
sumption, they were able to derive simple analytical results
which related h,, and hM with the growth rate as follows:
Vh 2 = K2/K J,
and [2]
vx], = K3, [31
where
K 3
is a constant parameter which depends on the
properties of the system only. The expression for
K 3
is
given in the Appendix. The range of stable eutectic spac-
ings is shown schematically in Figure 1.
The basic ideas of eutectic stability, first proposed by
Jackson and Hunt, have subsequently been examined by a
number of detailed stability analyses of eutectic inter-
faces. 19-161 The most extensive theoretical studies by Langer
and coworkers I14A5'161 show that the smallest stable spacing
corresponds to the minimum undercooling value. These
theoretical models of eutectic stability do not predict a
unique spacing value for a given growth rate. Datye and
LangertlSJ pointed out that the system might drift toward the
lower marginal stability point, given by
hm.
This sugges-
tion is somewhat analogous to that for the dendrite tip
selection where the experimental studies show that the den-
drite tip radius assumes the marginally stable radius based
on the isothermal or the isoconcentrate interface model. It
should be pointed out that the dendrite tip selects the largest
stable radius value, whereas the minimum undercooling
criterion predicts that the eutectic system will select the
smallest stable spacing value. It is not very clear as to why
these two different systems will drift in different directions.
Also, there is no proper justification for the marginal sta-
bility criterion. One of the current thoughts t~7'~8'~91 on the
dendrite growth problem is that the anisotropy of the inter-
facial energy controls the stable dendrite tip radius. For
eutectic growth, anisotropy effects do not appear to be im-
portant, at least for the case of nonfaceted eutectic phases.
One would thus expect that a range of eutectic spacings,
h,, < h < kin,
is possible. We shall now examine some of
the experimental studies which give some insight into the
actual observed spacings.
The variations in average eutectic spacing with growth
rate have been measured in many eutectic systems, and all
the results show that
Vh 2
= constant for low growth rates.
The values of the constants for different systems have been
summarized by Kurz and Sahm. t2~ These experimental re-
suits have often been interpreted as the justification for the
minimum undercooling principle. It is not well appreciated
in the literature that the largest stable spacing also gives a
similar result; see Eqs. [2] and [3]. The specific value of
the theoretical constants will, however, depend on whether
the spacing is established close to Am or h M. Often, the
system parameters are not known precisely and this makes
it difficult to interpret unambiguously where the operating
point of the eutectic spacing lies. For lead-tin and alu-
minum-copper systems, the system parameters have been
measured independently and accurately. In these systems
the average spacing is found to be close to but slightly
larger than
hm.t2~-241
A more detailed study on eutectic spac-
ings in the Pb-Sn system has been carded out by Jordan and
Hunt, t241 who observed a range of eutectic spacings for a
given velocity. This range is quite narrow at high veloci-
ties, although it is significantly larger at low velocities. In
this regard it is worthwhile to note the experimental studies
by Kurz and coworkers 12s-28~ on faceted-nonfaceted eutec-
tics in which the eutectic spacing oscillates between
hm
and hM.
The above experimental studies clearly indicate that
there is no sharp selection criterion for eutectic spacing.
It is then puzzling as to why the average eutectic spac-
ings tend to be closer to
hm
rather than midway between hm
and hM. In order to examine the eutectic spacing selection
problem in more detail, we have carried out critical experi-
mental studies in a transparent eutectic system. Initially,
steady-state eutectic lamellar structures were examined to
characterize precisely the distributions of spacings at differ-
ent velocities. These studies allowed us to characterize the
variations in the average, the smallest, and the largest spac-
ing with the growth rate. Subsequently, dynamical studies
were carried out to see how these observed distributions
evolve with time. Specifically, we have approached a given
velocity by initially stabilizing the eutectic spacings at a
smaller spacing
(i.e.,
higher velocity) or at a larger spacing
(i.e.,
lower velocity). In both cases, the spacing is found
to drift toward
hm
and stabilize at values slightly larger
than
h m .
A small hysteresis effect is found for the average
spacing value. In all cases spacings close to hM have been
found to drift toward the
h m
value, indicating that the range
of stable eutectic spacing is significantly smaller than that
proposed by Jackson and Hunt. [71
II. EXPERIMENTAL PROCEDURES
A. Materials and Cell Preparation
Experimental studies were carried out in carbon tetra-
bromide (CBr4)-hexachloroethane
(C2C16)
system which is
a transparent organic analog for nonfaceted, lameUar eutec-
tic structures. The equilibrium phase diagram for this sys-
tem has been determined by several authors.I29-321 Although
there are some discrepancies among these diagrams, par-
ticularly with respect to the maximum solid solubility of
C2C16 in CBr4 (c~ phase), there is a reasonably good agree-
ment on the following features: eutectic temperature =
356 K; eutectic composition = 8.4 wt pct
C2C16;
melting
point of CBr4 = 364.8 K, and melting point of
C2C16 :
459.6 K. Pure carbon tetrabromide solidifies with a face-
centered cubic structure with a lattice constant of 8.82/k.
It undergoes a transition to a monoclinic phase at 321.4 K.
Pure hexachloroethane also solidifies into a cubic form with a
lattice constant of 7.51/~ and undergoes a solid state transi-
tion to a triclinic phase at 344.3 K. Additional values of the
thermal and physical properties of the a and fl phases have
been measured by Kaukler. f321 Only the value of the inter-
diffusion coefficient has not been measured independently.
The commercial grade carbon tetrabromide was purified
by the vacuum sublimation process. This consisted of heat-
ing carbon tetrabromide at approximately 330 K under a
pressure of 100 mm Hg and allowing its vapors to condense
at -230 K. In contrast, the hexachloroethane could be sub-
limed at atmospheric pressure and at 320 K and then con-
densed at 290 K. These purified materials were mixed,
melted, and filled in specially fused glass cells which had
two narrow openings for filling the cell with the alloy.
Since the vapor pressures of both CBr4 and C2C16 are quite
appreciable at the eutectic temperature, it was necessary to
2956--VOLUME 19A, DECEMBER 1988 METALLURGICAL TRANSACTIONS A