Two-step percolation in aggregating systems
The two-step percolation behavior in aggregating systems was studied both experimentally and by means of Monte Carlo (MC) simulations. In experimental studies, the electrical conductivity, σ, of colloidal suspension of multiwalled carbon nanotubes (CNTs) in decane was measured. The suspension was...
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Інститут фізики конденсованих систем НАН України
2017
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| Цитувати: | Two-step percolation in aggregating systems / N. Lebovka, L. Bulavin, V. Kovalchuk, I. Melnyk, K. Repnin // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13602: 1–10 . — Бібліогр.: 53 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859847081346203648 |
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| author | Lebovka, N. Bulavin, L. Kovalchuk, V. Melnyk, I. Repnin, K. |
| author_facet | Lebovka, N. Bulavin, L. Kovalchuk, V. Melnyk, I. Repnin, K. |
| citation_txt | Two-step percolation in aggregating systems / N. Lebovka, L. Bulavin, V. Kovalchuk, I. Melnyk, K. Repnin // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13602: 1–10 . — Бібліогр.: 53 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The two-step percolation behavior in aggregating systems was studied both experimentally and by means of
Monte Carlo (MC) simulations. In experimental studies, the electrical conductivity, σ, of colloidal suspension of
multiwalled carbon nanotubes (CNTs) in decane was measured. The suspension was submitted to mechanical
de-liquoring in a planar filtration-compression conductometric cell. During de-liquoring, the distance between
the measuring electrodes continuously decreased and the CNT volume fraction ϕ continuously increased (from
10⁻³ up to ≈ 0.3% v/v). The two percolation thresholds at ϕ₁ . 10⁻³
and ϕ₂ ≈ 10⁻²
can reflect the interpenetration of loose CNT aggregates and percolation across the compact conducting aggregates, respectively. The
MC computational model accounted for the core-shell structure of conducting particles or their aggregates, the
tendency of a particle for aggregation, the formation of solvation shells, and the elongated geometry of the
conductometric cell. The MC studies revealed two smoothed percolation transitions in σ(ϕ) dependencies that
correspond to the percolation through the shells and cores, respectively. The data demonstrated a noticeable
impact of particle aggregation on anisotropy in electrical conductivityσ(ϕ) measured along different directions
in the conductometric cell.
Двосхiдцева перколяцiйна поведiнка в агрегованих системах була дослiджена експериментально за допомогою моделювання методом Монте Карло. В експериментальних дослiдженнях були проведенi вимiрювання електричної провiдностi, σ, колоїдних суспензiй багатошарових вуглецевих нанотрубок у деканi. Вимiрювання проводились в планарнiй фiльтрацiйнiй компресiйнiй кондуктометричнiй комiрцi при
механiчному вiджиманнi рiдини з суспензiї. При вiджиманнi вiдстань мiж електродами зменшувалась, i
об’ємна концентрацiя нанотрубок в ϕ в деканi збiльшувалася (вiд 10⁻³
до ≈ 0.3% v/v). Спостерiгалися два
перколяцiйнi переходи при ϕ₁ . 10⁻³
i ϕ₂ ≈ 10⁻²
, якi могли вiдповiдно вiдображати взаємопроникнення
рихлих агрегатiв нанотрубок i перколяцiю по компактних провiдних агрегатах. Обчислювальна модель
Монте Карло враховувала наявнiсть у провiдних частинок або їх агрегатiв структури типу ядро-оболонка,
тенденцiю частинок до агрегацiї, утворення сольватних шарiв на поверхнi частинок i наявнiсть подовженої геометрiї кондуктометричної комiрки. Комп’ютерна модель дозволила виявити наявнiсть двох розмазаних перколяцiйних переходiв в концентрацiйних залежностях σ(ϕ), якi вiдповiдали перколяцiї по
оболонках i ядрах. Спостерiгався значний вплив агрегацiї частинок на анiзотропiю електропровiдностi в
рiзних напрямках кондуктометричної комiрки.
