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\par\vspace*{44mm}    
\normalsize{\bf
\noindent COMPUTER SIMULATIONS OF \\
\noindent LARGE-GRAIN YBCO PROPERTIES}\\
\par
\vspace{9mm}
\normalsize
\hangindent=25mm
\hangafter=-1\noindent
I.A. Parinov, E.V. Rozhkov and C.E. Vassil'chenko \\
\par
\vspace{1mm}
\normalsize
\hangindent=25mm
\hangafter=-4\noindent
Mechanics and Applied Mathematics Research Institute, \\
     Rostov-on-Don, 344090, Russia \\
\par
\vspace{9mm}
\normalsize
{\bf
\noindent ABSTRACT}\\
\par
\vspace{0.1mm}
\normalsize
Based on a Monte-Carlo computer simulation technique the modelling of
large-grain superconductive $YBa_2Cu_3O_{7-\delta}$
 $(YBCO)$  microstructures is
developed taking into account of dispersed normal $Y_2BaCuO_5$ (211) phase
and using seedings. The possible toughening mechanisms acting in the 
$YBCO$ due to the existence of the particulate 211 phase are discussed.
Some effective conductive properties of modelled structures are 
estimated by percolation methods.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent INTRODUCTION}\\
\normalsize

    It is known that the melt-processing technologies have been used for
fabrication of   large-grain   superconductive  $YBa_2Cu_3O_{7-\delta}$
(123)
with high
critical current density of up to $J_c  =50000  A/cm^2$  at  77  K  and
1T.$^1$ In
the same time if the high values of intragrain currents have been 
attained the high intergrain ones are very difficult problem. This is 
caused by anomalous low value of the coherence length of $\xi$ in $YBCO$
( 5-10 A along c-axis and  20-50 A into a-b plane$^2$  ). Therefore, there
are two kinds of intercrystalline links, namely strength and weak ones.
The strength links are usual intergranular phase contacts which  being
similar to the intercrystalline bridges with value of $J_c$  that is
comparative with transgranular critical current density. The weak-link
behaviour is found by the phase continuity destructions which extent
of $l$ is compared with value of $\xi $. There are next causes of the phase
discontinuity: local violation of stoichiometry,$^1$  amorphous layers at
the grain boundaries$,^3$  and microcracks$.^4$ Moreover, the weak links in
$YBCO$ are caused by the high crystallographical anisotropy. Due to,
there are
a spatial misorintation of anisotropic grains and a diminution of $J_c$ in
superconductive plane a-b.$^5$

    One solution of high critical currents problem is the fabrication
of microstructures with relatively fine crystallites which are linked
other to other by strength links forming  a  solid  crystalline
cluster$.^5$
The second way is based on the use of seeds for directed material
formation$.^{1,6}$ Generally, the $YBCO$ fabrication using melt-processing
techniques exploits a peritectic reaction under increased pressure of
$Po_2:^{1,6}$
     $$
     YBa_2Cu_3O_{7-\delta } \to Y_2BaCuO_5 +L  \to YBa_2Cu_3O_{7-\delta}
                                                     \eqno (1)
     $$
Thus, a crystallized system possesses dispersed particles
of  normal 211 phase. Then, a formation new centres of initiation
and growth of the superconductive 123 phase is based on the inclusion
of the large single seeds from rare-earth analogues of 123 phase. (e.g.,
$SmBCO$ or $NdBCO$), which have more high melting temperature. The seed
placed on the YBCO pellet  before melt processing during cycle of
melting-crystallization  initiates (due to higher own melting temperature)
a formation of main phase at the crystallization front. This causes a
formation of large pseudomonocrystalline domains oriented in accordance
with seed and compared with sample sizes. However, it is necessary a
compromise between grain sizes and area of intergranular contacts,
because of abrupt rise of separate intergranular boundaries with
decrease of total intergranular surfaces to increases a microcracking
possibility and corresponding diminution of intergranular current.
Moreover, any variation of
technological parameters results the alteration of
 microstructural, strength and superconductive properties.

