CC BY
31
Харьков: Институт монокристаллов, 2001. С.137-145. 2. Бурачас С.Ф., Тилман Б.Л., Бондарь В.Г., Дубовик М.Ф. Изучение условий выращивания кристаллов методом Чохральского // Получение и исследование монокристаллов. 1978. С. 1-6. 3. Михальчук В.И., Ваше-рук А.В. Определение оптимальных тепловых условий и температурных градиентов монокристаллов кремния большого диаметра // Нові технології. 2003. №2(3). С. 30-34. 4. Эйденсон А.М., Пузанов Н.И. Выращивание совершенных монокристаллов кремния методом Чохральского // Неорган. материалы. 1997. Т.33 С. 272-279. 5. Тилман Б. .Л., Бурачас С. Ф. Анализ условий выращивания монокристаллов методом Чохральского // Физика и химия кристаллов. 1977. С. 01-04. 6. Вашерук А.В., Кротюк И.Г. Температурные поля на границе раздела двух фаз в процессе выращивания монокристаллов кремния методом Чохральского // Вісник Технологічного університету Поділля. 2003. Т. 2, №3. С. 146-148. 7. Багдасаров Х.С., Горяйнов Л.А. Состояние и задачи исследования процессов тепло- и массопереноса при выращивании монокристаллов из расплава // Инж.-физ. журн. 1983. Т. 44, № 2. С. 329-340. 8. Рейви К. Дефекты и примеси в полупроводниковом кремнии. М.: Мир, 1984. 176 с. 9. Туровский Б.М., Шендерович ИЛ, Попов А.И, Гришин В.П Особенности распространения кислорода в монокристаллах кремния, выращиваемых методом Чохральского с автоматическим регулированием диаметра // Научные труды Гиредмета. 1980. Т.102. С. 33-35. 10. Татарченко В.А. Устойчивый рост кристаллов. М.: Наука, 1988. 310с. 11. Пузанов Н.И., Эйденсон А.М. Опасные микродефекты в верхней части слитков кремния большого диаметра // Неорга-
нические материалы. 1997. Т.33, №8. С. 912 — 917. 12. Эйденсон А.М., Пузанов Н.И. Классификация ростовых микродефектов в кристаллах кремния, выращенных по методу Чохральского //Неорган.материалы. 1995. Т. 31, №4. С. 435-443. 13. Voronkov V.V. The Mechanism of Swirl Defect Formation in Silicon Crystals // J. Cryst. Growth. 1982. Vol.59, №3. P. 625-643. 14. Пузанов Н.И, Эйдензон А.М. Классификация микродефектов в без-дислокационном кремнии, выращенном по методу Чохральского // Цв. металлы. 1988. №4. С. 69-72.
Поступила в редколлегию 25.09.2004
Рецензент: д-р физ.-мат. наук, проф. Гордиенко Ю.Е.
Океании Анатолий Петрович, д-р техн. наук, профессор кафедры КСА ИЭНТ. Научные интересы: автоматизация процессов выращивания монокристаллов кремния. Увлечения и хобби: охота. Адрес: Украина, 39600, Кременчуг, ул. Пролетарская, 24/37, тел. (05366) 3-11-22.
Притчин Сергей Эмильевич, канд.техн.наук, доцент кафедры КСА ИЭНТ. Научные интересы: автоматизация процессов выращивания монокристаллов кремния. Увлечения и хобби: охота. Адрес: Украина, 39600, Кременчуг, ул. Пролетарская, 24/37, тел. (05366) 3-3038 (доп. 331).
Вашерук Алекеандр Ваеильевич, ст. преподаватель кафедры КСА ИЭНТ. Научные интересы: моделирование процессов выращивания монокристаллов кремния. Увлечения и хобби: литература. Адрес: Украина, 39600, Кременчуг, ул. Пролетарская, 24/37, тел. (05366) 3-30-38 (доп. 340).
