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\par\vspace*{44mm}    
\normalsize{\bf
\noindent COMPUTER SIMULATIONS OF \\
\noindent Bi-}\large{2223} \normalsize{\bf SINTERED BULK}\\
\par
\vspace{9mm}
\normalsize
\hangindent=25mm
\hangafter=-1\noindent
     Y.A. Kozinkina$,^{1}$ and I.A. Parinov$^{2}$\\
\par
\vspace{1mm}
\normalsize
\hangindent=25mm
\hangafter=-4\noindent
     $^{1}$Rostov State University  \\
     Rostov-on-Don, 344090, Russia \\
     $^{2}$Mechanics end Applied Mathematics Research Institute, \\
     Rostov-on-Don, 344090, Russia \\
\par
\vspace{9mm}
\normalsize
{\bf
\noindent ABSTRACT}\\
\par
\vspace{0.1mm}
\normalsize
     The models for $Bi-2223$ ceramic processing and fracture which
is obtained by hot-pressing are discussed. Computer simulation
is applied to phenomena occuring during sintering, cooling and
following fracture due to the macrocrack growth. The effects of
dispersed silver particles into $Bi-2223$ matrix on some strength
properties are studied.
Finally the numerical model for the ceramic conductivity investigation
is represented and found some effective characteristics.\\
\par
\vspace{2ex}
\normalsize
     {\bf
\noindent INTRODUCTION}\\
\normalsize

     The bulk  samples  of  $Bi_2  Sr_2  Ca_2  Cu_3  O_{10+\delta   }$
$(Bi-2223)$ have been applied in
the current leads and an increase of the superconductive properties
in particular critical current density $(J_c)$ is the main aim of
the studies. It is known, that the $Bi-2223$ phase in contrast to
$Bi_2 Sr_2 Ca_1 Cu_2 O_{8+\delta }$  $(Bi-2212)$
phase is stable in only a very narrow
temperature range, and the kinetics of its formation is so slow
that it is impossible to obtain the phase-pure material$.^1$

     In the case of $Bi-2223$ phase, it is impossible for the melt
process to be applied because $Bi-2223$ phase exists below the
partial melting point in the phase diagram$.^2$ An addition of Ag to
the calcined powder promotes to increase of $J_c$ $.^3$ The silver
dispersed grains induce a preferential formation of $Bi-2223$ phase
near silver grains, where most partial melting takes plase
during high temperature processing, facilitating diffusion of $Ca-Cu-O$
more easily along the $Ag/BSCCO$ interfaces. The $Ag$ particles decrease
porosity of the superconductive material improving its morphology
and mass density. These cause a better alignment of the superconducting
grains$.^4$ Then, a hot-pressing used for processing of $Bi-2223$ bulk,
removes voids in the material under pressure and high temperature.
The densification
of the materials improves the contact between $Bi-2212$ and the
secondary phase grains. Since the formation of the $Bi-2223$ phase
is based on the chemical reaction$:^5$
     $$
     Bi-2212 + secondary \  phase  \to  Bi-2223    \eqno  (1)
     $$
the increased contact area accelerates the chemical reaction and
formation of the $Bi-2223$ phase.

     There is problem of carbon segregation that is included in
the calcined powder, at the intergranular boundaries during
sintering. This decreases $J_c$ which causes a weak link preventing a
superconducting current among grains$.^2$

     The aim of the present investigation is the study by computer
simulations the effects of the hot-pressing, dispersed $Ag$ inclusions
and secondary phase segregation at the intergranular boundaries
on the some microstructural, strength and conductive properties
of the $Bi-2223.$\\
\par
\vspace{2ex}
\normalsize{\bf
\noindent MICROSTRUCTURE FORMATION BY PROCESSING}\\
\par
\normalsize

     Simulations of temperature fields in powder compacts and
powder recrystallization in the sintering field have been
studied in details for related materials $,^{6-8}$  basing on the sintering
theory for porous heterogeneous systems $.^9$ Here only
note, that the thermal treatment are carried out using the
finite difference method. The  rectangular sample is
modelled by two-dimensional square lattice with 2000 cells by
characteristic size of $ \delta$.