|
| first_indexed | 2025-12-07T15:40:20Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 1, 13602: 1–10
DOI: 10.5488/CMP.20.13602
http://www.icmp.lviv.ua/journal
Two-step percolation in aggregating systems∗
N. Lebovka1†, L. Bulavin2 , V. Kovalchuk2, I. Melnyk2, K. Repnin2
1 F.D. Ovcharenko Institute of Biocolloidal Chemistry of the National Academy of Sciences of Ukraine,
42 Acad. Vernadsky Blvd., 03142 Kyiv, Ukraine
2 Department of Physics, Taras Shevchenko Kyiv National University, 2 Acad. Glushkov Ave., 03127 Kyiv, Ukraine
Received December 3, 2016, in final form February 5, 2017
The two-step percolation behavior in aggregating systems was studied both experimentally and by means of
Monte Carlo (MC) simulations. In experimental studies, the electrical conductivity, σ, of colloidal suspension of
multiwalled carbon nanotubes (CNTs) in decane was measured. The suspension was submitted to mechanical
de-liquoring in a planar filtration-compression conductometric cell. During de-liquoring, the distance between
the measuring electrodes continuously decreased and the CNT volume fraction ϕ continuously increased (from
10−3 up to ≈ 0.3% v/v). The two percolation thresholds at ϕ1 . 10−3 and ϕ2 ≈ 10−2 can reflect the interpen-
etration of loose CNT aggregates and percolation across the compact conducting aggregates, respectively. The
MC computational model accounted for the core-shell structure of conducting particles or their aggregates, the
tendency of a particle for aggregation, the formation of solvation shells, and the elongated geometry of the
conductometric cell. The MC studies revealed two smoothed percolation transitions in σ(ϕ) dependencies that
correspond to the percolation through the shells and cores, respectively. The data demonstrated a noticeable
impact of particle aggregation on anisotropy in electrical conductivityσ(ϕ) measured along different directions
in the conductometric cell.
Key words: multiwalled carbon nanotubes, colloidal suspensions, anisotropy of electrical conductivity,
two-step percolation
PACS: 61.48.De, 64.60.ah, 72.80.Tm, 73.50.-h, 73.61.-r
1. Introduction
Classical percolation with one sharp transition from a non-conducting to a conducting state is com-
monly expected for composites filled with highly conducting particles. So far, a lot of different models
and equations were proposed for a description of the electrical conductivity behavior [1, 2].
However, in many experimental observations the percolation in composites is more complicated.
The presence of two-step [double percolation (DP)], several-step (multiple percolations) and even fuzzy
(smeared) type of percolation transitions has been reported bymany researchers [3–22]. Different mech-
anisms related with distribution of types of particles and types of the electrical contacts, geometrical ef-
fects, selective distribution of a conducting particle in multi-component media (e.g., in polymer blends),
the existence of static and kinetic network formation processes, as well as the core-shell structure of
particles may be responsible for the multiple percolation thresholds.
The superconducting DP has been experimentally observed and attributed to the distributions of par-
ticles within the composites [3, 4]. The DP has been described accounting for different types of electrical
contacts in the material: clean contacts with fraction α and insulator separated contacts with fraction
1−α [5]. This parameter α was a relevant function of the applied pressure and temperature, the type
of a matrix, the insulating layer on the particle’s surface, etc. For this model, the effective medium (EM)
theory predicted the existence of the DP at α < 1. An experimental realization for polymer/filler com-
posites with the multiple percolation thresholds has been explained accounting for the local variations
∗This work is dedicated to the 60th birthday of Professor Yurij Holovatch.
†Corresponding author, E-mail: lebovka@gmail.com.
© N. Lebovka, L. Bulavin, V. Kovalchuk, I. Melnyk, K. Repnin, 2017 13602-1
https://doi.org/10.5488/CMP.20.13602
http://www.icmp.lviv.ua/journal
N. Lebovka et al.
in filler concentration and/or irregularities in the shape and orientation of particles [6, 7]. The multiple
percolation was attributed to the presence of various geometric shapes and correlated arrangements of
particles.
For a two-dimensional (2D) square lattice, the DP has been simulated accounting for different edge-
edge nearest neighbor NN contacts and next-nearest neighbor NNN (von Neumann’s neighborhood) con-
tacts [8]. For NN (or NNN) and NN+NNN (Moore’s neighborhood, composed of a central cell and eight cells
that surround it) contacts, the percolation thresholds are 0.5927 and 1−0.5927 = 0.4073, respectively [9].
The existence of the DP due to a geometrical effect has been confirmed by experiments with angular and
rounded silicon carbide (SiC) particles in a polymer rubber [10]. For angular SiC grains with sharp edges
and rather flat surfaces, the DP was observed. However, for the rounded SiC grains, only one threshold
was observed. The value of conductivity at a plateau between two percolations was found to be close to
the geometric mean of the limiting conductivities at low (σi) and at high (σc) concentrations of particles.
The concentration dependence of electrical conductivity was simulated using the three-dimensional (3D)
impedance network model accounting for the presence of both edge and face contacts, where the model
also revealed the presence of the DP.
