dynamic physical
properties of multiphase materials and systems are determined by microscopic and macroscopic
spatial distribution of their phases. When the system or material is near a critical point or starts
to undergo a structural phase transition, the properties of such a system are best modelled by
continuum percolation with objects of a corresponding shape and size and are properly (in most
cases randomly) distributed, in contrast to site or bond percolation where sites or bonds are in a
discrete lattice and randomly occupied. In most systems inclusions of another structure are phy-
sically small and randomly distributed, which makes percolation the best tool for modelling the
phase transitions and critical behaviour in unified systems, containing the inclusions.
Percolation problems of this kind are known as three-dimensional (3D) continuum percolation
of hard-core or soft-core (permeable) geometrical objects and they were the object of persistent
research in the 80’s [1–6]. Due to computational complexity such percolation tasks, however, are
much less known than the well-known lattice percolation tasks, though the progress in computa-
tional techniques in the last decade has made it possible to achieve significant results in solving
some of them [2–4,7,8].
The percolation approach is widely applied to condensed matter problems solved in the way of
simulation. Common examples are doped materials [9] and polymers [10]. The simulation of etching
c© V.O.Zhydkov 193
V.O.Zhydkov
process in natural application for percolation simulations as well [11]. The present work is devoted
to the solution of some special percolation problems in order to characterize critical behaviour
resulting from volume-generated radiation damages in condensed matter. In particular, percolation
models of permeable spheres and sticks are used; the real physical object for application is LFCM
(Lava-like Fuel-Containing Materials), relating to the class of brittle materials; radiation defects
accumulation, as it will be shown below, can lead to critical change in its mechanical properties and
behaviour in future. From the practical point of view, LFCM mechanical steadiness to external
impacts is of especial interest, because LFCM mechanical destruction and transformation into
high-dispersive state, regarding its high specific radioactivity, is a potentially dangerous event.
2. Percolation problem on soft spheres
The mathematical formulation of the continuum percolation problem on the soft spheres is as
follows: there is a cube; it is filled with small (relative to the cube size) permeable spheres, their
centres (i.e. nodes) being randomly distributed. The critical concentration is to be reached when it is
possible to find a way through spheres from one side of the cube to the opposite side (so percolation
is reached). Such a task is known as three-dimensional percolation problem on random sites. It is
obvious that percolation is attained when infinite (in terms of percolation theory) cluster from the
sites is formed. The most suitable parameter to characterize the percolation process independently
from the linear dimensions, concentration and even the form of objects on which the percolation
is realized is average bond connections per node at percolation threshold Bc:
Bc = Nvex/V. (1)
Here N is the number of nodes, V is volume of the simulation cube, which is taken unitary for
simplicity, vex is an excluded geometric figure volume. The excluded volume is defined as a volume
surrounding and including a given geometric percolation figure (sphere in the present case), which
is inaccessible to another figure [12]. For spheres, this is a sphere of double size, where it is easy to
get the Bc for soft spheres:
Bc =
32
3
πNr3
c
/V, (2)
where rc is the critical spheres radius at which percolation is realised. The number of nodes should
be fairly large due to a systematic error (so-called finite-size effects) increasing with the size of
spheres, which slowly decreases with an increase of the number of nodes. To obtain the threshold Bc
in a “direct” way, finite-size effects should be excluded, which requires zero percolation radius, and,
consequently, an infinite number of percolation nodes. Nonetheless, the exact percolation threshold
with uncertainty as small as is required could be obtained by excluding a systematic error of finite-
size effect in the following way: a dependence of percolation threshold versus spheres radius should
be built, then approximated and extrapolated with power function. Extrapolation of Bc value to
zero percolation radius and a corresponding infinite number of nodes can be considered as a “true”
value for three-dimensional percolation threshold. The reverse of power in approximated power
function is commonly called a critical index ν [13].
A lot of algorithms for solving percolation problems have been developed [3,4,7,8,12,13]. On
having revised them, the author found out that they lacked speed and were doing a lot of un-
necessary calculations in the process, repeating them over and over again, and even the fastest
algorithms [4] are not free from certain drawbacks. Considering that the author has developed a
special algorithm for percolation on random sites, and for the first time it was tested for percolation
on spheres. Herein below there is a brief description of the main steps of the algorithm.