  In more detail we discuss the model of
abnormal grain growth caused by the sintering condions (i.e. pressure
and temperature$,^{10}$ ) and by existence of additions (e.g. $ CaCuO_2$  and
$CuO$ )  during  $Bi-2223$  formation$.^3$ Earlier for different
ceramics $,^{11,12}$
the Wagner-Zlyosov-Hillert's model$,^{13}$  has been used. In this modelling
it has been not accounted of the texture effect  on the grain growth.
  However as it has been shown  by  experiments  the
primary recrystallization,  as  rule,  to  cause  a texture initiation
$.^{14}$
Then here by secondary recrystallization modelling it is considered
dependence of the intergranular boundary energy and its mobility
from grain misorientation using correspondence texture component
for every grain. For simplicity we are limited by only two texture
components ($A$ and $B$) that it is quite sufficient for description of
 the considerable number of tests.  In particular, thus, it can be
modelled growth process along direction of
c and in the plane a-b for HTS.

     The mass transport between crystallites is based on the next
assumptions$:^{15}$\\
\noindent $(i)$ the migration rate of a boundary between two grains $i$  and $j$
is given by
     $$
    v_{ij}=m_{ij}p_{ij}=2\gamma   _{ij}(1/R_i-1/R_j)   \eqno (2)
     $$
where $p_{ij}$  being the driving force, $m_{ij}$ , $\gamma _{ij}$
are the mobility and
boundary energy,$(1/R_i-1/R_j)$
the average curvature for this
grain boundary,$M_{ij}=2m_{ij}\gamma _{ij}$
the grain growth diffusivity;
$(ii)$ all grains of the same size and orientation experience the
same growth rate (homogeneity condition); and $(iii)$ the grains around
a given grain are distributed randomly with respect to size and
orientation. Then, the grain growth rate of the size class $i$   with
texture component $A$ (or $B$) without stagnation has form