A particular type of blend double percolation (BDP) has been observed in immiscible polymer blends
filled with conducting particles [11]. For these systems, the conducting particles have different affinity to
polymer components A and B and are capable of finely dispersing only in one of them, e.g., in A. This
caused a selective spatial localization of conducting particles in A component. In such blends, the perco-
lation in electrical conductivity requires both percolation of conducting particles within the component
A and connectivity of component A within component B. The BDP has been frequently observed using
a carbon black (CB) conducting filler [12–15]. A fine regulation of electrical conductivity is possible by
changing the concentration of CB in phase A and the concentration of phase A in phase B. A general con-
cept of multiple percolation hierarchy for CB filled polymer blends was developed [16]. The percolation
concentration of CB in the blend may be rather low [17]. This conclusion was supported by the data of
computer simulation of the percolation behavior of composites containing small conducting particles be-
tween large isolating particles [18]. The BDP has been also observed in the polymers filled with a short
carbon fiber [19] and with Ketjenblack [20]. It was noted that carbon nanoparticles can affect the mor-
phology of the blends. The combination of fillers (graphite and carbon fiber) has been used to enhance
the interparticle connectivity and increase the electrical conductivity [21]. The effects were explained in
terms of bridged DP mechanism.
An interesting example of DP in carbon fiber reinforced cement-based materials has been experimen-
tally discovered [22]. The percolation was observed at ≈ 0.30−0.80 vol.% fibers in the paste portion and
at 70−76 vol.% carbon fiber cement paste in mortar.
In recent years, the composites filled with carbon nanotubes (CNTs) have attracted great attention.
Such composites have many fascinating properties due to their versatility of applications in antistatic
devices, electromagnetic interference shielding materials, capacitors, and sensors [23]. A significant re-
duction in percolation concentration of CNTs due to the BDP has been reported [24–35]. The DP in CNT
epoxy composites has been attributed to a static (at higher concentration) and to a kinetic network for-
mation (at lower concentration) processes [36]. The electrical conductivity plateau was attributed to the
presence of a superstructure of flocculated particles. Flocculation can noticeably affect the value of the
percolation threshold. The proposed model also accounted for the magnitude of individual interparticle
contact resistance. The DP in CNT filled liquid crystals medium has been attributed to a core-shell struc-
ture of conducting particles, where the EM theory has been used to explain the experimental data [37].
Percolation can be greatly affected by interaction between conducting particles and their aggregation.
The theory of correlated percolation predicts a strong dependence of the percolation threshold upon the
details of interaction [38, 39]. The effects of aggregation in multiple percolation are still far from being
completely understood. For well-dispersed CNTs (with small size of agglomerate) in polymeric compos-
ites, the increase in the electrical conductivity [40] and very small percolation thresholds [41] have been
observed. However, a shear-induced re-aggregation in well-dispersed systems facilitated the intercon-
nectivity of CNTs as well as could result in a decrease of the percolation threshold [42]. Regarding this
behavior two controversial effects can be important. On the one hand, a good dispersion is helpful in
forming well distributed paths. On the other hand, partial agglomerations are helpful in reducing the
distance between the tubes to the tunneling range.
13602-2
Two-step percolation in aggregating systems
The present work is devoted to experimental and computational studies of percolation in an aggre-
gating system. In the experimental part the electrical conductivity σ of suspensions of multiwalled CNTs
in decane was measured using a home-made filtration-compression conductometric cell. The decane was
continuously expressed from the suspension by pressure. During de-liquoring, the concentration of CNTs
increased the changes in σ encompassed by both the effects of changes in concentration and by agglom-
eration of CNTs. In the computational part, the Monte Carlo (MC)model was developed accounting for the
core-shell structure of conducting particles or their aggregates, the tendency of particle aggregation, the
formation of solvation shells, and the elongated geometry of the conductometric cell. Both experiment
and the MC data revealed DP. An impact of particle aggregation on anisotropy in electrical conductivity
measured along different directions in the conductometric cell is discussed. In section 2, the materials,
experimental methods, MC model, and details of simulation are presented. The obtained results are dis-
cussed in section 3. Conclusions and final remarks are formulated in section 4.
2. Materials and methods
2.1. Materials
Multiwalled CNTs, obtained by chemical vapour deposition of graphite in the gas phase with a cata-
lyst FeAlMo0.07 (Spetsmash, Ukraine), were used as a pristine material [43]. To separate the CNTs from
the catalyst and mineral impurities, the material was treated by aqueous solutions of alkali (NaOH) and
hydrochloric acid (HCl). The samples were filtered to remove the excess acid and repeatedly washed with
distilled water until the pH value of distilled water was reached. The studied CNTs have the outer diam-
eter d ≈ 20−40 nm and length l ≈ 5−10 microns. The density of CNTs was assumed to be the same as
the density of pure graphite, ρn = 2.1 g/cm3. Decane, CH3(CH2)8CH3 (TU 6-09-3614-74, Ltd. “Novohim”
Kharkiv, Ukraine) was used as a fluidic matrix. Pure decane has a fairly small electrical conductivity
(< 10−15 S/cm [44]), solubility of water in decane being also very low (7.2×10−3% [45]).
The CNT-decane colloidal suspensions were obtained by adding appropriate weights of CNTs to the
decane with subsequent 20−30 min sonication of the mixture using a UZD-22/44 ultrasonic disperser
(Ukrprylad, Sumy, Ukraine) at a frequency of 44 kHz and output power of 150 W. The measurements
were started immediately after sonification.