UDC 517.958: 536.71
SIMULATION OF DIFFUSION-CONVECTION PROCESSES IN MICROFLUIDIC CHANNELS
KLYMENKO O.V, OLEINICKA.I.,
AMATORE C.A, SVIR I.B.______________________
An application of the conformal mapping approach is proposed for the treatment of double band assemblies operating in a microfluidic channel using a quasi-conformal map for the numerical simulation of diffusion-convection processes occurring when the channel is submitted to a microfluidic flow. The latter applies also when such assemblies perform within a thin film of electrolyte. The results are presented for a generator-collector operation of the assembly but the methods apply also to other situations (for example, generator-generator mode). An optimization procedure for simultaneous fitting of any number of pairs of theoretical and experimental current curves is described. The latter can be used for recovering the values of parameters of the studied system from experimentally measured current data.
1. Introduction
Quasi-conformal mappings include conformal maps obtained by the Schwartz-Christoffel transformation and by superposition of analytic functions of complex variables [1-4]. The application of such mappings represents the most efficient and easiest way of simulation of accurate numerical solutions of two-dimensional electrochemical problems when isoconcentration and
flux lines are extremely curved as it occurs at microelectrodes and microelectrode assemblies. It often leads to the straightforward derivation of analytical limiting laws, but also is extremely powerful for the corresponding finite difference simulations since the extremely curved flux lines in the real space become almost flat in the conformal space, a fact which ensures efficiency and accuracy as well as speed of the simulations without resorting to ad-hoc highly variable dense meshes [1]. This enhanced performance has been exemplified on several two-dimensional diffusion problems occurring at single microelectrodes and microelectrode assemblies [4-11]. Furthermore, the transforms used generally allow analytical formulations of the back transformations so that the real flux lines and concentration profiles may be readily obtained from the ones generated in the transformed space [4, 7-11].
Here, we wish to propose an application of the quasiconformal mapping approach [4] to the simulation of diffusion-convection processes occurring at double band microelectrode assemblies operating within a channel submitted or not to a microfluidic flow. This is exemplified numerically for the parabolic flow but can be adapted to any type of flow. In particular, the approach is valid when the double band assemblies perform within a thin film of electrolyte [5,6] owing to the identity of their 2D-cross sections. It is readily adaptable for the simulation of the behaviour of assemblies involving an arbitrary number of microband electrodes performing at different or identical potentials. This includes both their arrays and a single microband electrode in a channel flow system.
РИ, 2005, № 1
47
Our conformal transformation for the numerical simulation is proposed in complex variables. This allows one to avoid sophisticated algebraic manipulations that do not necessarily lead to closed analytical expressions for the derivatives in the most general case. Furthermore, this simplifies the programming of complex diffusion-convection transport problems. We exemplify the performance of the method by considering its results when a paired band assembly is operated chronoamperometricaly in the generator-collector mode, but the method is readily adaptable to any other modes of electrochemical perturbation upon a simple modification of the electrochemical conditions applying at each microband surface. We relied on the Alternating Direction Implicit (ADI) method coupled with the conformal map since in our recent works this proved very efficient tandem [4, 7-11], and we used the nonuniform time grid proposed in [12] since this was shown to be fit to afford fast and accurate simulations of various microelectrode problems [4, 9, 13, 14].
2. Theory
2.1. Mathematical model in the real coordinates
Let us consider the 2D cross-section of a double band microelectrode assembly operating within a microfluidic channel or within a thin film of electrolyte [5, 6], and introduce the following dimensionless variables (Fig. 1a):
X = * • Y = y • W = w . G = g • h ’ h ’ h ’ h ’
g n c Dt hv*
X=— • C = — • t = — • V* (1)
w ’ co ’ h2 ’ * D , ( )
where x, y are the real spatial coordinates; h is the channel height w is the electrode width (both band microelectrode widths are considered equal) • g is the gap between the electrodes; c is the concentration of the
reacting species and co is its bulk concentration at the channel entrance; t is the real time and d is the diffusion coefficient. Whenever the solution is submitted to a microfluidic flow, vx is the flow velocity along the x-direction in the channel (note that this may or may not vary along the vertical coordinate, y, depending on the flow velocity profile). With these notations, the real 2D cross-section of the assembly is represented in the
dimensionless real space zj = X + iY as shown in Fig.1a.