$$
  dR^{A(B)}_i/dt=M^{A(B)}_*[1/R^{A(B)}_*-1/R^{A(B)}_i] \eqno (3)
$$
where  $M^{A(B)}_*$  and $R^{A(B)}_*$
are the integrated diffusivity and integrated
critical radius for the component $A$ (or $B$) controlling its growth,
respectively. Note, that value of $R^{A(B)}_*$    divides the grain
sizes of
component $A$ (or $B$) in such which grow and which shrink, and
$M^{A(B)}_*$   determines the rate of these processes$^{15}$
$$
  M^A_*=\frac{F^A\langle R^2\rangle ^AM^{AA}+F^B\langle R^2\rangle
  ^BM^{AB}}{F^A\langle R^2\rangle ^A+F^B\langle R^2\rangle ^B};
  M^B_*=\frac{F^A\langle R^2\rangle ^AM^{BA}+F^B\langle R^2\rangle ^B
  M^{BB}}{F^A\langle R^2\rangle ^A+F^B\langle R^2\rangle ^B} \eqno (4)
$$
$$
  R^A_*=\frac{F^A\langle R^2\rangle   ^AM^{AA}+F^B\langle   R^2\rangle
  ^BM^{AB}}{F^AM^{AA}\langle R\rangle ^A+F^BM^{AB}\langle R\rangle ^B};
   R^B_*=\frac{F^A\langle R^2\rangle   ^AM^{BA}+F^B\langle   R^2\rangle
  ^BM^{BB}}{F^AM^{BA}\langle R\rangle  ^A+F^BM^{BB}\langle  R\rangle ^B}
\eqno(5)
$$
where $M^{HK}$ ($H$, $K=A$ or $B$)   is the grain growth diffusivity of
a boundary between grains of the orientation classes $H$ and $K$.
The grain fraction from size class    with orientation A (or B)
is defined by relations
$$
  F^{A(B)}_i=n^{A(B)}_i/N_G;\,\sum _{i=1}^{N_S}(n^A_i+n^B_i)=N_G;\,
  \sum ^{N_S}_{i=1}(F^A_i+F^B_i)=1;\,F^{A(B)}=\sum ^{N_S}_{i=1}F^{A(B)}_i
     \eqno(6)
$$
where $N_S$ and $N_G$   being the total
number of size classes  and grains,
$n^{A(B)}_i$   the number of grains per unit volume from size
class $i$   and orientation one $A$ (or $B$);$\langle R\rangle $ and
$\langle R\rangle ^{A(B)}$ are the mean grain radius in the system and
with orientation
component $A$ (or $B$)
$$
   \langle R\rangle=\sum_{i=1}^{N_S}F_iR_i;\,\langle R\rangle ^{A(B)}=
   \sum_{i=1}^{N_S}F^{A(B)}_iR^{A(B)}_i/F^{A(B)}   \eqno (7)
$$
$$
   \langle R^2\rangle=\langle  R\rangle ^2+\sigma ^2;\, \langle R^2\rangle
^{A(B)}=[\langle R\rangle ^{A(B)}]^2+[\sigma ^{A(B)}]^2      \eqno(8)
$$
$\sigma $ and $\sigma ^{A(B)}$   denote the standard deviations. Belonging to size
class for considered grain is found by cell number composing
this grain. The condition for abnormal growth in the grain size
class    with orientation $A$ (or $B$) in the space $(R,t)$ taking
into account of the relation (3) is found in the form
$$
     \mid 1/R_*^{A(B)}-1/R_i^{A(B)}\mid >I_R/2  \eqno (9)
$$
where$I_R=6f_v/(\pi  r)$ is the value of the grain growth stagnation                                                      ,
$f_v$ and r  are the volume fraction and mean radius of the secondary
phase particles. It is assumed that the stagnation parameter is
not depended on the grain orientation and in the calculations
one is found as before $.^{12}$ This parameter as well as critical
radiuses of components $A$ and $B$ governs the abnormal grain growth.
This circumstance allows to define the HTS ceramic properties in
dependence on the secondary phase characteristics.

     The size parameters necessary for calculations have been
 found by the primary recrystallization. The orientations $A$
and $B$ have been distributed between grains using  Monte-Carlo
procedure and these are not altered during growth. A mass
transport between grains of different orientations is absent.
As example, we consider next case $:^{15}$ $M^{AB}=M^{BA}=2M^{AA}=2M^{BB}$ .
Such enough arbitary
selection is explained by the absence of the reliable
data for the $Bi-2223$ ceramic.  The mass transport between  grains  has
been modelled in accordance to the grain growth mechanism at the
non-singular surfaces$.^{10}$

     The numerical algorithm for abnormal grain growth consists of
the next stages:
$(i)$ the distribution of orientations $H$, where
$H = A$ or $B$, between grains which are formed  after primary recrystallization;
$(ii)$ the definition of all neighbours for every grain of both
orientation classes;
$(iii)$ the determination of the adjacent grains couple in the every
orientation class ($i^H$,$j^H$) with
$\begin{array}{cc}
max &  \mid  1/R_i^H-1/R^H_j\mid  \\  1\le  i^H,j^H\le  N_S & \end{array}$;
 \   $(iv)$ the growth of greater grains from ($i^H$,$j^H$) at the expense of
less ones;
$(v)$ the checking of the conditions $\mid 1/R^H_*-1/R^H_i\mid \le I_R/2$,
where $i^H=1\cdots N_s$;
$(vi)$ the grain growth end in the corresponding component $H$ due to
fulfilment of the condition $(v)$ or alteration of parameters into
relations (4) - (8) with following repetition of the stages $(ii)$-
$(vi)$.