2.2. Filtration-compression conductometric cell
Figure 1 shows the sketch of home-made vertical planar filtration-compression conductometric cell.
CNT-decane suspension was placed into calibrated cylinder between two electrodes. The electrode sur-
faces were covered by the plates of porous nickel. A meshed bottom electrode permits filtration of a
dispersed medium. The displacement of the upper electrodes was realized by the screw-down press and
was controlled by cathetometer MK-6 (LOMO, Saint Petersburg) with a precision of ±0.01 mm. The fil-
ter liquor was collected within the bottom container. An initial volume concentration of suspension was
ϕi = 0.1% v/v. The electrical conductivity was measured using AM 3003 conductivity meter (Data.com) at
the temperature of 293 K, frequency of 1 kHz and voltage of 0.25 V. The choice of 1 kHz made it possible
to avoid the effects of near-electrode polarization and migration of CNTs in an external electric field. To
avoid CNTs’ sedimentation the conductometric cell was intensively shaked before each measurement.
2.3. Microstructure of suspensions
Optical microphotographs of CNT-decane suspensions were obtained using a microscope Biolar 03-
808 (Warsaw, Poland). Suspensions were placed in a flat cell with the layer thickness of 70 µm. All mea-
surements were done at 293 K. Figure 2 shows examples of micro-photos of CNT-decane suspensions at
different CNT-decane suspensions at different volume fractions of CNTs, ϕ. The CNT aggregates became
visible at low concentration (ϕ ≈ 0.01%) and they grown in size with an increase of ϕ. Finally, a large
spanning aggregate with the size exceeding the microscope visual field was formed at ϕ≈ 0.025%.
13602-3
N. Lebovka et al.
Suspension
Filter liquor
Electrodes
Membrane
Compression
Figure 1. (Color online) Sketch of a vertical planar filtration-compression conductometric cell. The
meshed bottom electrode was placed onto the membrane. The CNT-decane suspension between the up-
per and bottom electrodes was submitted to mechanical de-liquoring, the decane was filtered through
the membrane, and filter liquor was collected within the container.
0.05 %0.025 %
0.10 % 0.15 %
500 µm
Figure 2.Micro-photos of CNT-decane suspensions at different volume fractions of CNTs, ϕ, and temper-
ature of 293 K.
2.4. Monte Carlo computational model
The MC simulation was used to imitate the changes of electrical conductivity measured in the pla-
nar filtration-compression cell during mechanical de-liquoring of colloidal suspension. The tendency of
particle aggregation was accounted for using the previously described model of the interactive cluster-
growth [46, 47]. A 2D model on a square lattice was used. All sites were initially empty and have small
electrical conductivity, σi(= 1). The empty sites were randomly filled with conducting particles of much
larger electrical conductivity, σc(= 106).
The core-shell structure of conducting particles was assumed. The cores of conducting particles were
covered with a conducting shell of intermediate electrical conductivity σs(= 103). The shells can account
for the presence of solvation shells around the conducting species of a smaller electrical conductivity.
The probability of filling a new empty site pr was dependent on the state of the NNs in the Moore
neighborhood (it is composed of a central cell and eight cells that surround it):
• In the absence of direct contacts (core-core or core-shell) the probability of site filling was 1/r ,
where r Ê 1 is a factor of aggregation [figure 3 (a:I)].
• In the presence of a direct contact, the probability was f for core-core contacts [figure 3 (a:II)] and
13602-4
Two-step percolation in aggregating systems
σs
σi
pr=1/r pr=f pr=1-f
σc core-core core-shell
core-shell
ϕ=0.01 ϕ=0.032
ϕ=0.051
ϕ=0.120
ϕ=0.501
y
x
a)
b)
I) II) III)
Figure 3. (Color online) Description of the aggregation model of core-shell particles (a) and examples of
simulated aggregation patterns during the filtration-compression process at different volume fractions
of conducting particles ϕ (b).
1− f for core-shell contacts [figure 3 (a:III)]. Here, f is a solvation factor (0 < f É 1) that accounts for
core-core and core-shell interactions. The cases f = 1 and f = 0 correspond to the strong core-core
and core-shell contacts, respectively.
The volume fraction of occupied sites ϕ was determined as a ratio of the number of filled sites N and
the total number of sites Lx ×Ly . At an initial state, Lx = Ly and ϕ=ϕi. In all calculations, the long lattice
side was Lx = 512 and the initial concentration of conducting particles was ϕi = 0.01.
Figure 3 (b) shows an example of evolution of aggregation patterns during the MC simulation of
filtration-compression process. The data are presented for a large factor of aggregation, r = 1000, and
weak solvation effects, f = 1 (only core-core contacts are allowed). Before compression, the size of the
lattice was Ly = Lx = 512. In the course of compression the lattice anisotropy a = Lx /Ly (a = ϕ/ϕi) and
the volume fraction of particles increase. At high values of ϕ= 0.501, the electrical closure of contacts in
vertical direction y was visually observed.