The diffusion-convection equation for the species produced at the generator electrode is:
SC d2C 52C ^ SC
— =----------v- Vx— (2)
Sr aX2 5Y2 x SX (2)
in the dimensionless real coordinates and this is associated with the initial condition (% = 0):
VX, Y : C = 0 (3)
When the double band assembly (Fig. 1a) is operated in the generator-collector mode under the mass transport
limited conditions, the system obeys the series of boundary conditions in eqs. (4) (%> 0 ). In formulating the set of eqs. (4), we found convenient to separate the channel into two sub-compartments I (generator compartment) and II (collector compartment). Their common limit is arbitrarily fixed at the plane perpendicular to the channel at the middle of the gap to introduce an exact symmetry. For the same sake of symmetry, we use the same abscissa notation, X, albeit each abscissa is different in each compartment their origin being set at the center of each electrode. This will allow a facile formulation of the quasi-conformal transformations (see below).
Fig. 1. Schematic representation of a channel equipped with a double band microelectrode assembly (a) in the real space and (b) in the quasi-conformal space (eqs. 5) developed for the simulation of transient responses
Generator compartment (I):
X ^ -да , 0 < Y < 1, C ^ 0 (channel entrance) (4a)
- “ < X < -W , Y = 0 , = 0 (insulator)
WW
- — ^ X ^ —, Y = 0 , C = 1 (generator)
W ^ v W + G SC ,
— < X < —2— , Y = 0 , = 0 (insulator)
X W + G
X = —;—, 0 < Y < L
SC
SX
_ SC
W+G _0 " 5X
W+G+0 (exit from I)
2
2
(4b)
(4c)
(4d)
(4e)
- да < X <
W + G 2
Y = 1,
48
РИ, 2005, № 1
dC
SY
0 (ceiling of the channel)
Collector compartment (II):
X = -
W + G 2
0 < Y < 1,
(4f)
dC
dX
W+G 2
- 0
dC
dX
W+G 2
+0
(entrance into II)
(4g)
independent of x and corresponds to a constant h value (h = hg) as shown in Fig. 1b. We will pursue the description of the method assuming this is the case to simplify the presentation, but the transition to the general case is straightforward. The two half compartments (Fig. 1b) labeled I and II are arbitrarily separated by the vertical segment located at the middle of the gap in the real space. In each compartment the horizontal axis origin is set at the center of the electrode active in this compartment.
The back transformation is given by
W + G W SC
---2— < X < “2 ’ Y = 0 , ^Y = 0 (insulator) (4h)
WW
^ X ^ —, Y = 0 , C = 0 (collector) (4i)
* Л
1 = 1 ; л = —4;
і+p
* * * і
ra =q + ip = — arccos n
a - cos(irczi)
b
(6a)
(6b)
W
2
<X <да , y = 0
SC
SY
0 (insulator)
X^ да,0 < Y < 1, C^0 (channel exit)
(4j)
(4k)
W + G 2
< X < да
Y = 1,
— = 0 (ceiling of the channel) (4l)
2.2. Quasi-conformal map for numerical simulation
Let us introduce the following conformal transform which maps the юі-plane (юі = § + ip , see Fig. 1b)
onto the zi -plane (see Fig. 1a) in each channel subcompartment:
zi =—arccos n
a - bcosfrcro )
(5a)
i
2.3. Mathematical model in the quasi-conformal coordinates
Upon using the above quasi-conformal map the diffusion-convection equation (eq. 2) is transformed in the op space into:
— = D dx
S2c, S2C — +
2
Sp
c SC SC
? dz, л Sp •
where
(7a)
D =
2 JVC 2
SX ! і SY,
—
df
-1-2
dp
F, = V
x SX ’
(7b)
(7c)
(7d)
where
* * *
ю + ip =^ + if (p),
1
a = — 2
cos I “2"W
+1
(5b)
(5c)
Fn =
( 2 d2f " df'
v dh 2 dp
-|-3
D + Vx
Sp
SX
df _ 1
dh (1 -p)2 .
(7e)
(7f)
b = -2
cos| ™W I-1
(5d) The inverse Jacobian (J 1) of the transformation in eqs. (5) is:
f(h)
h
1 -p ’
(5e)
so that the ®1 space consists of the two adjacent finite boxes containing each electrode and interconnected by a common boundary located at 0 <^< 1, h = hg (generator compartment) or at -1 <^< 0, h = hg (collector compartment) respectively. This dual boundary in the quasi-conformal space (Fig. 1b) features the
segment [X = +(W + G)/2 , 0 < Y < 1 ] of the real dimensionless space (Fig. 1a). Whenever the gap is sufficiently large compared to the height and width of the channel, this common boundary is almost
J-1
s^
SX SY
Sp Sp
.SX SY.
