- Email: [email protected]
Critical velocities for particle pushing by moving solidification fronts
Am mefall. Vol. 37,No. 7.pp. 2085-2091,1989 Printed in Great Britain. All rights reserved
CRITICAL
OOOI-616OjSS 63.00+ 0.00
Copyright C 1989Pergamon Press plc
VELOCITIES FOR PARTICLE PUSHING MOVING SOLIDIFICATION FRONTS
BY
R. SASIKUMAR, T. R. RAMAMOHAN and B. C. PAI Regional Research Laboratory, Industrial Estate P.O., Trivandrum 695 019, India (Receioed 1 July 1988)
Abstract-A theory is developed for evaluation of critical velocities for pushing of particles by solidification fronts. A numerical simulation is done in which the critical velocity is evaluated as a function of particle size, thermal conductivity, temperature gradient as the solidification front, the Hamaker constant of interaction between the particle and the front, solid-liquid interfacial energy and melt viscosity. The shape of the solidification front under steady state pushing conditions as well as the changes in shape that take place during the non-steady state engulfment process can be determined using this theory. The critical velocity is shown to be closely linked to the shape of the solidification front and hence to all the parameters that affect the shape of the front. R&urn&-On diveloppe une theorie pour Cvaluer les vitesses critiques pour que des particules soient joussCes par les fronts de solidification. Une simulation numtrique est proposee dans laquelle la vitesse critique est &al&e en fonction de la taille des particules, de la conductivitt thermique, du gradient de temptrature au front de solidification, de la constante d’Hamaker d’intkraction entre la particule et le front, de l’knergie de l’interface solids-liquide et de la viscositt du bain fondu. La forme du front de solidification sous des conditions de poussCe stables ainsi que le changement de forme qui se produit pendant le processus instable d’engloutissage peuvent 6tre di?terminCs g&e B cette thborie. On montre que la vitesse critique est ttroitement like g la forme du front de solidification et done B tous les paramttres qui affectent la forme du front. Zusammenfassung-Eine Theorie zur Auswertung kritischer Geschwindigkeiten fiir das Vorwlrtstreiben von Teilchen durch eine Erstarrungsfront wird entwickelt. Mit einer numerischen Simulation ergibt sich die kritische Geschwindigkeit in Abhlngigkeit von der TeilchengriiDe, der thermischen Leitfihigkeit, dem Temperaturgradienten an der Erstarrungsfront, der Hamaker-Konstanten der Wechselwirkung zwischen dem Teilchem und der Front, der fest-fliissigen GrenzflCchenenergie und der ViskositLt der Schmelze. Die Form der Erstarrungsfront unter stationiiren Bedingungen des Vorwiirtstreibens wie such die VerHnderrungen der Form, die wSihrend des nicht-stationlren Verzehrprozesses auftreten, k6nnen mittels dieser Theorie bestimmt werden. Es zeigt sich, da13 die kritische Geschwindigkeit eng mit der Form der Erstarrungsfront und damit mit all den Parametern. die die Form beeinflussen, zusammenhlngt.
1. INTRODUCTION
When a liquid containing a dispersion of particles is solidified, the solidification front can exert a repulsive force on the particles and push them along with it. thereby increasing the particle concentration in the last solidifying liquid. It has been known that this phenomenon of particle pushing occurs only below a certain velocity of the advancing solidification front known as the critical velocity. If the velocity of the growth front is greater than the critical velocity. the particle is engulfed by the front. The critical velocity depends on the properties of the matrix material and the particle as well as the experimental conditions. Theories for evaluation of the critical velocities have been developed by various authors [l-4]. All the theories are based on the concept of molecular surface forces repelling the particle from the growth front and thus facilitating the entry of liquid into the gap between the growing solid and the particle. The pushing of the particle is considered as a steady state phenomenon in which the
particle and the solidification front move with the same velocity and the gap width is maintained constant. The critical velocity is identified as the maximum velocity above which the steady state cannot be maintained. The most rigorous of these theories is that of Chernov et al. [4] who determine the shape of the interface and use this shape to calculate the forces on the particle and the steady state velocity. Very recently the present authors have developed a theory [5] which follows the approach of Chernov but improves upon it by:
2085
(1) including the effect of the thermal conductivity of the particle being different from that of the melt. (2) performing a rigorous numerical solution of the equations describing the shape of the front. rather than assuming the shape to be described as a combination of a plane and a paraboloid intersecting at a certain point, and (3) enabling calculation of non-steady state shapes also.
