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Aggregation and Densification of Cohesive Granular
Materials
Cohesive granular materials have been the focus of only a small fraction
of recent research into the granular state. Yet cohesive granular materials
will surely draw increasing attention from scientists and engineers, if
only because they are used in numerous applications. Conspicuous examples
are the forming of ceramic parts, powder metallurgy components and pharmaceutical
tablets by compaction of fine powders. The cohesiveness of powders
stems from the large surface-to-volume ratio of their constitutive particles,
which enhances the effect of attractive van der Walls forces among the particles.
In other applications, e.g., the stabilization of soils, the cohesiveness
is due to the presence of liquid menisci among the particles.
Particle Aggregation: Experiments
When, preceding compaction, a cohesive granular material is poured into
a container, the mobility of the particles reaching the bottom of the container
is hindered by the cohesive forces. To investigate this phenomenon we filled
a narrow Plexiglas container (of thickness 1.9 mm) with monosized glass beads
(of diameter 1.7 mm). Before pouring them into the container, we wetted the
beads with water in order for menisci to form among the beads. These
menisci provided the required cohesion. The figure shows a number of stages
during the experiment, and should be read clockwise starting from the left
top (Uribe and Gioia, unpublished, 2001). As a result of the cohesive
forces, a low-density, open particle aggregate obtains within the
container. Open particle aggregates are locked in local energy minima, far
from the global minimum.
Particle Aggregation: Computations
We performed computer simulations of aggregation using ballistic aggregation
methods. These methods originated in the study of processes such as coagulation
of colloids and high-energy homoepitaxial growth. In our implementation
the particles are sequentially deposited along randomly chosen vertical paths,
and then allowed to roll down to a stable position in the presence of attractive,
short-ranged, center-to-center cohesive forces any two particles. More specifically,
each particle rolls down until it contacts two points (at least) on the surface
of the growing aggregate. In spite of its simplicity, the results
of this computation match the experimental observations quite well.
The obtained aggregates may be conveniently used as initial conditions for
the computational simulation of densification processes.
It has long been known that when compacted open particle aggregates densify
by particle rearrangement at low pressure. Particle rearrangement can be
readily identified in the compaction curves of cohesive powders,
where it takes the form of a region of large compliance known as Region I.
After rearrangement has been completed, further densification requires the
particles to deform. In compaction curves the deformation stage takes the
form of a region of small compliance which is customarily called Region II.
Two-Phase Densification by Particle Rearrangement
The recent X-ray tomography experiments of Lannutti et al. cast a critical
light on the prevalent view that particle rearrangement is a spatially
homogeneous process. These authors studied the densification of ceramic
powders and documented the development of density fields with rather well
defined high- and low-density regions. They concluded that ``densification
occurs in a step-wise fashion,'' in the form of ``a wave generated
at the advancing ram'' (C. M. Kong, C.M., and Lannutti, J.J., J. Amer.
Ceram. Soc. 83, 685, 2000).
To elucidate the nature of Kong and Lannutti's `wave' we compacted an
open particle aggregate using a ram. The first photographic sequence
in the figure shows three stages during the experiment. A high-density region
(dark gray) and a low-density region (light gray) are discernible in the
three stages. (The ram is black.) By direct visual inspection we verified
that no particle rearrangement occurs within these regions; in fact,
the densities of the high- and low-density regions remain constant throughout
the experiment. We conjecture that the high-density region is composed of
a configurational phase H wherein rearrangement has taken place
already; and that the low-density region is composed of a configurational
phase L wherein the initial open aggregate remains essentially unchanged.
In a narrow vicinity of the H-L interface or rearrangement front
we could clearly observe the collapse by snap-through buckling of successive
layers of rings of particles. We have documented this process in the
second photographic sequence in the figure, which should be read clockwise
starting from the top left. We conclude that densification occurs by growth
of the volume fraction of H at the expense of the volume fraction
of L.
Our interpretation of these experimental results is that densification
by particle rearrangement occurs in the form of a phase transformation
L->H.
The Micromechanism of Particle Rearrangement
To substantiate our interpretation of the experimental evidence we start
by turning our attention to the micromechanism of particle rearrangement.
It has been proposed (Kuhn, L.T., McMeeking, R.M., and Lange, F.F.,
J. Amer. Ceram. Soc. 74, 682, 1991) that particle rearrangement is
due to the collapse by snap-through buckling of the `rings
of particles' of the cohesive aggregates. (In two dimenions, the rings of
particles are composed of six or more particles; these rings are a pervasive
feature in experimentally obtained quasi two-dimensional aggregates, and
also in computationally simulated aggregates; see the figure.) Snap-through
buckling occurs when one of the particles in the ring of particles jumps
to the center of the ring; see the figure.
It is possible to show that the snap-through buckling mechanism leads
to nonconvex strain energy functions. Nonconvex energy functions are characteristic
of systems which undergo phase transformations. We conclude that our interpretation
of the experimental evidence is consistent with the micromechanism of
particle rearrangement.
A Model of Compaction
We have formulated a model of compaction based on the energetics of particle
rearrangement. In the figure we compare the density distribution measured
by Kong and Lannutti in a ceramic compact (Op. Cit., 2000) with the predictions
of our model (Gioia and Cuitiño, unpublished, 2001). The model accounts
for the effects of container roughness and particle deformability. Because
of container roughness, L->H densification at the rearrangement front
may take place concurrently with further rearrangement and inelastic particle
deformations within the high-density region. This calls into question the
classical distinction between Regions I and II of the compaction curve,
which was tied to a view of densification as a homogeneous process. Provided
that the physical nature of densification is properly understood, however,
the concept of Regions I and II can still play a useful role, for example
when comparing the overall behavior of different powders.
References
Gioia, G., Cuitiño, A.M., Zheng, S., and Uribe, T.
Two-Phase Densification of Cohesive Granular
Aggregates
Phys. Rev. Lett. 88, 204302, 2002.
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This project is supported by grants from the UIUC Research Board and
a UIUC Critical Research Initiative