Re
ОТ
dp
df
dz
\
( _ *Л
dz J - Im dz V
(л *Л ОТ i dp „ f Зю —Lx Re —
df
Sz
v /
(8)
РИ, 2005, № 1
49
The initial condition (x = 0) becomes:
, V-q , C = 0 (9)
while the boundary conditions (x > 0) in the generator compartment (I) are:
-1 <^< 0, q = 1, C = 0 (channel entrance) (10-Ia)
§ = -1, 0 <q< 1, — = 0 (insulator) (10-Ib)
d%
-1 <^< 1, q = 0 , C = 1 (generator) (10-Ic)
2.4. Currents
The time-dependent currents at the generator and collector electrodes expressed in the quasi-conformal space are given by
Ig(T)
1
nFD^L |-
-1
dgdC df дц
r|=0
d|; (generator)
(11a)
IcW = nFDc0L
j-dgdC •^df 8ц
r|=0
d|, (collector)
(11b)
яг
| = 1, 0 <q<qg , — = 0 (insulator) (10-Id)
BC
дц
0 <|< 1, q = qg: cC
^g
-0 ^
^g
+0
(exit from I) (10-Ie)
5 = +0 , CT<h<hg,
BC
— = 0 (ceiling of the channel) (10-If)
^ = ~° , ст<р< 1,
5C
— = 0 (ceiling of the channel) (10-Ig)
and those in the collector compartment (II) are:
-1 <|< 0 , q = qg ,
where n is the number of electrons transferred; F is the Faraday constant; l is the length of both bands and equal to the channel width.
2.5. Dimensionless channel parameters
To complete the problem formulation, let us introduce an additional dimensionless parameter:
Pe =
Vavgh2
gD
(12)
where Vavg is the average flow velocity in the vertical
cross-section of the channel (see below, Eq. 13). In fact this parameter is the Peclet number fitted to the problem at hand, since it relates the diffusion and convection transports to one of the cell dimensions. Since three independent dimensions (g , w and h) describe the channel geometry completely, this number (Pe) must be associated with the two other parameters G and X (see eq.1).
In the following we wish to restrict ourselves to the parabolic flow so that:
— = — (entrance into II) ^%g + 0 5%g -0 (10-IIa)
BC ^ = -1, 0 <P<Pg , — = 0 (insulator) d% (10-IIb)
-1 <^< 1, q = 0 , C = 0 (collector) (10-IIc)
BC \ = 1, 0 <q< 1, — = 0 (insulator) (10-IId)
0 <^< 1, q = 1, C = 0 (channel exit) (10-IIe)
| = +0 , CT<q< 1,
BC — = 0 (ceiling of the channel) BE, (10-IIf)
5 = -0 , CT<h<hg,
BC — = 0 (ceiling of the channel) (10-IIg)
Whenever the gap is small compared to the channel height or electrode widths, qg is a symmetrical function of x (see above).
Vx = 6VavgY(1 - Y). (13)
Yet, this condition may be replaced by any other adequate one featuring the flow under consideration.