1. Random distribution of the nodes within the simulated cubic region is made, the nodes
positions are set to the proper database with X,Y and Z axes oriented along the cube axes;
2. The positions of the spheres are sorted in the positions database index in order to quickly
figure out the neighbouring spheres around the chosen one. For example, in one index they
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3D continuum percolation approach and its application to LFCM behaviour forecast
are sorted by their X coordinate in ascending manner. It means that at the stage of defining
distances between the nodes one has to take into account the nodes only within a known range
of corresponding index. A similar procedure of indexing is done for Y and Z coordinates.
3. For each node a distance to nearby nodes is determined. Step 2 makes the process much
faster. For each separate node, its neighbours within a certain range on X, Y, and Z axes
can be quickly determined. All neighbours are automatically placed together in the database,
because they are sorted by corresponding X,Y or Z coordinates.
Let us explain the item 3 more in detail:
3.1 First, the program gets coordinates of the current node and its position in the database.
Next, in the database, the first index array of all the nodes is sorted by X coordinate
with the current node as starting point. Then, the program goes up on the sorted array
starting from the current node position, picking the numbers of the nodes that are close
to the current one by X coordinate. This process stops when the difference between X
coordinate of the current node and the other node becomes too large, i. e., more than
some limiting distance (more than the distance at which percolation surely occurs). The
same procedure is performed, but the array is run down by X coordinate. So, in the end
we got recorded an array of nodes which are close to the current node by X coordinate.
The same procedure is performed for Y and Z coordinates, and only those nodes are
checked for distance that are close enough to X, Y, and Z coordinates simultaneously.
By choosing the correct limiting radius, the number of nodes finally checked for the
distance can be reduced to the average of 4–5, instead of running the full array of nodes
required to be checked for distance.
Therefore, time consuming radius and distance calculations are replaced by a rather
quick picking up the node numbers from the database. It runs much faster, and, in
addition, its calculation time is roughly proportional to N 4/3, or even to N for heavily-
optimized case, with time for initial sorting and indexing roughly proportional to NlnN
(quick sorting); N here is the number of nodes. For comparison, calculation time is
proportional to N2 if one uses a “classical” approach without sorting. Therefore, for
huge node numbers ( > 100, 000) the calculation scheme proposed can be very effective.
Actually, for N > 500, 000, the time required for calculation of relative positions was
less than all other steps of algorithm taken together.
Thus, for a large ( > 10, 000) number of nodes, the preliminary indexing makes sense
despite the time required for the indexing procedure. All the procedures at the start
take a lot of memory for indexing, for arrays, etc., but for the current situation, memory
was sacrificed for the sake of speed.
3.2 The distances are calculated and recorded for corresponding pairs of nodes. After the
procedure, the database of indexes and even positions can be eliminated: the operation
turns from three-dimensional problem into simple logical pathfinding problem.
4. After turning the assembly of nodes from spatial structure into a connected graph, a special
pathfinding algorythm, which can be classified as conditional propagation pathfinding algori-
thm with some author’s modification, is applied. It finds a path through the nodes from one
edge of the cube to the opposite one with optimization by a minimum distance between the
nodes on the path. Technically, this is realized in the following way: every node has a specific
parameter assigned: an individual “percolation radius”, e.g., characteristic radius of spheres,
at which the current node will be percolated.
At the starting iteration, all the nodes have the “percolation radius” equal to their distance
from the starting edge of the cube, e.g., they percolate when the sphere centred on the node
“touches” the starting wall of the cube. Then, each pair of nodes is checked in the following way:
if the maximum of values of “percolation radius” of the first node and half the distance between
nodes is less than “percolation radius” of the second node, the percolation radius of the second
195
V.O.Zhydkov
node becomes equal to the maximum of the two values. This process is illustrated schematically
in figure 1.
Figure 1. Schematic drawing for percolation radius calculation.
The left-hand node in figure 1 has “percolation radius” which is equal, naturally, to the distance
from the node to the cube wall, which is a starting one for pathfinding. The node will percolate
if the sphere radius is sufficient to touch the starting wall. The left and middle nodes will touch
each other if the percolation radius reaches half the distance between the nodes, and “percolation
radius” for the left node is less than half the distance between the left and the middle nodes. So,
the middle node will percolate at the sphere radius equal to half the distance between the left
and the middle node. Thus, “percolation radius” of the middle node is assigned as being equal to
half the distance between the left and the middle node. The right node can percolate from the
middle node only, and its “percolation radius” is equal to the radius of the middle node, because
half-distance between the nodes is less than “percolation radius” of the middle node. So, the right
node will percolate at the sphere radius equal to the radius of the middle node.