2086
SASIKUMAR
et al.: PARTICLE PUSHING BY SOLIDIFICATION
This paper presents the results of extensive numerical simulations based on the above theory. The equations in the previous paper have been scaled and correlations giving the critical velocity of all the relevant material and experimental parameters have been developed. Special attention has been paid to the effect of distortion of the temperature field by the particle on the critical velocity, which has not been treated till now by the previous theories. It is shown that as the solidification front ap proaches the particle, the particle velocity increases, goes through a maximum and then decreases. If the particle velocity becomes equal to the front velocity before the maximum is reached, the front cannot approach any closer to the particle and steady state pushing results. If the velocity of the front is greater than the maximum particle velocity, steady state never occurs, the particle velocity crosses the maximum, decreases and engulfment results. Hence the maximum particle velocity is identified as the critical velocity. All changes in the critical velocity can be linked to the shape of the solidification front at the position resulting in the maximum particle velocity and hence to any parameter affecting the shape of the front. Evaluation of non steady state shapes assumed by the front for velocities above the critical velocity gives an understanding of the physical reasons for the existence of the maximum in particle velocity.
2. THE THEORY 2.1. The shape of the solid$cation
front
The details of the derivation are given in Ref. [S]. The basic assumption is that the solidification front takes such a shape that the actual temperature and every point is equal to the equilibrium temperature for co-existence of solid and liquid. The equilibrium temperature is T,,[Zi (r), rl
= TM- AT,- ATsutiace force
(1)
where
FRONTS
r
t-l Fig. 1. The coordinate system.
where B = Hamaker
constant
h(r) = gap width. A negative B implies a repulsive force between the particle and the front and hence a lowering of the equilibrium temperature. The temperature field can be assumed to be linear with gradient G far away from the particle. From Fig. 1, the actual temperature at any point of the interface far away from the particle (r >>R,,) is T(H,r)=T,-GH,
rmR,.
(2)
In the vicinity of the particle, the temperature field will be distorted if the thermal conductivity of the particle is different from that of the melt. This distortion is shown schematically in Fig. 2. If the neglect transverse flow of heat we can write T(Z,, r) = TO- G, ,/v - G(Z, - ,/m),
r C R,
(3a)
TM = freezing point
AT, = Gibbs Thomson undercooling
=
2urc~/As
where G, is the gradient inside the particle
and CI= solid liquid interface energy AS = entropy of fusion R = molar volume K = mean curvature of the front Z” Z” Z’ = 2[1 + Z’Z]W + 7 + 7 A Curfaceforces = undercooling forces [4] =-
-BR h:,, AS
due to molecular surface
1, = thermal conductivity ductivity of particle and
of melt; 1, = thermal con-
T(Z,, r) = TO- GZr),
r > R,.
W)
If transverse heat flow is considered, the distortion, [i.e. the difference in temperature between a point (Z,, r c RP) and (Z,, r -)a)] will be less than that predicted by equation (3). Hence use of equation (3) gives an upperbound on the extent of distortion that can be expected.
SASIKUMAR
et al.:
PARTICLE
PUSHING BY SOLIDIFICATION
2087
FRONTS
z=o
2
--
3
e ai
Undistorted Distorted
temperature temperature
E” @!
Fig. 2. Distorted temperature field for I., # I.,
Substituting equations (1) and (3) into the equation, T,, [Z,(r), r] = T[Z,(r), r] gives the equation for the shape of the solidification front as Bt2 cw213’2 J_ {(Z,
z,, = 211 + I
nu
+ AS[G(Z, - H)
x [
Z”
Z: ---L, r
O
r
=
I
1 -:Jl
(
Zf’ r*
z,,
Z:--
2C,C,
C: ZY’ r* ’
i I
-rrr2
>I
r* > 1
211+ (-w213’* ASG (Z, - H) Ra
z;
----
Zi3
r
r > R,
r ’
r* =0
BR
1 (Z, -
zy = is
+
4557)’
As[G(Z, -H)
where C, = H/R,
- G(l - &,,/L,)R,]
,
r = 0.
(4)
For doing a numerical simulation it is more convenient to scale these equations. This gives Z:l’ = (1 +
c;z72)3,2
C2= B/Ria C, = ASGR”,/BQ c, = 1 - l,/l,
i$ I
1
X
i
(Zf
-
$72)3 1
The shape of the solidification depends on:
front
therefore
(1) the distance of the plane region of the interface from the plane Z = 0 as depicted by C, ;
2088
SASIKUMAR
et al.:
PARTICLE PUSHING
(2) the extent to which the Gibbs-Thomson effect balances the deflection due to the repulsive surface force, as depicted by C,; (3) the extent to which the temperature gradient and the consequent penetration of the solidification front into a colder region balances the deflection due the repulsive surface force, as depicted by C3; and (4) the distortion brought in due to the difference in thermal conductivities of the particle and the melt, as depicted by C,.