The MC model used makes it possible to account for the tendency of particles aggregation (aggrega-
tion factor r ) and formation of shells in the vicinity of conductivity particles (solvation factor, f ). The
value of r controls the degree of aggregation. At r = 1, the aggregation is absent. However, it can be pro-
nounced at r ≫ 1. The shells have intermediate electrical conductivity. They are “active” and can also
control aggregation. For example, for the case f ≪ 1 (strong solvation), the deposition of a newcomer
in the vicinity of the core of the previously deposited particle is unlikely to take place and the core-shell
contacts are mainly realized. In another limiting case f = 1 (weak solvation), the core-core contacts are
mainly realized. At small values of f (≪ 1), the formation of checkerboard patterns with loose density
is observed. In the limit of f → 1, the model predicts the formation of more compact and less spatially
extended aggregates.
To simulate the filtration-compression process, the system was compressed along the vertical direc-
tion y by sequentially removing the upper rows and redistributing the filled sites in these rows within the
rest of the system. The fraction of occupied sites continuously increased with an increase of compression
13602-5
N. Lebovka et al.
ϕ= aϕi, where a = Lx /Ly is a lattice anisotropy. The compression was stopped atϕ= 1 that corresponded
to the final lattice anisotropy of a = 1/ϕi. Periodical boundary conditions were used in the horizontal x
direction.
The Hoshen-Kopelman algorithm was used for labeling different clusters [48]. The value of the per-
colation threshold ϕc corresponds to the minimum fraction of occupied sites at which an infinite cluster
formed in the infinite lattice. Electrical conductivity of the system was calculated using Frank and Lobb’s
algorithm [49]. This algorithm utilizes a repeated application of a sequence of series, parallel and star-
triangle Y−∆ transformations to the square lattice bonds. The final result of this sequence of transforma-
tions is a reduction of a finite portion of the lattice to a single bond that has the same conductance as the
entire lattice portion. We used a scheme of four equivalent resistors (see, e.g. [50]) with high, σc = 106,
and low, σi = 1, conductivity for the occupied and empty sites, respectively.
2.5. Statistical analysis
The experiments were replicated 3−5 times. In the MC calculations, the number of independent runs
was 100. The mean values and the standard deviations were calculated. The error bars in all the figures
correspond to the confidence level 95%. The least square fitting of the experimental dependencies, deter-
mination of the fitting parameters and the coefficient of determination, were provided using Table Curve
2D software (Jandel Scientific, USA).
3. Results and discussion
3.1. Electrical conductivity of CNT-decane colloidal suspensions
Figure 4 presents experimental data on electrical conductivity of CNT-decane suspension, σ, versus
the volume concentration of CNTs, ϕ. The DP with thresholds at ϕ1 . 10−3 and ϕ2 ≈ 10−2 (estimated
from inflection point in figure 4) was observed. For large concentrations above ϕ ≈ 0.1, the σ(ϕ) curve
saturated. Such a DP behavior of electrical conductivity can reflect the complexity of electrical contacts
in the studied suspensions. A similar behavior was previously observed in the liquid crystalline medium
filled with CNTs [37]. The simplest hypothesis explaining the DP behavior can be based on the model of
shell-core structure of CNT particles or their aggregates. In this model, the CNT particles with electrical
10-3 10-2 10-1
10-5
10-4
10-3
10-2
10-1
σ
, S
/c
m
ϕ
ϕ1
ϕ2
Figure 4. (Color online) Electrical conductivity σ versus volume concentration ϕ of CNTs in decane. The
value of σ was measured using the vertical planar filtration-compression conductometric cell. The sys-
tems demonstrated the presence of two percolation thresholds at ϕ1 . 10−3 and ϕ2 ≈ 10−2.
13602-6
Two-step percolation in aggregating systems
conductivity σc are surrounded by solvation shells with electrical conductivity σs that is intermediate
between the conductivity of CNTs, σc, and continuous medium, σi. The percolation threshold at a small
concentration (at ϕ1 ≈ 10−3) can be explained by interpenetration of the shells of loose CNT aggregates.
With a further increase of ϕ above ϕ1, the external pressure can cause transformation of large loose
aggregates into the more compact aggregates. Thus, the threshold at ϕ2 ≈ 10−2 can reflect the conducting
path across the more compact aggregates with higher effective electrical conductivity. It is interesting to
note that the saturation value of electrical conductivity σ≈ 0.4 S/cm for CNT-decane suspensions was still
much smaller (approximately by one order of magnitude) compared to the experimentally measured
value of σ ≈ 3 S/cm for the pressed CNT-air system. It can reflect the formation of a tightly bound low
conducting thin solvation layer of decane near the surface of CNTs.