     Then it has been modelled the intergranular cracking of ceramic
by cooling basing on the known $Bi-2223$ parameters$^{16,17}$ and elsewhere
developed approach for ceramics with grain thermal expansion
anisotropy (TEA)$.^{7,8,11,12}$ The latter defines a critical size of
cracked boundary   $l_c^s$  through  grain  boundary  energy
$(\gamma_{gb})$, Young's
modulus $(E_0 )$ and thermal boundary strain $(\varepsilon )$. Note, that carbon
segregation at the intergranular boundaries during $Bi-2223$ processing
causes the boundary brittleness, i.e. weak links formation. Thus
the value of  $\gamma_{gb}$  decreases  causing  the  diminution  of
$l_c^s$ .\\
\par
\vspace{2ex}
{\bf
\noindent $Bi-2223$ TOUGHENING BY SILVER DISPERSION}\\
\normalsize

     Obviously, that the fracture features of microstructure directly
define electrical and magnetic properties. However the experimental
and theoretical studies of the $Bi-2223$ strength and fracture
exist at the initial stage.
     It may be assumed that addition of the silver dispersed ductile
phase to the $Bi-2223$ ceramic  causes considerable increase of
fracture resistance to compare with untoughened matrix. Then it is
known that the main mechanism responsible for the enhanced toughness
of brittle composites with ductile particles appears to be crack
bridging by  ductile  phase $.^{18}$ The bridging effects have
been studied by modelling of the
YBCO ceramic  fracture$,^{12,19}$  and  ferroelectric  ceramics
failure$.^{20}$
As the structural elements constraining the macrocrack, the brittle
grains-bridges have been considered.Detailed bridging
models for brittle composites with ductile inclusions based on
distributions of    non-linear    springs    have   been   considered,
too$.^{21,22}$
The approach using plane strain slip-line solution $^{23}$  has been
represented for analysis  of plastical deformed bridging ligaments
at stretch as well as finite  element method$.^{18}$

     Fractographic examination of toughened composites and the
experiments establish several distinct types of behavior for the
constrained bridging particles$.^{18}$ Here we are
limited by case in which the toughness rise is independent on the
particle size and ductile strength. This corresponds to the moment
when the ductile flux has been occured into considerable zone
near crack tip. Then, taking into account of Budiansky's analysis
for the toughness increase due to the ductile particles (stationary
crack case) we have$:^{24}$
     $$
       \sqrt{1-\nu ^2_{ef}}(K_c/K_0)=\sqrt{3}[1+10(1-\nu
^2_{ef})f_p/(7-5\nu_{ef})(1-f_p)]^{1/2} \eqno (10)
     $$
where $K_c$ and  $K_0$ are the fracture toughness with and without toughening,
respectively,$f_p$ is the ductile particulate concentration defined
by fraction  of  this  phase  intercepted  by the macrocrack path,$\nu
_{ef}$
    is effective Poisson's ratio found by the fractions of the
$Bi-2223$ and Ag phases, and by the intergranular microcracking
during processing. Under conditions of the modified cubic model
we obtain$:^{25}$
$$
  \nu _{ef}=(1-f^{2/3}_m )\nu _c+f^{2/3}_m \frac{(f^{1/3}_m/E_m)\nu
_m+[(1-f^{1/3}_m)/E_{cr}]\nu _c}{f^{1/3}_m/E_m+(1-f^{1/3}_m)/E_c} \eqno (11)
$$
where indexes $m$ and $c$ correspond to the metal inclusions and ceramic
matrix. For the cracked matrix with microcrack density of $\beta _{cr}$,
Poisson's ratio ($\nu _c$)   and Young's modulus ($E_c$)  are expressed as
follows$:^{26}$