Equations (5) are solved numerically to determine Z,(r) at all r for any specified value of C, i.e. any distance H or the interface from the plane Z = 0. The numerical method followed is described in Ref. [5]. 2.2. Forces on the particle and the critical velocity The repulsive force exerted by the solidification front on the particle is given by [4] FR = 2xB
m rdr
(6)
s 0 h)(r)’
The movement of the particle due to this force necessitates a flow of liquid into the gap, which exerts a drag force on the particle of the form [2]
ss
FA = 127c2r/V,,
RP
r dr
0
where
z!!L=yD
o h3(p)
p
FA
c7)
VP= velocity of the particle
q = viscosity of the melt D,, = 12n29
(8)
The gap width at any point h(r) can be determined knowing Z,(r) from the previous section. Thus FR and FA can be determined for any position of the interface, as determined by H. 2.2.1. The steady state. Due to the forces on the particle, the particle is accelerated according to the equation
BY SOLIDIFICATION
FRONTS
If at any value of H, V,(H) becomes equal to V, dH/dt becomes zero, FR becomes equal to - FA and steady state pushing of the particle occurs. 2.2.2. Determination of VP as a function of H. In equation (9), since m is generally very small, IFR-FAI<
N IFAI
(12)
even under non-steady state conditions. From equation (7) we can then write V,(H)=
--.
FR (H)
(13)
DFA(H)
As H decreases, i.e. the unaffected plane region of the solidification front advances forward, the region affected by the interaction also advances, thereby decreasing the gap width for all values of r and increasing both 1FRj and )DF, /. Initially the net force will be repulsive and hence VPincreases with decrease of H. However because of the high undercooling due to the repulsive force in the region r x 0, the gap width in this region cannot decreases indefinitely. Hence this region of the solidification front slows down considerably while the unaffected region still advances at velocity V. This situation is shown schematically in Fig. 3. This results in the solidification front enveloping the particle more closely. Only the very small values of h(r) in the region r x 0 contribution to DFA comes from even large values of h(r). Hence, when the interface starts curving round the particle, keeping h(0) nearly constant, D,, increases more than FR with change of H and VP starts decreasing. VP as a function of H thus goes through a maximum as shown in Fig. 4. If V is less than VP,, , steady state is possible at two positions H,, and H,, of the solidification front. If v is greater than VP,,, steady state is never possible; H keeps on decreasing [equation (lo)], VP goes through the maximum and starts decreasing as the solidification front envelopes the particle more and more closely and engulfment results.
dVP mdt=FR+FA where m = mass of the particle.
Consider a solidification front moving with an imposed velocity V towards the particle. The plane region of the interface which is unaffected by the particle continues to move with velocity V even after the interaction with the particle becomes effective. Hence H varies as dH = VP- v. dt Equation (9) can be rewritten as rn$j
[V,(H)
- V] = F,(H)
+ F,(H).
(11)
Fig. 3. Shapes assumed by the solidification approaching the particle.
front on
SASIKUMAR
vp
et al.:
PARTICLE
PUSHING
BY SOLIDIFICATION
FRONTS
max = Vc
H
Hc
s2
H91
HFig. 4. Variation
of particle
velocity
with the position
r The existence of the critical velocity is thus a consequence of existence of the maximum in the particle velocity with respect to the position of the solidification front. This in turn is a consequence of the fact the gap width cannot decrease significantly beyond a certain value because of the repulsive interaction. The value of the minimum gap width depends on the extent to which the undercooling due to the repulsive interaction can be balanced by the Gibbs-Thomson undercooling and the effect on the temperature gradient. Thus high values of the Gibbs-Thomson Coefficient or the temperature gradient lead to low values of the minimum gap width and vice versa. It can be seen from equation (10) and Fig. 4 that the steady state at the position HS2 of the solidification front is an unstable one. Any fluctuation which results in an increase of H will bring it to the steady state position HS, and fluctuations leading to decrease of H will lead to engulfment of the particle.
(schematic).
It is evident from the correlations
that
Vc oc R P -‘.2.