3.2. Monte Carlo simulations
Figure 5 presents examples of calculated dependencies of relative electrical conductivity σ/σi versus
concentration of particlesϕ in horizontal (x, solid lines) and vertical (y , dashed lines) directions. The data
are presented for r = 100 (a) and r = 1000 (b) and for different values of solvation factor, f .
The electrical conductivity along short vertical direction y , σy , always exceeded the value σx along
horizontal x direction. Anisotropy of electrical conductivity was absent at r = 1 and became noticeable
at large values of r (≫ 1). The rapid grow of electrical conductivity can be explained by the formation of
electrical closures in a vertical direction cased by the presence of large aggregates [figure 3 (b)].Moreover,
at small values of f ( f ≪ 1, strong solvation), a clear DP behavior with a sharp increase of electrical con-
ductivity for the isolator-shell (at ϕ=ϕis) and shell-conductor (at ϕ=ϕsc) transitions was observed (fig-
ure 5). However, at high values of f ( f → 1, weak solvation), the electrical conductivity transitions were
smeared over a wide interval of concentrations. To characterize the percolation behavior in this work,
the values ϕis andϕsc along horizontal x and vertical y directions were roughly estimated as geometrical
concentrations that correspond to the mean geometrical conductivities
p
σiσs and
p
σsσc , respectively
[figure 5 (a)]. Note that for the 2D systems with equal concentrations of the phases and their random dis-
tribution, the theory predicts a geometric conductivity at the percolation threshold [51]. MC simulation
predicted the geometrical concentration to be rather close to the percolation threshold [52].
Figure 6 presents these geometrical concentrations for the isolator-shell, ϕis, and shell-conductor, ϕsc,
transitions in horizontal (x, solid lines) and vertical (y , dashed lines) directions versus the probability of
the face-to-face contact f at a fixed factor of aggregation, r = 100 (a) and versus a factor of aggrega-
tion r at a fixed factor of solvation, f = 0.5 (b). In all cases we observed ∆ϕis = ϕis(x)−ϕis(y) > 0 and
0.2 0.4 0.6 0.8 1
100
101
102
103
104
105
106
ϕ
σ
/σ
i
y
x
r=100
f=
0.001
0.5
1.0
ϕis
ϕsc
a)
siσσ
csσσ
0.2 0.4 0.6 0.8 1
100
101
102
103
104
105
106
ϕ
σ
/σ
i
y
x
r=1000
f=
0.001
0.5
1.0
b)
Figure 5. (Color online) Dependencies of relative electrical conductivity σ/σi versus concentration of
particles ϕ in horizontal (x, solid lines) and vertical (y , dashed lines) directions for different values of
solvation factor, f , and for the factor of aggregation r = 100 (a) and r = 1000 (b).
13602-7
N. Lebovka et al.
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
f
ϕ
is
,ϕ
sc
r=100
a)
ϕis
ϕsc
x
x
y
y
100 101 102 1030.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r
ϕ
is
,ϕ
sc
f=0.5f=0.5
ϕis
ϕsc
x
x
y
y
b)
Figure 6. (Color online) Geometrical concentrations for the isolator-shell, ϕis, and shell-conductor, ϕsc ,
transitions in horizontal (x, solid lines) and vertical (y , dashed lines) directions versus the factor of sol-
vation f at r = 100 (a) and versus the factor of aggregation r at f = 0.5 (b).
∆ϕsc =ϕsc(x)−ϕsc(y) > 0, i.e., percolation was more enhanced in a vertical direction y as compared with
horizontal direction x.
At a fixed value of r , ϕis (isolator-shell transition) increases for both x and y directions with an in-
crease of f [figure 6 (a)]. However, the valueϕsc (shell-conductor transition) increases for x and decreases
for y directions with an increase of f . At a fixed value of f , the differences ∆ϕis and ∆ϕsc increase with
an increase of the aggregation factor r [figure 6 (b)]. In horizontal direction x, the both geometrical con-
centrations ϕis and ϕsc increase with an increase of r . It reflects suppression effect of aggregation on the
percolation in x direction. In vertical direction y , the opposite effect of aggregation is observed for shell-
conductor transition, and the value of ϕsc decreases with an increase of r . However, the aggregation has
practically no effect on geometrical concentrations for the isolator-shell, ϕis, transition.
4. Conclusions and final remarks
The experimental data obtained for electrical conductivity, σ, of colloidal CNT-decane suspension
measured in the planar filtration-compression conductometric cell evidenced the presence of two perco-
lation thresholds at ϕ1 . 10−3 and ϕ2 ≈ 10−2. The experimentally observed DP can be explained on the
basis of shell-core model of CNT particles or their aggregates. The percolation threshold at ϕ1 and ϕ2 can
reflect the interpenetration of low conducting shells of loose CNT aggregates and percolation across the
more compact conducting aggregates, respectively.