$$
   \frac{\nu _c}{\nu _0}=\frac{1+[(16/45)(1-\nu ^2_0)/(2-\nu _0)]\beta
_{cr}}
     {1+[(16/45)(1-\nu ^2_0)(10-3\nu _0)/(2-\nu  _0)]\beta  _{cr}}
\eqno (12)
$$
\newpage
\vspace*{40mm}
\par
\small
\noindent {\bf Figure 1}.(a) \small A scheme of the crack (1) bridging caused by intercepted
ductile particles (4), process (2) and stretch (3) zones of plastically
distorted particles (h and L are the corresponding zone sizes;
 $u^*$  the residual crack opening at ligament failure and $\sigma_0$
the bulk yield strength)$,^{24}$ (b) a model fragment of the $Bi-2223/Ag$
composition with
$Bi-2223$ grains (5) and cooling microcracks (6).\\
\normalsize
$$
   \frac{E_c}{E _0}=1/[1+\frac{16(1-\nu  ^2_0)(10-3\nu   _0)}{45(2-\nu
_0)}\beta _{cr}]    \eqno (13)
$$
where $\nu _0$  and $E_0$ are the intrinsic elastic moduli.
     Next, due to  metallic inclusions have greater thermal expansion
than ceramic matrix $,^{27}$ then the residual tension  $\sigma
_R$
exists in the metal and the ceramic is compressed.
The residual stress affects the toughness ,because the compressive
stress in the matrix must be exceeded before crack opening beginning within
the bridging zone. An additive increment in
toughness has form$:^{24}$
$$
  \Delta G_R \simeq \alpha f_p \sigma _Ru^*  \eqno (14)
$$
where $\alpha   (\sim   0.25)$ is a factor that depends on the precise nature
of the $\sigma (u)$ function,$ u^*$ is the total crack opening when
the ductile material fails (Figure 1,a$.^{24}$). For a cylindrical
metal dispersion in the case of non-hardening material, the
axial residual stress of $\sigma _R^z$   may be obtained by Hsueh-Evans'es
solution$:^{24}$
$$
     \frac{\sigma _R^z}{E_m\Delta \alpha \Delta  T^*}=\frac{3}{3(1-2\nu
_m)+2(1+\nu _c)(E_m/E_c)}   \eqno (15)
$$

     In the computer simulations it has been proposed that the
$Ag$ inclusions are localized at the triple junctions where usually
there are sites of defects$^{28}$   healed by the silver. Necessary
parameters for $Ag$ particles are selected elsewhere$.^{29}$  Optimum $Ag$
volume concentration in the $Bi-2223$ bulk is assumed $(f_m =0.2).^{27}$
Finally, the intergranular macrocrack path (see, Figure 1,b) has
been modelled using Vitterbi's algorithm for graphs     and taking
into account  of  the   grain   structure   and   processing
microcracks$.^{7,30}$\\
\par
\vspace{2ex}
{\bf
\noindent CERAMIC CONDUCTIVITY MODELLING}\\
\normalsize

     Obviously, the HTS superconductivity is found by the
inter- and transgranular values of $J_c$ . The high magnitudes of
critical current into grain have been approached . However, due
to the percolating character of current properties a specific
contribution is caused by the intergranular values of $J_c$ .
     It is known, that the electrical conductivity (i.e., value
being inverse to specific resistance) in non-ordered media is
proportional to self-diffusion factor and thus the mean square
displacement of the separate  liquid particles in absence of the
external force$.^{31}$ Consider the model structure of the $Bi-2223$
 as percolating cluster with cells occupied by grains. The percolating
(or conducting) properties are caused by intergranular boundaries
lattice, microcracks and texture\\
\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 defining connectivity of neighbouring
grains. The model structure in any case possesses a joining percolating
cluster due   to  the  fulfilment  of  inequality  $f_b\ll p_c$,  where
$f_b=l_g/l_l$
is the ratio of the cracked facets to the total number of boundaries
between cells of the joining cluster (obviously,$f_b<\beta _{cr}=l_g/l_i$,
where $l_i$ is the total length of intergranular boundaries, because of
$l_i<l_l$),$p_c = 0.5927$ is the percolation threshold for square lattice.
     For estimate of the model structure electrical conductivity
we modify  known algorithm entitled "ant into maze" applied for
description of diffusion in irregulated media$.^{31}$
Consider a chance movement
on square lattice cells. At the 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 considered cell is separated from the main one by intergranular
boundary and both grains have the same orientations, then its
possible number is decreased by a 0.1 (for designation of the
predominant cluster growth within grain). If these grains have
different orientations,  then  additional  generated  possible  number
is multiplied on the corresponding value of the considered cell,
so one is taken into account of the grain connectivity. Further,
if an intergranular boundary has been replaced by microcrack,
then corresponding possible number is found by zero. 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. initial cell is surrounded by microcracks or all $p_k<p_c$ ),
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. An example of cluster growth is
shown in  Figure 2.\\
\par
\vspace{2ex}
{\bf
\noindent DISCUSSION AND CONCLUSIONS}\\
\normalsize