In the process of successive steps, if the maximum value obtained is larger than “percolation
radius” of the current node, “percolation radius” naturally does not get assigned to that maximum.
Therefore, in successive procedure of “percolation radius” re-assignment, the modified path-
finding algorithm, in a few iterations, re-assigns the “percolation radius” to every node. As the
final result of iterations, all “percolation radius” values became minimal, e.g., if the sphere radius
is less than “percolation radius” the node will certainly will not be percolated.
Thus, to obtain the radius at percolation threshold one should get the “percolation radius”
from the nodes at the remote side of the cube only. Then, knowing N and the resulting percolation
radius rc, one can easily calculate Bc.
The algorithm is rather general and can be applied to a system of virtually any number of
dimensions and geometric shapes. This algorithm is also very convenient in defining the dependence
of infinite cluster power on the sphere radius. For accounting one should only build a dependence
on the numbers of spheres whose “percolation radius” is equal or less than the sphere radius.
The method turned out to be rather robust and reliable. So, the percolation threshold determi-
nation procedure after initial tests underwent slight changes: several initial percolating nodes were
placed in the centre of the cube with “percolation radius” property assigned to zero, and, after a
few iterations, it is easy to get a dependence of cluster power on the radius and thus to obtain the
percolation threshold by power-law approximation.
For each given number of nodes, a few thousand variants of node positions were taken initially
for every simulation, Bc calculated for every variant, and, in the end, they were averaged. The
spread of percolation thresholds for different realizations can be estimated from 2: the probabi-
lity of percolation realization is essentially an integral of percolation event probability density.
Subsequently, a steeper curve corresponds to a narrower distribution.
196
3D continuum percolation approach and its application to LFCM behaviour forecast
As a conclusion, this algorithm has the following advantages: no calculation time wasting on
adding another node and on re-calculating all the node assembly after every increment; only nearby
nodes are taken into account which are simply taken from the database without any additional
calculations; rc and Bc is found in just one pathfinding session. Compared to common algorithms,
[1,2,4], in practice this is about 10× increase in productivity for 100,000 nodes, further increas-
ing with the nodes number. Con is rather obvious: the algorithm is correctly applicable to fully
permeable objects only.
For objects of non-spherical shape, one should understand property of “percolation radius” as
a scaling factor applied to all geometrical figures, provided their position kept constant and figures
are convex. In such a fashion, one can easily apply this algorithm even to the objects with various
but proportional sizes, and even to objects of different shape.
The dependence of probability of percolation (ratio of events where percolation occurred to
the total number of events) on B (the average number of bond connections per node which is
proportional to density) is presented in figure 2. The curves were calculated for different node
numbers to evaluate the error given by finite concentration.
Figure 2. Dependence of percolation probability W on the average bond connections. Calcu-
lations for different node concentrations per simulation box were performed from 1,000 up to
100,000 spheres.
Figure 3. Plot of percolation threshold Bc versus critical radius of spheres rc. The simulation
cube is of unitary size.
197
V.O.Zhydkov
For every N , finite size effect suggests different Bc as well as rc. In figure 3 the plot of Bc versus
rc is presented, each point representing the results for its own N . Error bars are actually about
the size of a dot and thus they are not shown. The dependence very well fits to the power type.
One can extrapolate the true Bc at the point of rc = 0, and at this point Bc is 2.734± 0.005. The
power is 1.11 ± 0.02. It corresponds to the percolation critical index of ν = 0.89 ± 0.02, similar
to the results established in [13]. Compared to results established earlier, the current realization
presents a more efficient algorithm only, which, using appropriate calculational power, can be used
for a more precise determination of percolation parameters.
3. Percolation task on soft sticks
The mathematical formulation of the task is as follows: there is a cube; it is randomly and
isotropically filled with fully permeable objects, having the form of capped cylinders. The perco-
lation is realized on the sticks from one wall of the cube to the other. The first solutions of such
a task are in [1,12]. The solution revealed the analytical approach to the solution of such a task,
known as an excluded volume rule. However, their simulation was not quite correct, as it is revealed
in further work [2] due to the flaw in the mathematical model built: the distribution of the stick’s
direction was not truly isotropic.