This implies that larger particles are more easily engulfed than smaller particles, 3.2. Effect of temperature jield distortion on critical velocity When non-zero values of C,, were considered, the critical velocity was found to increase for negative values of C,(&, > 2,) and decrease for positive values of C, (L,,,< 1,) (Fig. 5). The results can be represented within an error of 1% by the correlation v: = v: (C, = 0) - 10-3 c;‘I c4 in the range C, from lo3 to 10’ and C, from - 10 to + 1. C, can actually vary from - a to + 1. However for large values of C,C,, the distortion of the temperature field is too high to warrant neglecting the transverse heat flow. 3.3. Observations on the shape of the solidiJication
3. RESULTS
front
3.1. Undistorted temperature field Numerical simulations were done for values of C, ranging from -10m6 to -10-r and CX from lo3 to 1013. We found that the dependence of the critical velocity on C, is different for C, C,cc 1 and for Cz C, >>1. The results could be represented (within an error of 10%) by the correlation
or
of the interface
(14) I’,* = 0.72 ]C21-‘--’C,‘.” C,C,<< 1 1
where the dimensionless critical velocity UC* = t’c (6qRi)/B C, C, >>1 generally corresponds to larger particle sizes for most practical systems. When CZC, <<1 the deflection due to the molecular surface force is balanced mainly by the Gibbs-Thomson curvature effect. When C, C, >>1 the effect of temperature gradient becomes prominent. This is reflected in the correlations [14].
Figure 6 shows schematically the shapes assumed by the solidification front for the cases 1, > 1,1, = LP and 1, < 1,. It should be noted that the shape of the front for the case 1, > 1, cannot be represented by Chernov’s approximation of the intersection of a paraboloid and a plane. The depression at the tip of the bump develops as the interface approaches closer in the particle and the repulsive force becomes effective. The critical velocity can be linked to the shape assumed by the solidification front at the position H, corresponding to the maximum particle velocity. For any value of H, the corresponding value of H, [Z,(r = 0), Fig. l] is determined by the balancing of the undercooling due to the repulsive interation by the Gibbs-Thompson undercooling and the effect of the temperature field. Decrease of H (advancement of the solidification front) leads to corresponding decrease of H,,, till H, assumes a value of H, from which it changes very little on further decrease of H.
2090
SASIKUMAR
et al.:
PARTICLE PUSHING
BY SOLIDIFICATION
FRONTS
C,=0.64x106
. C, = 0.1736
x 10’
13 C, = 0.63
12
x IO’
A
\
Fig. 5. Effect of temperature field distortion on critical velocity.
This is the position corresponding to the maximum particle velocity. The ratio H&/H, can be correlated to the value of the critical velocity. This ratio is a measure of the extent to which the solidification front envelopes the particle at the position corresponding to the maximum particle velocity. Larger values correspond to closer envelopment and hence to larger values of D,, and thus to lower values of VP,,, or the critical velocity and vice versa. Table 1 shows the values of H,, H,/H, and V,l for different values of c,.
When C, is positive (I,,, < &), due to the enhancement of the backward deflection, the value of Ho reaches H, only after the solidification front has advanced forward beyond the edge of the particle (note the lowering of H, in Table 1). Hence the ratio H,/H, is larger and the critical velocity smaller than for C., = 0. When C, is negative (A,,,> A,), and the solidification front develops a bump, the critical value Hoc is reached for relatively large values of H, and the ratio H,,,/H, is smaller and the critical velocity larger than for C, = 0. 4. CONCLUSIONS
Table 1. G 1.0 0.8 0.2 0 -1 -2 -10
H,Olm) 1.7 1.75 1.0 I .95 2.15 2.4 4.0
Ho. (am) 2.0286 2.0284 2.0287 2.0288 2.0260 2.0264 2.0215
HdH, 1.19 1.16 1.07 1.04 0.93 0.84 0.51
v: 0.50 x 0.52 x 0.55 x 0.57 x 0.62 x 0.68 x 1.2 x
102 102 101 lop lo2 IO* 10’
1. This theory enables calculation of the critical velocity as a function of all the relevant experimental and material parameters of the system. 2. The non-steady state engulfment of the particle is a consequence of the fact that the gap width cannot reduce significantly below a certain value because of the repulsive interaction.
Fig. 6. Effect of temperature field distortion on the shape of the solidification front.
SASIKUMAR
et al.:
PARTICLE
PUSHING
3. The critical velocity is shown to be linked to the shape of the solidification front and hence to any parameter affecting the shape.
Acknowledgements-The authors sincerely thank Dr A. D. Damodaran. Director Regional Research Laboratory, Trivandrum who nucleated the activity on theoretical modelling of solidification and Dr K. G. Satyanarayana. Head, Materials Division, R.R.L., Trivandrum, for encouragement of the work.
BY SOLIDIFICATION
FRONTS
2091
REFERENCES 1. D. R. Uhlmann, B. Chalmers and K. A. Jackson, J. appl. Phys. 35, 2986 (1964). 2. G. F. Boiling and J. Cisse. J. Cryst. Growth 10, 55 (1971). 3. R. R. Gilpin, J. CON. Inrerface Sci. 74, 44 (1980). 4. A. A. Chernov. D. B. Temkin and A. M. Melnikova, Sotliet Phvs. Crvstallopr. 21, 369 11976). 5. R. Sasikumar, T. R. Ramamohan‘and B. C. Pai, Proc. of Indo-US Workshop on Solidification and Materials Processing. India, (1988).