The aim of present study is establishment of dependences between 
different properties of large-grain melt $YBCO$ using joint
investigations of ceramic fabrication and fracture, computer
simulations and percolation theory.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
MODEL REPRESENTATION}\\
\normalsize

    Two main computer simulations approaches, namely Monte-Carlo
simulations$^{7-10}$    and phase field method$^{11-13}$ have
been applied to study
of microstructural
changes  which accompany the $YBCO$ fabrication.
In the present study it is used a modified Monte-Carlo scheme. The 
$YBCO$ precursor microstructure simulation is based on the
recrystallization
model for ferroelectric ceramics$^{14,15}$  and on the study of thermal
conductivity in mixes and heterogeneous solids$.^{16}$  Formation of the final
$YBCO$ microstructures using seeds is  found  by  Monte-Carlo
techniques$.^{17-19}$
    Thus, an initial microstructure after $YBCO$ pellet microstructure
modelling is
represented by discrete lattice with 2000 square cells. Every lattice
site is assigned a number between 1 and $Q$ corresponding to the orientation
of the grain in which it is embedded. All grains have different 
orientations defined during pellet microstructure simulation in which 
lesser numbers correspond to earlier initiated crystallites. Then, the
cells which have neighbours with unlike orientation lie at the grain
boundary in the other case they are placed into grain. The grain 
boundary energy is specified by defining an interaction between nearest
neighbouring lattice sites and has form$:^{17-19}$
     $$
       E=-J\sum_{nn}(\delta_{S_iS_j}-1)        \eqno (2)
     $$
where $S_i$  is  one  of  the $Q$ orientations on site $i$ $(1<S_i<Q)$ and
$\delta _{ij}$  is the
Kronecker delta. The sum is taken over all nearest neighbour sites, J is
a positive constant that sets the energy scale of the simulation. Note,
that the orientation dependence of the energy of a straight boundary segment
exhibits a ratio of the maximum to minimum boundary energy of $2/\sqrt{3}$
 for triangular lattice, and $\sqrt{2}$ for square one. Both ratios
 are close to unity,
i.e. the grain boundary energy is nearly isotropic and only weakly 
lattice  dependent$.^{17}$
    In simulation of boundary motion kinetics by Monte-Carlo technique, a\\
\newpage
\vspace*{50mm}
\par
\small
\noindent {\bf Figure 1}. \small
Three models of YBCO microstructure evolution used in computer
simulations (a) dispersion of 211 phase (black) in 123 matrix, (b)
large-grain  seed (white) into 123 grain matrix, and (c) 211 
dispersion and grain-seed in 123 matrix.\\
\par
\normalsize
\noindent lattice cell is selected at random, and a new trial orientation is 
randomly changed to one of the other grain orientations. Then the 
energy alteration caused by change in orientation ($\Delta E$) is
evaluated. The transition probability, $P$, is given by$:^{17}$ 

     $$
     P=\left\{
              \begin{array}{lclcc}
    \exp(-\Delta E/k_BT), \hspace{2cm} \Delta E >0\\
     1 , \hspace{4.6cm} \Delta E\le 0\\
     \end{array}
     \right.                                  \eqno (3)
     $$
where $k_B$ is  Boltzman constant, $T$ is the temperature. A
reorientation
of a site at a grain boundary corresponds to boundary migration.
Next, a boundary segment moves with velocity related to the local
chemical potential difference,  $\Delta E_i$ as$:^{17}$
     $$
      v_i =C[1-exp(- \Delta E_i/k_B T)]       \eqno (4)
     $$
where factor of $C$ is found by the boundary mobility and symmetry of
the lattice. The simulations for cases of $T\approx 0$ and $T\approx T_m$
(where $T_m$ is
the melting  point)  have shown similar results$,^{18}$ therefore we use
$T=0$.

The $(N-N_p)$ attempts of reorientations (where $N$ is the number of lattice
cells and $N_p$ is the total particle number) is arbitrarily used as unit
of time and it is defined as 1 Monte-Carlo step (MCS) per site. In order to 
incorporate  particles of  211 phase into model,
 corresponding cells
are assigned by orientation which distinct from all grain orientations.
The particle concentration and sites initially arbitrarily selected
are fixed during simulations. The insertion of the crystalline seed into
microstructure is done by replacing the grains at the centre of the
microstructure with one large square grain with characteristic size considerably
greater than mean grain radius of the initial pellet.
    Computer simulations are applied to three cases of $YBCO$ microstructural
evolution after primary recrystallization (see, Figure 1):
(i) particle dispersion of 211
phase into matrix of 123 phase grains, (ii) insertion of large grain (seed)
into structure of 123 grains, and (iii) insertion large seed grain into
matrix of 123 phase grains  with dispersed 211 phase. As condition of
computation stop in the first and third cases it is selected an existence 
one particle at the any intergranular boundary, at least (this corresponds
to complete  pinning  of   microstructure   with   dispersed 
  particles$^{18,19}$). In the
second case the computations are finished when any grain radius (in the
microstructure without seed grain) attains seed grain size.
    A necessary number of the computer realizations is established on the
basis of the stereological method$.^{20}$ Then, an use of comparatively small
grain aggregates together with stereological approach allows to accelerate
computations due to the operation with lesser arrays of variables and
obtain necessary statistics.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
EFFECT OF 211 PARTICLES ON YBCO FRACTURE}\\
\normalsize