    Statistical reliable result during computer simulations
has been caused by application of the stereological method$.^{32}$
For determination more clear estimations for effects of dispersed
Ag particles on the strength  properties it has been assumed
an absence other toughening mechanisms.  The numerical results
have been found for values of $I_RD_0=2.0$ and 1.2 (first and
second line in Table 1, respectively), where $D_0$   is the grain size
before grain growth. After foundation of $l_c^s$ it is selected
$\delta =l_c^s$.
In the macrocrack modelling which propagates along intergranular
boundaries of the $Bi-2223$ matrix the value of $f_p$  is found
taking into account of the macrocrack length, number of triple
junctions (or $Ag$ particles) at the crack path and relation between
defect size  (2a)  and  critical  boundary  size  ($l_c^s$),  namely
$a=0.1l_c^s$
which being for ceramic with TEA$.^{28}$
The value  of $f_p$  is  proportional to the toughness difference of $\Delta
G_R$
(see, formula (14)). A decrease of $I_R$   causes an increase of
matrix grain size with corresponding growth of the microcracking
density $(\beta_{cr}$). Moreover, the triple junctions number is diminished
too, with rise of relative macrocrack size ($l_{cr}/b$,
where $l_{cr}$  is the
macrocrack size, b the sample width) due to the necessary path
around greater grains. These cause a decrease of $f_p$  and corresponding
diminution of $\Delta G_R$. Finally, less value of $\beta _{cr}$   finds greater elastic
moduli and in addition to increased concentration of $f_p$ increases
 $K_c/K_0$    with grain size decrease.
     The percolation properties have been estimated at the $t=50$.
Computed relative increase of the mean square displacement ($R/R_0$)
and of the joining cluster area $(S/S_0 )$ with decrease of total
intergranular boundary length $(l_i/l_i^0 )$ in the case of the greater
grain growth coincide with experimental data, e.g. for large-grain
melted HTS$.^1$  More dense lattice of intergranular boundaries
and especially a misorientation of adjacent grains affect in more
degree a degradation of conductivity in the fine grain case to
compare with some increased microcracking in the large-grain
structure.
Thus, a greater toughening of the $Bi-2223$ bulk by ductile
$Ag$ particles (a fine grain case) and increased conductivity
( a large grain case) demand optimization both contributions in
estimation of the HTS properties. Other
direction of studies is the modelling of the intergranular
boundary brittleness by carbon additions which being in development.\\

\begin{figure}
\begin{center}
             {\bf Table 1.} Numerical results.
\end{center}
\begin{center}
\begin{tabular}{cccccccc}
\hline
$D/\delta$ & $\beta _{cr}$ & $l_{cr}/b$ & $f_p$ & $K_c/K_0$ & $l_i^0/l_i$ & $R/R_0$
& $S/S_0$  \\
\hline
2.40 & 0.15 & 1.51 & 0.12 & 1.96 & 1.00 & 1.00 & 1.00  \\
2.84 & 0.22 & 1.62 & 0.10 & 1.92 & 1.16 & 1.85 & 1.52 \\
\hline
\end{tabular}
\end{center}
\end{figure}
{\bf
\noindent ACKNOWLEDGMENT}\\
\normalsize

     This work was supported by the Russian Foundation of Fundamental
Research under  Grant  No.95-01-00072.Y.A.K.  thanks the International
Soros Educational Program for financial support under 1997  Soros
Student Grant (No S97-2077).\\

\par
\vspace{2ex}
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\vspace{1ex}
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