The MC model proposed accounted for the core-shell structure of conducting particles, the tendency
of particle aggregation, the formation of solvation shells, and an elongated geometry of the conducto-
metric cell. The studies revealed DP transitions that correspond to the percolation through the shells and
cores. The MC data also demonstrated a noticeable impact of particle aggregation on anisotropy in σ(ϕ)
dependencies measured along different directions in the conductometric cell. Simulation data predicted
a rather complex behavior for electrical conductivity curves in dependence on aggregation, r , and sol-
vation, f , factors. The impact of r and f on the percolation behavior can be explained accounting for
the differences in the structure of aggregates. An easier percolation along the shorter vertical axis y for
strongly aggregated systems is expected. The impact of solvation factor f on the percolation behavior
can reflect the changes in the compactness of aggregates. For strong solvation effects, f ≪ 1, loose ag-
gregates with small density were formed while for weak solvation effects, f ≈ 1, the aggregates became
more compact and less spatially extended. However, quantitative comparison between experiments and
the present model is difficult due to fairly artificial assumptions of the model accounting for the tendency
13602-8
Two-step percolation in aggregating systems
of particles aggregation and formation of low conducting shells in the vicinity of conductivity particles.
Similar difficulties have been recently stated in the studies of percolation of disordered segregated com-
posites [53]. Future work should consider amore realistic model with account for the tunneling transport
between conducting particles and an elongated shape of CNTs.
Acknowledgements
This work was partially funded by the National Academy of Sciences of Ukraine, Projects No. 2.16.1.4
and No. 43/17-H.
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Двосхiдцева перколяцiя в агрегованих системах
М.I. Лебовка1, Л. Булавiн2, В. Ковальчук2, I. Мельник2, K. Репнiн2
1 Iнститут бiоколоїдної хiмiї iм. Ф.Д. Овчаренка НАН України,
бульв. акад. Вернадського, 42, 03142 Київ, Україна
2 Фiзичний факультет, Київський нацiональний унiверситет iм. Тараса Шевченка,
пр. акад. Глушкова, 2, 03127 Київ, Україна
Двосхiдцева перколяцiйна поведiнка в агрегованих системах була дослiджена експериментально за до-
помогою моделювання методом Монте Карло. В експериментальних дослiдженнях були проведенi вимi-
рювання електричної провiдностi, σ, колоїдних суспензiй багатошарових вуглецевих нанотрубок у дека-
нi. Вимiрювання проводились в планарнiй фiльтрацiйнiй компресiйнiй кондуктометричнiй комiрцi при
механiчному вiджиманнi рiдини з суспензiї. При вiджиманнi вiдстань мiж електродами зменшувалась, i
об’ємна концентрацiя нанотрубок в ϕ в деканi збiльшувалася (вiд 10−3 до ≈ 0.3% v/v). Спостерiгалися два
перколяцiйнi переходи при ϕ1 . 10−3 i ϕ2 ≈ 10−2, якi могли вiдповiдно вiдображати взаємопроникнення
рихлих агрегатiв нанотрубок i перколяцiю по компактних провiдних агрегатах. Обчислювальна модель
Монте Карло враховувала наявнiсть у провiдних частинок або їх агрегатiв структури типу ядро-оболонка,
тенденцiю частинок до агрегацiї, утворення сольватних шарiв на поверхнi частинок i наявнiсть подовже-
ної геометрiї кондуктометричної комiрки. Комп’ютерна модель дозволила виявити наявнiсть двох роз-
мазаних перколяцiйних переходiв в концентрацiйних залежностях σ(ϕ), якi вiдповiдали перколяцiї по
оболонках i ядрах. Спостерiгався значний вплив агрегацiї частинок на анiзотропiю електропровiдностi в
рiзних напрямках кондуктометричної комiрки.