CRITICAL
OOOI-616OjSS 63.00+ 0.00
Copyright C 1989Pergamon Press plc
VELOCITIES FOR PARTICLE PUSHING MOVING SOLIDIFICATION FRONTS
BY
R. SASIKUMAR, T. R. RAMAMOHAN and B. C. PAI Regional Research Laboratory, Industrial Estate P.O., Trivandrum 695 019, India (Receioed 1 July 1988)
Abstract-A theory is developed for evaluation of critical velocities for pushing of particles by solidification fronts. A numerical simulation is done in which the critical velocity is evaluated as a function of particle size, thermal conductivity, temperature gradient as the solidification front, the Hamaker constant of interaction between the particle and the front, solid-liquid interfacial energy and melt viscosity. The shape of the solidification front under steady state pushing conditions as well as the changes in shape that take place during the non-steady state engulfment process can be determined using this theory. The critical velocity is shown to be closely linked to the shape of the solidification front and hence to all the parameters that affect the shape of the front. R&urn&-On diveloppe une theorie pour Cvaluer les vitesses critiques pour que des particules soient joussCes par les fronts de solidification. Une simulation numtrique est proposee dans laquelle la vitesse critique est &al&e en fonction de la taille des particules, de la conductivitt thermique, du gradient de temptrature au front de solidification, de la constante d’Hamaker d’intkraction entre la particule et le front, de l’knergie de l’interface solids-liquide et de la viscositt du bain fondu. La forme du front de solidification sous des conditions de poussCe stables ainsi que le changement de forme qui se produit pendant le processus instable d’engloutissage peuvent 6tre di?terminCs g&e B cette thborie. On montre que la vitesse critique est ttroitement like g la forme du front de solidification et done B tous les paramttres qui affectent la forme du front. Zusammenfassung-Eine Theorie zur Auswertung kritischer Geschwindigkeiten fiir das Vorwlrtstreiben von Teilchen durch eine Erstarrungsfront wird entwickelt. Mit einer numerischen Simulation ergibt sich die kritische Geschwindigkeit in Abhlngigkeit von der TeilchengriiDe, der thermischen Leitfihigkeit, dem Temperaturgradienten an der Erstarrungsfront, der Hamaker-Konstanten der Wechselwirkung zwischen dem Teilchem und der Front, der fest-fliissigen GrenzflCchenenergie und der ViskositLt der Schmelze. Die Form der Erstarrungsfront unter stationiiren Bedingungen des Vorwiirtstreibens wie such die VerHnderrungen der Form, die wSihrend des nicht-stationlren Verzehrprozesses auftreten, k6nnen mittels dieser Theorie bestimmt werden. Es zeigt sich, da13 die kritische Geschwindigkeit eng mit der Form der Erstarrungsfront und damit mit all den Parametern. die die Form beeinflussen, zusammenhlngt.
1. INTRODUCTION
When a liquid containing a dispersion of particles is solidified, the solidification front can exert a repulsive force on the particles and push them along with it. thereby increasing the particle concentration in the last solidifying liquid. It has been known that this phenomenon of particle pushing occurs only below a certain velocity of the advancing solidification front known as the critical velocity. If the velocity of the growth front is greater than the critical velocity. the particle is engulfed by the front. The critical velocity depends on the properties of the matrix material and the particle as well as the experimental conditions. Theories for evaluation of the critical velocities have been developed by various authors [l-4]. All the theories are based on the concept of molecular surface forces repelling the particle from the growth front and thus facilitating the entry of liquid into the gap between the growing solid and the particle. The pushing of the particle is considered as a steady state phenomenon in which the
particle and the solidification front move with the same velocity and the gap width is maintained constant. The critical velocity is identified as the maximum velocity above which the steady state cannot be maintained. The most rigorous of these theories is that of Chernov et al. [4] who determine the shape of the interface and use this shape to calculate the forces on the particle and the steady state velocity. Very recently the present authors have developed a theory [5] which follows the approach of Chernov but improves upon it by:
2085
(1) including the effect of the thermal conductivity of the particle being different from that of the melt. (2) performing a rigorous numerical solution of the equations describing the shape of the front. rather than assuming the shape to be described as a combination of a plane and a paraboloid intersecting at a certain point, and (3) enabling calculation of non-steady state shapes also.