3. Automatic fitting procedure
In order to apply the simulation approach described above to the analysis of experimental current data we need to develop a procedure for the solution of the so called inverse problem. A double band electrode assembly can be used for simultaneous measuring of the concentration and diffusion coefficient of electroactive species which we consider here. In this case the inverse
problem consists of recreating the values of c0 and D from experimentally measured current transients by iterative adjustment of c0 and D until a good fit is attained between theoretically predicted and measured currents. Formally, this inverse problem is presented by a minimization problem for the function:
SSQ(c0,D) = £ j[j(t,c0,d)- jP^fw^x (14)
j=g,c 0
where Ijh (t,c0,d) is the theoretical computed time-
50
РИ, 2005, № 1
dependent current for either the generator (j = g) or collector (j = c) electrode, Iexp (t) is the corresponding experimental current, w j (t) is an electrode-dependent weighting function (vide infra) and t is the dimensionless duration of the observation period. Note that the suggested approach allows for simultaneous fitting of up to two pairs of current curves. In the case when a single curve is to be fitted the weighting functions are used as
switches: wc(t) = 0 when only the generator current is
to be fitted and Wg(x) = 0 when only the collector
current is to be fitted. Otherwise the weighting functions are defined as follows:
w
5M=|1
W
л/т
T [igxp tf
4-1
dC
(15)
for the generator electrode since this ensures that points in the rapidly changing part of the current-time curve have greater weights than points in the flattened part of the curve (these have nearly unitary weights) and
(
w
:(T) =
1 +10
be
b( t-to)
(e''( >->°) +J2
T Ho]2
4-1
dC
(16)
for the collector electrode, since its current-time curve has a sigmoidal shape. The parameters b and To are obtained from the best forced-fitting of the accordingly normalised experimental collector current to the sigmoidal
function eb x_x 0)/(eb(x_x^ +1 • The factor of 10 was
introduced to additionally enhance the corresponding weights of points on the sloping part of the current curve.
The Nelder-Mead (or downhill simplex) minimisation algorithm [14-16] was applied for minimising the
function SSQ(co, d) . The advantage of this algorithm is its easy implementation and the fact that the method does not require evaluation of function derivatives and converges rather quickly provided that a reasonable initial guess is supplied.
4. Simulations and Results
All programs were written in Borland Delphi 7 Enterprise Edition and executed on a PC equipped with Intel Pentium 4 processor at 3 GHz and 512 Mb of RAM.
First, we tested the convergence properties of the suggested simulation method that utilises the conformal mapping technique. To this end we performed a series of simulations with increasing grid size for different values of the Peclet number, i.e. for different relative speeds of diffusion and convective mass transport of species in the channel. Convergence rates were then estimated for both generator and collector currents. The exact values of currents were estimated by extrapolating the data computed on grids of different size and then relative errors were computed for each grid size. The rate of convergence is equal to the gradient of the log-log plot of the error against the coordinate step size. These results are given in Table 1.
Table 1
Convergence rates for simulated currents
Peclet number Convergence rate
Generator Collector
0,01 1,11 0,82
0,1 1,45 1,17
0,5 0,98 1,13
It is clear from Ta rle 1 that convergence rates are close
to unity which shows efficiency of the conformal mapping approach. Indeed, since the initial and boundary value problem under study has singularities at the electrode ends one would expect to have convergence rates of about 12 when using finite difference approximations of second order [17]. The convergence rates observed here are significantly higher than expected ones which proves that the conformal mapping (5), (6) is fit to the convection-diffusion process at double channel microband electrodes.
It should be noted that for obtaining stable solutions it is necessary to apply the upwind discretisation of the convective terms. Since the direction of flow in the compartments (-1 <|< 0 ) and (0 <|< 1) is different the formulations of the finite difference equations are different in each compartment. Computational grid N£x Np = 121 x 120 and the non-uniform time grid parameters Дто = 10_5 and ц = 5 x 10_4 were used in all our calculations [9, 12-14].
Fig. 2 demonstrates the consistency and precision of the method since it establishes that the generator currents tend steadily towards their steady-state values depending linearly on the parameter p defined by [10, 11]: w 1
p=T^Dt. (14)
This is presented (Fig. 2) in a normalized form by dividing each current by their common exact analytical steady state value § (dimensionless current, see ref. [18]) achieved at infinite times so that у /g ^ 1 for p ^ 0 (horizontal dashed-dotted line). The dashed line tangent to the generator current curve when p ^ 0 represents the linear limiting behaviour of the generator current at long times (see text and [5]).
Fig. 2. Variations of the transient dimensionless generator (Vg) and collector currents (yc ) with p (see eq.14) for G и 4,12 and X = 3,5
РИ, 2005, № 1
51
Lastly, the automatic fitting procedure (Section 3) was applied to recover the values of the bulk concentration and the diffusion coefficient. To test the performance of the fitting approach we emulated pseudo-experimental current data by simulating current-time curves for
chosen values of cq and d and adding to them Gaussian noise with the amplitude of 1% of the steady state generator current to all simulated current curves in order to imitate experimental noise. The latter means that the relative amplitude of noise on the collector current is higher than that for the generator current (due to lower values of steady state currents at the collector), while the absolute amplitude of noise is the same for both generator and collector pseudo-experimental currents. The resulting spiked data were introduced into the program instead of real experimental data and the automatic fitting procedure was run to find best-fit values
of cq and D .