We have excluded the flaw in our model in principle by generating direction vectors in the
following way: a random radius vector was generated within the volume-centred cube with edge
length twice the unitary. If the length of the radius vector exceeded the unitary length, thus taking
place out of sphere with unitary radius, the vector was generating again until a vector of length
no more than unitary was obtained. So, the endpoints of the vectors distributed uniformly inside
the sphere with the starting point in its centre and, therefore, isotropic. Afterwards, the vector
was divided on its length. As a final result, vectors of unitary length and isotropic distribution
were obtained. The process is roughly illustrated in figure 4a: the radius-vector shorter than the
unitary, stays, longer than the unitary rejected. All that is left to do is to normalize them to have
a the unitary length.
(a) (b)
Figure 4. Illustration for isotropic vector generation procedure. On the left you can see (a) the
choice of isotropic radius-vector and on the right (b) the generated radius-vectors with random
spherical coordinates.
If one takes all the radius-vectors within the cube, for example, the ones directed to the corners
of the cube it is more probable that the distribution will not be isotropic. In work [2] the distribution
was on the spherical coordinates, so the random distribution looked like in figure 4b i. e., denser at
the poles of spherical coordinates. The method described above eliminates the flaws of that kind
in principle. The computational algorithm is similar to the one used with permeable spheres; the
radius of cylinders, at which pair of capped cylinders at a given position and directions came in
contact while maintaining R/L ratio, was taken as “percolation radius” property. Here L is the
cylinder length and R is the cylinder radius, as shown in figure 5. Apart from adding the property
198
3D continuum percolation approach and its application to LFCM behaviour forecast
of direction and somewhat complex algorithm of determining “percolation radius” to every pair of
nodes, percolation algorithm required no changes in comparison with percolation on spheres.
Figure 5. Elementary geometrical object for percolation: a capped cylinder.
Figure 6. S = Bc − 1 as a function of the R/L aspect ratios of sticks presented on the left
graph. On the right graph, the dependence of S on R/L for the range of R/L < 0.01 is shown
in the log-log scale. The straight line is the power function approximation shows the power of
0.58 ± 0.02.
Figure 7. Dependence of percolation threshold on sticks grade. Triangles correspond to R/L =
0.02, circles correspond to R/L = 0.01 and squares correspond to R/L = 0.005 Percolation is
performed inside a cube of unitary size.
The simulations were run with different ratio of the capped cylinders radius R to their length
L. For each R/L value, the simulation of percolation was performed with different concentration of
the capped sticks N . Then, like in our percolation with spheres, a power extrapolation was made
as shown in figure 7 and the dependence of extrapolation on R/L ratio is presented in figure 6.
Thus, each point on the graph is an effective approximation of percolation threshold by different
numbers of objects in volume.
In the work we have made a slight advance in R/L ratio in comparison with [2], so we can
observe that power-law at low R/L maintains up to R/L equal to 0.001 and the power is equal to
199
V.O.Zhydkov
0.58 ± 0.02. In general, the curve presented in figure 6 lies slightly below the curve presented in
[2], which can be explained by more accurate determination of Bc.
4. Application to the real material: LFCM behaviour forecast
LFCM (Lava-like Fuel-Containing Materials) are the materials formed as a result of the well-
known heavy nuclear accident in Chornobyl in 1986. As it is accepted for now [14], the LFCM are
glass ceramic alkaline-earth silicate compositions (devitrified glasses) containing 5–10 mass percent
of irradiated uranium nuclear fuel in their volume in company with high-radioactive fission prod-
ucts. The review of the main LFCM physical properties is presented in [14]. Approximately 1,000
tons of this material, accompanied both by the other core debris and the building constructions
of the destroyed Chornobyl NPP 4-th unit, is located in the so-called “Shelter” site, which had
been erected quickly after 1986 accident as a forced measure directed to primary prevention of
radioactive substances dissemination in the environment. The “Shelter” site is not a hermetically
sealed construction and there are no doubts that a large quantity of high-radioactive compositions,
such as LFCM and irradiated nuclear fuel itself do have a direct air contact with the environment.