2086
SASIKUMAR
et al.: PARTICLE PUSHING BY SOLIDIFICATION
This paper presents the results of extensive numerical simulations based on the above theory. The equations in the previous paper have been scaled and correlations giving the critical velocity of all the relevant material and experimental parameters have been developed. Special attention has been paid to the effect of distortion of the temperature field by the particle on the critical velocity, which has not been treated till now by the previous theories. It is shown that as the solidification front ap proaches the particle, the particle velocity increases, goes through a maximum and then decreases. If the particle velocity becomes equal to the front velocity before the maximum is reached, the front cannot approach any closer to the particle and steady state pushing results. If the velocity of the front is greater than the maximum particle velocity, steady state never occurs, the particle velocity crosses the maximum, decreases and engulfment results. Hence the maximum particle velocity is identified as the critical velocity. All changes in the critical velocity can be linked to the shape of the solidification front at the position resulting in the maximum particle velocity and hence to any parameter affecting the shape of the front. Evaluation of non steady state shapes assumed by the front for velocities above the critical velocity gives an understanding of the physical reasons for the existence of the maximum in particle velocity.
2. THE THEORY 2.1. The shape of the solid$cation
front
The details of the derivation are given in Ref. [S]. The basic assumption is that the solidification front takes such a shape that the actual temperature and every point is equal to the equilibrium temperature for co-existence of solid and liquid. The equilibrium temperature is T,,[Zi (r), rl
= TM- AT,- ATsutiace force
(1)
where
FRONTS
r
t-l Fig. 1. The coordinate system.
where B = Hamaker
constant
h(r) = gap width. A negative B implies a repulsive force between the particle and the front and hence a lowering of the equilibrium temperature. The temperature field can be assumed to be linear with gradient G far away from the particle. From Fig. 1, the actual temperature at any point of the interface far away from the particle (r >>R,,) is T(H,r)=T,-GH,
rmR,.
(2)
In the vicinity of the particle, the temperature field will be distorted if the thermal conductivity of the particle is different from that of the melt. This distortion is shown schematically in Fig. 2. If the neglect transverse flow of heat we can write T(Z,, r) = TO- G, ,/v - G(Z, - ,/m),
r C R,
(3a)
TM = freezing point
AT, = Gibbs Thomson undercooling
=
2urc~/As
where G, is the gradient inside the particle
and CI= solid liquid interface energy AS = entropy of fusion R = molar volume K = mean curvature of the front Z” Z” Z’ = 2[1 + Z’Z]W + 7 + 7 A Curfaceforces = undercooling forces [4] =-
-BR h:,, AS
due to molecular surface
1, = thermal conductivity ductivity of particle and
of melt; 1, = thermal con-
T(Z,, r) = TO- GZr),
r > R,.
W)
If transverse heat flow is considered, the distortion, [i.e. the difference in temperature between a point (Z,, r c RP) and (Z,, r -)a)] will be less than that predicted by equation (3). Hence use of equation (3) gives an upperbound on the extent of distortion that can be expected.
SASIKUMAR
et al.:
PARTICLE
PUSHING BY SOLIDIFICATION
2087
FRONTS
z=o
2
--
3
e ai
Undistorted Distorted
temperature temperature
E” @!
Fig. 2. Distorted temperature field for I., # I.,
Substituting equations (1) and (3) into the equation, T,, [Z,(r), r] = T[Z,(r), r] gives the equation for the shape of the solidification front as Bt2 cw213’2 J_ {(Z,
z,, = 211 + I
nu
+ AS[G(Z, - H)
x [
Z”
Z: ---L, r
O
r
=
I
1 -:Jl
(
Zf’ r*
z,,
Z:--
2C,C,
C: ZY’ r* ’
i I
-rrr2
>I
r* > 1
211+ (-w213’* ASG (Z, - H) Ra
z;
----
Zi3
r
r > R,
r ’
r* =0
BR
1 (Z, -
zy = is
+
4557)’
As[G(Z, -H)
where C, = H/R,
- G(l - &,,/L,)R,]
,
r = 0.
(4)
For doing a numerical simulation it is more convenient to scale these equations. This gives Z:l’ = (1 +
c;z72)3,2
C2= B/Ria C, = ASGR”,/BQ c, = 1 - l,/l,
i$ I
1
X
i
(Zf
-
$72)3 1
The shape of the solidification depends on:
front
therefore
(1) the distance of the plane region of the interface from the plane Z = 0 as depicted by C, ;
2088
SASIKUMAR
et al.:
PARTICLE PUSHING
(2) the extent to which the Gibbs-Thomson effect balances the deflection due to the repulsive surface force, as depicted by C,; (3) the extent to which the temperature gradient and the consequent penetration of the solidification front into a colder region balances the deflection due the repulsive surface force, as depicted by C3; and (4) the distortion brought in due to the difference in thermal conductivities of the particle and the melt, as depicted by C,.