    Effects of microstructural features and various toughening
mechanisms on fracture of gradient-sintered $YBCO$ ceramics have been
studied$.^{7-10,21}$ As shown by experiments especial influence on the 123 matrix
fracture may  be  exerted  by  dispersed  inclusions  of  the 211
phase$.^{22-25}$ The
laminated structure of 123 superconductive phase grains covering during
solidification from melting via a zipper-like mechanism the 211 single
inclusions promotes to formation at the phase interface the
defects-faults and increased dislocation concentration$.^{5,26}$
Therefore, the existence
of secondary phase inclusions causes the strength property alterations
indirectly and directly. In the first case, these influence formation
of certain microstructure during material fabrication with corresponding
fracture resistance. In the second case, these define acting toughening 
mechanisms, fracture toughness and strain energy release rate differences.
Then possible toughening mechanisms may be following ones.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
Crack deflection}\\

  Due to the 211/123 interface strength being less then $YBCO$  matrix
one, it is possible a crack deflection around dispersed particles  in the
form of crack tilt and twist. Then a crack driving force which orients
randomly to the main tension vector is described by the local stress
intensity factors (SIF) of $k_1$ , $k_2$ and $k_3$  corresponding
to the opening,
sliding and tearing. The crack driving force governed by the strain 
energy release rate has form$:^{27}$
     $$
       EG=k^2_1(1-\nu ^2)+k^2_2(1-\nu^2)+k^2_3(1+\nu ) \eqno (5)
     $$
where $E$ and $\nu$  is Young's modulus and Poisson's ratio, respectively.
In the case of
applied SIF  of Mode $I (K_I )$ the local SIF's for tilted crack $(k_i^t)$
are given as$:^{27}$
     $$
       k_1^t=F_{11}(\Theta )K_I;  k^t_2=F_{21}(\Theta )K_I \eqno (6)
     $$
     $$
        F_{11}(\Theta )=\cos^3(\Theta /2); F_{21}(\Theta )=\sin (\Theta /2)
     \cos^2(\Theta /2)                                    \eqno (7)
     $$
For twisted crack the local SIF's $k^T_i$ have forms$:^{27}$
     $$
        k_1^T=F_{11}(\Phi )k_1^t+F_{12}(\Phi )k^t_2;
k^T_3=F_{31}(\Phi )k_1^t+F_{32}(\Phi )k^t_2                \eqno (8)
     $$
     $$
     F_{11}(\Phi )=\cos^4(\Theta /2)[2\nu \sin^2\Phi + \cos^2(\Theta /2)\cos^2\Phi ]
     $$
     $$
     F_{12}(\Phi )=\sin^2(\Theta /2)\cos^2(\Theta /2)[2\nu
\sin^2\Phi +3\cos^2(\Theta /2)\cos^2\Phi ]
     $$
     $$
     F_{31}(\Phi )=\cos^4(\Theta /2)[\sin\Phi \cos\Phi (cos^2(\Theta /2)-\nu )]
     $$
     $$
     F_{32}(\Phi )=\sin^2(\Theta /2)\cos^2(\Theta  /2)[\sin \Phi
\cos\Phi (3\cos^2(\Theta /2)-2\nu )]      \eqno (9)
     $$
     Then computation of toughening due to the crack deflection is the
statistical problem which connected with averaging of a crack driving force
in all possible tilt and twist angles. The effect of particle volume
fraction of $V_f$  which causes an initial crack path alteration is equal to
     $$
        G_c /G^m_c =1+0.87 V_f                     \eqno (10)
     $$
where $G^m_c$  is the matrix fracture toughness. The inclusions with higher
characteristic ratio have more effects, i.e. spherical particles create
a lesser tougheninng to compare with disk-shape and red-shape ones.
Toughening effect attains saturation at the $V_f =0.1$. An increased
dimension of loading causes the crack driving force increase due to the complexity
of stress state and toughening decrease.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
Crack pinning by particles}\\