LFCM themselves look like the coloured glass ceramics and one can classify them like brown (8–
10% fuel content), multicoloured (6–7%) and black (4% fuel as usually). These materials do have a
lot of interesting properties, among which there is a spontaneous self-sputtering of its surface [15].
The percolation models presented above can be successfully applied to the ensemble of radiation
damages behaviour modelling. The LFCM do have disordered regions (DR) of spherical form in
its volume, induced by heavy recoil nuclei [16]. The DR spatial distribution is quite random, and
the neighbouring DR can be overlapped in a great measure, which just coincides with the model
of permeable spheres. Therefore, the model of percolating permeable spheres looks physically ad-
equate. If we apply the above results of the modelling to LFCM, we will acquire the results given
below. First of all, we can establish the approximate time when the infinite cluster of disordered
regions will be formed. Taking into account the variations in LFCM specific radioactivity and rate
of defect formation, one can approximately determine the DR concentration and, therefore, the
calendar year, when the infinite cluster will be formed.
Figure 8. Probability of percolation event versus calendar year plot for LFCM containing 10%
of fuel.
Taking the DR diameter to be 25 nm [16], knowing formula (2) and Bc for spheres, calculated
earlier, we can derive the critical concentration of DR: (2.5 ± 0.5) · 1016 cm−3. The most of the
error arises from variations in DR diameter, which, as it was established in [16], is about 25±2nm.
Considering the rate of defects generation (8 ± 2) · 1014 cm−3/year for 10% fuel content [16], the
formation of such a concentration corresponds to the calendar year of 2015 ÷ 2045 depending on
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3D continuum percolation approach and its application to LFCM behaviour forecast
the variations in alpha-active nuclides concentration. Probability of such a percolation event in the
calendar year for LFCM containing 10% of fuel is presented in figure 8.
Finite size effects expressed in different curves for different numbers of defects in this case
do have its own physical sense: when there is a huge block of LFCM, a curve for the greatest
number of defects should be taken into account. With time, however, LFCM will undergo further
destruction into individual dust particles of a smaller grade, containing lesser number of defects,
so the curve for 10,000 or even for 1,000 should be appropriate. Thus, in accordance with figure 8,
the process is likely to develop in the following way: initially, large (macroscopic) LFCM fragment
will be fractured into cluster particles of the order of tens micron grade. Then (this will occur fast
enough) such fragments will fall apart into “elementary” particles with the grade of one or a few
DR. This final stage of destruction means conversion of the whole LFCM volume into submicronic
dust, just similar to that observed in experiments on LFCM surface self-sputtering [15].
The above description of fracture process is based on fundamental concepts of fracture me-
chanics [17]. The volume fracture of brittle materials starts with cracks development, which prop-
agate along the border, splitting the areas with different mechanical properties. Such a kind of
border one may identify as a cluster edge. This version has a confirmation in experiments, when
spontaneous dust productivity of brittle dielectrics (LFCM) was observed: the shape of dust par-
ticles was identified as clustered disordered regions, i. e., mechanical destruction of the surface
occurred along the borders dividing DR from the surrounding material [15].
With the increase of the density of defects, the mean cluster spanning size increases as well,
obviously leading to an increase in the average lengths of the cracks. At this stage, however, there
will be no radical changes in material properties, since the crack propagation will be limited by the
nearest disordered region. The situation will change drastically, however, when the infinite cluster
is formed.
In light of the fact that cracks appear at the borders of the disordered regions, there is no doubt
that the infinite cluster of disordered regions formation in LFCM means the formation of the main
crack along the infinite cluster border, penetrating the macroscopic fragment of material thorough
as a whole. Considering percolation properties of the infinite cluster (spanning all along the system,
occupying most of the volume), LFCM after percolation threshold will likely no longer possess any
macroscopic mechanical properties: it will be reduced to fine grade sand. The approximate LFCM
volume dust productivity increase one can roughly estimate as 20× minimum.
Figure 9. Plot of total spheres clusters specific surface versus defects density.
The further additional changes in the material structure may occur when the area of the
clustered DR surface reaches its maximum - this will cause the leavings of LFCM to be especially
susceptible to mechanical loads and other external factors. The clustered DR surface parameter is
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V.O.Zhydkov
of great interest due to its effect on mechanical steadiness of materials beyond the critical points.