Equations (5) are solved numerically to determine Z,(r) at all r for any specified value of C, i.e. any distance H or the interface from the plane Z = 0. The numerical method followed is described in Ref. [5]. 2.2. Forces on the particle and the critical velocity The repulsive force exerted by the solidification front on the particle is given by [4] FR = 2xB
m rdr
(6)
s 0 h)(r)’
The movement of the particle due to this force necessitates a flow of liquid into the gap, which exerts a drag force on the particle of the form [2]
ss
FA = 127c2r/V,,
RP
r dr
0
where
z!!L=yD
o h3(p)
p
FA
c7)
VP= velocity of the particle
q = viscosity of the melt D,, = 12n29
(8)
The gap width at any point h(r) can be determined knowing Z,(r) from the previous section. Thus FR and FA can be determined for any position of the interface, as determined by H. 2.2.1. The steady state. Due to the forces on the particle, the particle is accelerated according to the equation
BY SOLIDIFICATION
FRONTS
If at any value of H, V,(H) becomes equal to V, dH/dt becomes zero, FR becomes equal to - FA and steady state pushing of the particle occurs. 2.2.2. Determination of VP as a function of H. In equation (9), since m is generally very small, IFR-FAI<
N IFAI
(12)
even under non-steady state conditions. From equation (7) we can then write V,(H)=
--.
FR (H)
(13)
DFA(H)
As H decreases, i.e. the unaffected plane region of the solidification front advances forward, the region affected by the interaction also advances, thereby decreasing the gap width for all values of r and increasing both 1FRj and )DF, /. Initially the net force will be repulsive and hence VPincreases with decrease of H. However because of the high undercooling due to the repulsive force in the region r x 0, the gap width in this region cannot decreases indefinitely. Hence this region of the solidification front slows down considerably while the unaffected region still advances at velocity V. This situation is shown schematically in Fig. 3. This results in the solidification front enveloping the particle more closely. Only the very small values of h(r) in the region r x 0 contribution to DFA comes from even large values of h(r). Hence, when the interface starts curving round the particle, keeping h(0) nearly constant, D,, increases more than FR with change of H and VP starts decreasing. VP as a function of H thus goes through a maximum as shown in Fig. 4. If V is less than VP,, , steady state is possible at two positions H,, and H,, of the solidification front. If v is greater than VP,,, steady state is never possible; H keeps on decreasing [equation (lo)], VP goes through the maximum and starts decreasing as the solidification front envelopes the particle more and more closely and engulfment results.
dVP mdt=FR+FA where m = mass of the particle.
Consider a solidification front moving with an imposed velocity V towards the particle. The plane region of the interface which is unaffected by the particle continues to move with velocity V even after the interaction with the particle becomes effective. Hence H varies as dH = VP- v. dt Equation (9) can be rewritten as rn$j
[V,(H)
- V] = F,(H)
+ F,(H).
(11)
Fig. 3. Shapes assumed by the solidification approaching the particle.
front on
SASIKUMAR
vp
et al.:
PARTICLE
PUSHING
BY SOLIDIFICATION
FRONTS
max = Vc
H
Hc
s2
H91
HFig. 4. Variation
of particle
velocity
with the position
r The existence of the critical velocity is thus a consequence of existence of the maximum in the particle velocity with respect to the position of the solidification front. This in turn is a consequence of the fact the gap width cannot decrease significantly beyond a certain value because of the repulsive interaction. The value of the minimum gap width depends on the extent to which the undercooling due to the repulsive interaction can be balanced by the Gibbs-Thomson undercooling and the effect on the temperature gradient. Thus high values of the Gibbs-Thomson Coefficient or the temperature gradient lead to low values of the minimum gap width and vice versa. It can be seen from equation (10) and Fig. 4 that the steady state at the position HS2 of the solidification front is an unstable one. Any fluctuation which results in an increase of H will bring it to the steady state position HS, and fluctuations leading to decrease of H will lead to engulfment of the particle.
(schematic).
It is evident from the correlations
that
Vc oc R P -‘.2.
This implies that larger particles are more easily engulfed than smaller particles, 3.2. Effect of temperature jield distortion on critical velocity When non-zero values of C,, were considered, the critical velocity was found to increase for negative values of C,(&, > 2,) and decrease for positive values of C, (L,,,< 1,) (Fig. 5). The results can be represented within an error of 1% by the correlation v: = v: (C, = 0) - 10-3 c;‘I c4 in the range C, from lo3 to 10’ and C, from - 10 to + 1. C, can actually vary from - a to + 1. However for large values of C,C,, the distortion of the temperature field is too high to warrant neglecting the transverse heat flow. 3.3. Observations on the shape of the solidiJication
3. RESULTS
front
3.1. Undistorted temperature field Numerical simulations were done for values of C, ranging from -10m6 to -10-r and CX from lo3 to 1013. We found that the dependence of the critical velocity on C, is different for C, C,cc 1 and for Cz C, >>1. The results could be represented (within an error of 10%) by the correlation
or
of the interface
(14) I’,* = 0.72 ]C21-‘--’C,‘.” C,C,<< 1 1
where the dimensionless critical velocity UC* = t’c (6qRi)/B C, C, >>1 generally corresponds to larger particle sizes for most practical systems. When CZC, <<1 the deflection due to the molecular surface force is balanced mainly by the Gibbs-Thomson curvature effect. When C, C, >>1 the effect of temperature gradient becomes prominent. This is reflected in the correlations [14].