Ключовi слова: багатошаровi вуглецевi нанотрубки, колоїднi суспензiї, анiзотропiя електричної
провiдностi, двосхiдцевий перколяцiйний перехiд
13602-10
https://doi.org/10.1039/C4RA13689F
https://doi.org/10.1016/j.compscitech.2006.02.037
https://doi.org/10.1103/PhysRevE.92.012502
https://doi.org/10.1103/PhysRevA.32.506
https://doi.org/10.1103/PhysRevB.29.387
https://doi.org/10.1016/j.polymertesting.2013.03.005
https://doi.org/10.1016/j.polymer.2016.06.004
https://doi.org/10.1016/j.polymer.2009.11.047
https://doi.org/10.1007/s11167-005-0420-y
https://doi.org/10.1103/PhysRevA.38.4198
https://doi.org/10.15407/ujpe60.09.0910
https://doi.org/10.1103/PhysRevB.14.3438
https://doi.org/10.1103/PhysRevB.37.302
https://doi.org/10.1140/epjb/e2010-00089-2
https://doi.org/10.1103/PhysRevE.94.042112
https://doi.org/10.1103/PhysRevE.79.020104
Introduction
Materials and methods
Materials
Filtration-compression conductometric cell
Microstructure of suspensions
Monte Carlo computational model
Statistical analysis
Results and discussion
Electrical conductivity of CNT-decane colloidal suspensions
Monte Carlo simulations
Conclusions and final remarks
|
| id | nasplib_isofts_kiev_ua-123456789-156554 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T15:40:20Z |
| publishDate | 2017 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Lebovka, N. Bulavin, L. Kovalchuk, V. Melnyk, I. Repnin, K. 2019-06-18T16:30:44Z 2019-06-18T16:30:44Z 2017 Two-step percolation in aggregating systems / N. Lebovka, L. Bulavin, V. Kovalchuk, I. Melnyk, K. Repnin // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13602: 1–10 . — Бібліогр.: 53 назв. — англ. 1607-324X PACS: 61.48.De, 64.60.ah, 72.80.Tm, 73.50.-h, 73.61.-r DOI:10.5488/CMP.20.13602 arXiv:1703.10373 https://nasplib.isofts.kiev.ua/handle/123456789/156554 The two-step percolation behavior in aggregating systems was studied both experimentally and by means of Monte Carlo (MC) simulations. In experimental studies, the electrical conductivity, σ, of colloidal suspension of multiwalled carbon nanotubes (CNTs) in decane was measured. The suspension was submitted to mechanical de-liquoring in a planar filtration-compression conductometric cell. During de-liquoring, the distance between the measuring electrodes continuously decreased and the CNT volume fraction ϕ continuously increased (from 10⁻³ up to ≈ 0.3% v/v). The two percolation thresholds at ϕ₁ . 10⁻³ and ϕ₂ ≈ 10⁻² can reflect the interpenetration of loose CNT aggregates and percolation across the compact conducting aggregates, respectively. The MC computational model accounted for the core-shell structure of conducting particles or their aggregates, the tendency of a particle for aggregation, the formation of solvation shells, and the elongated geometry of the conductometric cell. The MC studies revealed two smoothed percolation transitions in σ(ϕ) dependencies that correspond to the percolation through the shells and cores, respectively. The data demonstrated a noticeable impact of particle aggregation on anisotropy in electrical conductivityσ(ϕ) measured along different directions in the conductometric cell. Двосхiдцева перколяцiйна поведiнка в агрегованих системах була дослiджена експериментально за допомогою моделювання методом Монте Карло. В експериментальних дослiдженнях були проведенi вимiрювання електричної провiдностi, σ, колоїдних суспензiй багатошарових вуглецевих нанотрубок у деканi. Вимiрювання проводились в планарнiй фiльтрацiйнiй компресiйнiй кондуктометричнiй комiрцi при механiчному вiджиманнi рiдини з суспензiї. При вiджиманнi вiдстань мiж електродами зменшувалась, i об’ємна концентрацiя нанотрубок в ϕ в деканi збiльшувалася (вiд 10⁻³ до ≈ 0.3% v/v). Спостерiгалися два перколяцiйнi переходи при ϕ₁ . 10⁻³ i ϕ₂ ≈ 10⁻² , якi могли вiдповiдно вiдображати взаємопроникнення рихлих агрегатiв нанотрубок i перколяцiю по компактних провiдних агрегатах. Обчислювальна модель Монте Карло враховувала наявнiсть у провiдних частинок або їх агрегатiв структури типу ядро-оболонка, тенденцiю частинок до агрегацiї, утворення сольватних шарiв на поверхнi частинок i наявнiсть подовженої геометрiї кондуктометричної комiрки. Комп’ютерна модель дозволила виявити наявнiсть двох розмазаних перколяцiйних переходiв в концентрацiйних залежностях σ(ϕ), якi вiдповiдали перколяцiї по оболонках i ядрах. Спостерiгався значний вплив агрегацiї частинок на анiзотропiю електропровiдностi в рiзних напрямках кондуктометричної комiрки. This work was partially funded by the National Academy of Sciences of Ukraine, Projects No. 2.16.1.4 and No. 43/17-H. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Two-step percolation in aggregating systems Двосхiдцева перколяцiя в агрегованих системах Article published earlier |
| spellingShingle | Two-step percolation in aggregating systems Lebovka, N. Bulavin, L. Kovalchuk, V. Melnyk, I. Repnin, K. |
| title | Two-step percolation in aggregating systems |
| title_alt | Двосхiдцева перколяцiя в агрегованих системах |
| title_full | Two-step percolation in aggregating systems |
| title_fullStr | Two-step percolation in aggregating systems |
| title_full_unstemmed | Two-step percolation in aggregating systems |
| title_short | Two-step percolation in aggregating systems |
| title_sort | two-step percolation in aggregating systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156554 |
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