    In the  case  of small-scale crack bridging in which bridging zone
size 
to be small to compare with crack length, sample sizes and distance from
crack up to the sample boundaries it is possible a crack pinning by elastic
particles with size of 2a. Then, the toughening  factor of $\Lambda$
  may be found as$:^{28}$
     $$
      \Lambda \equiv      \frac{K/K_m}{\sqrt{\omega
 (1-c)}}=\left[1+\frac{\pi S^2ac(1-\sqrt{c})}{2K^2_m}\right]^{1/2},
     \omega =\frac{E(1-\nu ^2_m)}{E_m(1-\nu ^2)}, c=(a/b)^2
                                                      \eqno (11)
     $$
where $K$ and $K_m$  are the critical SIF's for crack growth in composite and
into matrix, respectively, $S$ is the particle strength,
 b is the radius of penny-shape  crack  with
central pinning particle, $E$ and $\nu$ are effective elastic moduli of
 composite ($E_m$  and $\nu _m$   are the elastic matrix constants).
Thus, the toughening is increased with inclusion size and strength.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
Toughening due to periodically distributed inclusions}\\
\normalsize

The internal stress state caused by periodically distributed inclusions
is almost sinusoidal at the mean plains between inclusion layers where
a finite crack places along x-axis. Dislocation techniques have been applied
to inclusions modelled by  centres of shear, compression, antiplane shear and
pressure. There are next results$:^{29}$
     $$
       \Delta K_{II}=1.6\mu V_fe^T_{xy}\sqrt{\lambda}/(1-\nu ) , for \
       shear\ centres     \eqno (12)
     $$
     $$
     \Delta K_I=1.6\mu V_fe ^T_{xx}\sqrt{\lambda}/(1-\nu )
                    , for\ horizontal\ compression\ centres  \eqno (13)
     $$
     $$
     \Delta K_I=2.2\mu V_fe ^T_{xx}\sqrt{\lambda}/(1-\nu )
                    , for \ vertical \ compression \ centres    \eqno (14)
     $$
     $$
     \Delta K_{III}=0.4\mu V_fe ^T_{yz}\sqrt{\lambda}
                    , for \ antiplane \ shear \ centres     \eqno (15)
     $$
     $$
     \Delta K_I=0.8EV_fe^T\sqrt {\lambda}/(1-2\nu )
                    , for \ pressure \ centres     \eqno (16)
     $$
Thus, SIF's difference of correspondent Mode for periodically distributed
inclusions is proportional to the elastic modulus, inclusions volume fraction,
deformation mismatches $(e_{\alpha \beta }^T)$ and square root  of inclusion
spacing $(\lambda )$. One
is independent on the crack length, i.e. dispersion toughening is more 
effective at the crack initiation to compare with its growth.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
YBCO CONDUCTIVITY MODELLING}\\
\normalsize

    Obviously, the $YBCO$ superconductivity in above case is found by an
existence of the normal (211) and superconducting (123) phases, and
intergranular boundary lattice. Then, the conductive properties of the
$YBCO$ microstructure may be modelled using percolation theory$^{30}$
(see also$^{31}$ ).
Briefly, consider at the square lattice $(p_c =0.5927$ is the percolation
threshold) the $YBCO$ model structure as percolating cluster with cells
occupied by grains of the 123 superconductive phase or particles of the 
211 normal phase. For estimate of the model structure electrical 
conductivity we modify known
\newpage
\vspace*{40mm}
\par
\small
\noindent {\bf Figure 2.} \small An example of the percolating cluster growth. Numbers
show possibilities of cell occupation (in the brackets there 
are considered possibilities)
which compare with percolation threshold,  $ p_c=0.5927$. A cross
corresponds to the cluster cell.\\
\par
\normalsize
\noindent
algorithm entitled "ant into maze" applied
for description of diffusion in irregulated media$.^{30}$ Consider a chance 
movement on square lattice cells. At  any time, the possible numbers of
$p_k\in [0,1]$ (where k=1...4) are generated for all cells which adjacent to
main one. In the case when neighebouring cell is occupied by the 211
particle then its possible number is found by zero. If considered cell
is separated from the main one by
intergranular boundary, then its
possible number is decreased by a 0.1 (for designation of the predominant 
cluster growth within grain). The cluster growth occures due to the
occupation of cell with most possible number $p_k \ge p_c$. In the case of
impossibility of the cluster growth (i.e. at the $p_k <p_c$ ), its increase at
this stage is absent. Next, all process is repeated. At the every stage 
including a marking time, the value of $t$ is increased on one. In the
time $t=0$ a movement begins from some chance cell of the joining cluster.
In the time of $t$ it is calculated a square of distance between initial
and final points. Then modelling is repeated several times and found the 
mean square displacement of $R$ associated with conductivity of the
considered model structure. As above, a number of computer realisations
is estabilished  by  the  stereological  method$.^{20}$ An  example of
cluster growth is shown in Figure 2.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
DISCUSSION AND CONCLUSIONS}\\
\normalsize