Figure 6 shows schematically the shapes assumed by the solidification front for the cases 1, > 1,1, = LP and 1, < 1,. It should be noted that the shape of the front for the case 1, > 1, cannot be represented by Chernov’s approximation of the intersection of a paraboloid and a plane. The depression at the tip of the bump develops as the interface approaches closer in the particle and the repulsive force becomes effective. The critical velocity can be linked to the shape assumed by the solidification front at the position H, corresponding to the maximum particle velocity. For any value of H, the corresponding value of H, [Z,(r = 0), Fig. l] is determined by the balancing of the undercooling due to the repulsive interation by the Gibbs-Thompson undercooling and the effect of the temperature field. Decrease of H (advancement of the solidification front) leads to corresponding decrease of H,,, till H, assumes a value of H, from which it changes very little on further decrease of H.
2090
SASIKUMAR
et al.:
PARTICLE PUSHING
BY SOLIDIFICATION
FRONTS
C,=0.64x106
. C, = 0.1736
x 10’
13 C, = 0.63
12
x IO’
A
\
Fig. 5. Effect of temperature field distortion on critical velocity.
This is the position corresponding to the maximum particle velocity. The ratio H&/H, can be correlated to the value of the critical velocity. This ratio is a measure of the extent to which the solidification front envelopes the particle at the position corresponding to the maximum particle velocity. Larger values correspond to closer envelopment and hence to larger values of D,, and thus to lower values of VP,,, or the critical velocity and vice versa. Table 1 shows the values of H,, H,/H, and V,l for different values of c,.
When C, is positive (I,,, < &), due to the enhancement of the backward deflection, the value of Ho reaches H, only after the solidification front has advanced forward beyond the edge of the particle (note the lowering of H, in Table 1). Hence the ratio H,/H, is larger and the critical velocity smaller than for C., = 0. When C, is negative (A,,,> A,), and the solidification front develops a bump, the critical value Hoc is reached for relatively large values of H, and the ratio H,,,/H, is smaller and the critical velocity larger than for C, = 0. 4. CONCLUSIONS
Table 1. G 1.0 0.8 0.2 0 -1 -2 -10
H,Olm) 1.7 1.75 1.0 I .95 2.15 2.4 4.0
Ho. (am) 2.0286 2.0284 2.0287 2.0288 2.0260 2.0264 2.0215
HdH, 1.19 1.16 1.07 1.04 0.93 0.84 0.51
v: 0.50 x 0.52 x 0.55 x 0.57 x 0.62 x 0.68 x 1.2 x
102 102 101 lop lo2 IO* 10’
1. This theory enables calculation of the critical velocity as a function of all the relevant experimental and material parameters of the system. 2. The non-steady state engulfment of the particle is a consequence of the fact that the gap width cannot reduce significantly below a certain value because of the repulsive interaction.
Fig. 6. Effect of temperature field distortion on the shape of the solidification front.
SASIKUMAR
et al.:
PARTICLE
PUSHING
3. The critical velocity is shown to be linked to the shape of the solidification front and hence to any parameter affecting the shape.
Acknowledgements-The authors sincerely thank Dr A. D. Damodaran. Director Regional Research Laboratory, Trivandrum who nucleated the activity on theoretical modelling of solidification and Dr K. G. Satyanarayana. Head, Materials Division, R.R.L., Trivandrum, for encouragement of the work.
BY SOLIDIFICATION
FRONTS
2091
REFERENCES 1. D. R. Uhlmann, B. Chalmers and K. A. Jackson, J. appl. Phys. 35, 2986 (1964). 2. G. F. Boiling and J. Cisse. J. Cryst. Growth 10, 55 (1971). 3. R. R. Gilpin, J. CON. Inrerface Sci. 74, 44 (1980). 4. A. A. Chernov. D. B. Temkin and A. M. Melnikova, Sotliet Phvs. Crvstallopr. 21, 369 11976). 5. R. Sasikumar, T. R. Ramamohan‘and B. C. Pai, Proc. of Indo-US Workshop on Solidification and Materials Processing. India, (1988).