Therefore, due to the practical importance of the mentioned parameter, the dependence of such
an area per unit volume was calculated; corresponding results are shown in the figure 9. The task
was calculated accepting DR diameter to be 25 nm, like it takes place in LFCM. One can see from
figure 9 that specific border surface will reach its maximum at 6.5 · 1016 cm−3 DR concentration,
which corresponds to the calendar years of 2050÷ 2080.
If one tries to apply the theory of Coulomb explosion [18] and thermal spikes [19] to the LFCM,
we will get the percolation task on permeable sticks, since Coulomb explosion forms stick-like
heavily disordered regions in LFCM. Considering density of sticks as equal to the DR density,
application of the percolation based on the soft sticks calculation results to the example yields the
percolation threshold at the concentration of 9.4·1019 cm−3, which is much larger than the numbers
for the model of permeable spheres, but much less than the known numbers of steadiness for
silicate glass under alpha-irradiation. Such a concentration corresponds to 110,000 years of internal
irradiation action. Nonetheless, such a structure of intersecting percolation stick can effectively
serve as mechanical stress concentrators, thus making the material investigated rather susceptible
to mechanical loads.
5. Conclusions
The 3D percolation problem on random sites with permeable spheres or sticks as percolating
objects is an appropriate way of stimulating the behaviour of radiation damages ensemble in LFCM.
Physical considerations, coming from the basics of the fracture mechanics of brittle materials,
make it possible to conclude that LFCM will eventually mechanically degrade into highly dispersed
state. The current results of computer simulations based on percolation approach indicate that the
potentially dangerous process of LFCM destruction will be urgent in observable future.
Analysis of the results obtained indicates that the destruction of LFCM will undergo in two
stages: first, LFCM will be transformed into fine grain sand (having typical grade of a few microns);
second, the sand will quickly turn into submicronic dust. Quantitative calculation results allow to
predict that for brown LFCM with most of fuel content(8%) the destruction will be actual starting
2015–2020; for black LFCM with less fuel content(4%) this may be actual after year 2045.
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14. Zhidkov A.V., Problems of Chornobyl, 2001, 7, 23–40.
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17. Hellan K. Introduction to Fracture Mechanics. McGraw-Hill, 1984, 364 p. (in Russian).
18. Zhydkov V., Condens. Matter Phys., 2004, 7, No. 4(40), 845–858.
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3D continuum percolation approach and its application to LFCM behaviour forecast
Тривимiрне перколяцiйне наближення та його застосування
до прогнозування поведiнки лавоподiбних паливовмiсних
матерiалiв
В.О.Жидков
Iнститут Проблем Безпеки АЕС НАН України
Отримано 9 лютого 2009 р., в остаточному виглядi – 25 травня 2009 р.
Ця робота присвячена теоретичному вивченню перколяцiйних задач на елементарних проникаючих
об’єктах та його застосуванню до реальних фiзичних об’єктiв. Сфери та iзотропно орiєнтованi скру-
гленi палички були обранi як елементарнi геометричнi об’єкти для моделювання перколяцiї, фiзично
адекватнi для опису поведiнки крихких матерiалiв, що мiстять радiацiйнi дефекти, зокрема до так
званих лавоподiбних паливовмiсних матерiалiв (ЛПВМ), де цi дефекти суттєвим чином впливають
на механiчну стiйкiсть. ЛПВМ являють собою високорадiоактивне скло, що сформувалося у пере-
бiгу активної фази добре вiдомої важкої ядерної аварiї, що вiдбулася на Чорнобильськiй Атомнiй
Електростанцiї у 1986 роцi. Фiзичнi процеси, що мають мiсце у цих матерiалах, становлять великий
практичний iнтерес. Застосовуючи перколяцiйнi моделi до ЛПВМ, можливо створити приблизний
прогноз їх поведiнки. З результатiв моделювання випливає, що фiзичнi властивостi ЛПВМ корiнним
чином змiняться в перiод 2015 ÷ 2045 року, в залежностi вiд вмiсту палива в них.
Ключовi слова: тривимiрне протiкання, опромiнене ядерне паливо, радiацiйнi ушкодження
PACS: 64.60.Ak, 78.70.-g
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