Here we chose the bulk concentration of cq = 10 _6molcm_3 and the diffusion coefficient of
__с ___2
D = 10 cm s to generate pseudo-experimental currents which are shown in Fig. 3. The other parameters have the following values: w = 20pm , g = 80pm , h = 20pm, L = 500pm , vavg = 0.1 cm s_1. After loading the above data into the program the fitting procedure was started with the initial guess for the “unknown” parameters cq and d as cq = 5 x10_6molcm_3 and
D = 2 x 10 _6cm_2s . The results of fitting the currenttime curves are presented in Table 2, which contains the values of cq and d reconstructed using i) only the generator pseudo-experimental current, ii) only the collector pseudo-experimental current or iii) generator and collector pseudo-experimental currents simultaneously. The best-fit currents are not shown because they are visually indistinguishable from the pseudo-experimental data in Fig. 3.
Fig. 3. Pseudo-experimental generator and collector electrode currents
As can be seen from the Table 2 the deviations of the reconstructed values of cq and d obtained from fitting only the collector current from the expected values
( cq = 10_6mol cm_3 and D = 10_5cm_2s respectively) are smaller than the deviations of the values obtained
Table 2 Fitting results
Fitted curves Cq , 10-6 mol cm-3 Cq error, % D, 10-5 cm2 s-1 D error, %
Generator 1,1218 12,18 0,83689 16,31
Collector 1,0429 4,29 0,96105 3,90
Generator- collector 1,0072 0,72 0,98844 1,16
from fitting only the generator current. This is explained by the fact that the collector current carries more information than the generator current because it integrates the information provided by the generator current (because the collector detects electrogenerated species produced at the generator) and additional information about the flow contained in the delay before the appearance of the collector current wave and the wave shape. However, it is clear that the best reconstruction efficiency is obtained when both the generator and collector currents are fitted at the same time. In this case all available experimental information is used which makes the method more sensitive to small changes of unknown parameters.
5. Conclusions
This work exemplifies the simplicity of formulating diffusion-convection problems occurring at microband assemblies performing within microfluidic channels or within thin layers of electrolyte, through the quasiconformal mapping approach presented here since this affords fast and accurate simulations. For the sake of simplicity these approaches have been presented only for the generator-collector dual microband assembly but it can obviously be readily adapted to other experimental situations involving either other electrochemical operation modes (viz., ECL, generator-generator, etc.) or to a single band. Similarly, it can be adapted by simple recurrence to more complex electrochemical arrays based on microband electrodes.
It has also been shown that when using experimental data for recovery of the parameter values all available data must be used simultaneously for efficient and accurate results. Thus, fitting of current-time responses of both generator and collector electrodes at the same time provided the best performance during simultaneous recovery of the values of the bulk concentration and diffusion coefficient of electroactive species as compared to fitting single current curves.
Acknowledgements
Authors from Kharkov wish to thank NATO for the Reintegration Grant (981488) allowed to Dr. Klymenko for supporting this research project. In Paris, this work was supported in part by CNRS, ENS (BQR), UPMC and the French Ministry of Research (UMR 8640). Irina Svir acknowledges ENS for providing the Invited Professor position for a period of one month during which a part of this work was performed.
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Поступила в редколлегию 18.03.2005
Рецензент: д-р техн. наук, проф. Хаханов В.И.
Klymenko Oleksiy Victorovich, PhD, Senior Scientist of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in mathematical physics, development of scientific software. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue. Email:
klymenko@kture.kharkov.ua.
Oleinick Alexander Igorevich, Cand. Tech. Sci., Senior Scientist of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in mathematical physics, programming. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue.
Amatore Christian Andre, Professor of Ecole Normale Superieure (ENS), Academician of French Academy of Science, Head of the Department of Chemistry of ENS. Scientific interests: mathematics, physics, analytical chemistry, biochemistry. Address: ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France.
Svir Irina Borisovna, Chief Scientist, Dr. Tech. Sci., Head of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in science. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue.
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