    Numerical results presented in Table 1 have been found for some
microstructural and percolation properties in all possible cases that
is found by 211 particle concentration ($f=0.1$ and $0.2$) and seed size
($S_s\simeq 2S_0^m$   and $3S_0^m$    , where $S_s$
is the seed area, and $S_0^m$    is the
maximum grain one after primary crystallization before beginning of the 123
grain growth). The final microstructures with 211 dispersed particles 
show dependences which like with $\=S \simeq 4a/f$, where $\=S$ is the
 mean pinned grain
area, $a$ is the particle one. This corresponds to topological ideas
developed in computer simulations of the grain growth pinning by dispersed
particles$.^{18}$ Behaviour of  outstripping grain growth  around seed
coincides well with modelling results of large grain growth into microstructure
of normal grain growth$,^{19}$ and growth morphology of $YBCO$ specimens with seed
solidified at unfavourable temperatures$.^6$ Moreover, the present computer
simulations may be characteristic to 211 particle pushing processes by
the propagating 123 growth front$.^6$ Numerical results show a correlation 
between grain area $(\=S)$, superconductive field per grain $(\=S_{sc})$ and ratio
of the total intergranular boundary length to the total number of 
boundaries between cells of the joining cluster $(l_i /l_l )$. The percolation
properties have been estimated at the $t=100$. Behaviour of the mean square
displacement $(R/R_0 )$ and joining cluster area $(S/S_0 )$ as well as a change
of the  value  of  $\={S}_{sc}$  coincide  with  experimental
data$.^{5,6}$ It is confirmed, that
$YBCO$ superconductive properties decrease with rise of the 211 normal phase
concentration and with decrease of seed. In the same time, toughening 
mechanisms, above considered are intensified with increase of the 211
particle size and concentration into 123 matrix. Finally, it should be 
noted that in the present study the cooling intergranular  microcracking
and intergranular boundary misorientations are not taken into account 
during percolation modelling. While these indirectly are defined by dispersed
phase and directly cause the $YBCO$ superconductive properties.These
problems will be duscussed in the next study.
Thus, a creation
of large-grain melted YBCO demands to account of all contrast tendencies
for improvement of material properties.\\
\begin{figure}
\begin{center}
             {\bf Table 1.} Numerical results.
\end{center}
\begin{center}
\begin{tabular}{cccccc}
\hline
Variants & $\=S/\delta^2$ & $\=S_{sc}/\delta$ & $l_i/l_l$ & $R/R_0$ &
 $S/S_0$\\
\hline
$f=0.1, S_s=0$ & 45.45 & 40.91 &  0.133  &  0.649  & 0.929\\
$f=0.2, S_s=0$ & 25.01 & 20.05 & 0.136 & 0.452 & 0.893 \\
$f=0, S_s\simeq 2S_0^m$ & 13.89 & 13.89 & 0.325 & 0.803 & 0.979 \\
$f=0, S_s\simeq 3S_0^m$ & 23.81 & 23.81 & 0.235 & 1.000 & 1.000 \\ 
$f=0.1, S_s\simeq 2S_0^m$ & 50.03 & 45.08 & 0.116 & 0.816 & 0.957\\
$f=0.1, S_s\simeq 3S_0^m$ & 62.50 & 56.25 & 0.111 & 0.865 & 0.979 \\
$f=0.2, S_s\simeq 2S_0^m$ & 29.41 & 23.53 & 0.099 & 0.473 & 0.900 \\ 
$f=0.2, S_s\simeq 3S_0^m$ & 31.25 & 25.00 & 0.083 & 0.544 & 0.921 \\
\hline
\end{tabular}
\end{center}
\end{figure}
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent
ACKNOWLEDGMENT}\\
\normalsize

    This work was supported by the Russian Foundation of Fundamental
Research under Grant No 95-01-00072 and Russian Goskomvuz (Ural Metallurgy
Program).\\

\par
\vspace{2ex}
\normalsize{\bf \noindent REFERENCES}\\

\small
\vspace{1ex}
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