This work examines the diffusional growth of discrete phase particles dispersed within a matrix. Engineering materials are microstructurally heterogeneous, and the details of the microstructure determine how well that material performs in a given application. Critical to the development of designing multiphase microstructures with long-term stability is the process of Ostwald ripening. Ripening, or phase coarsening, is a diffusion-limited process which arises in polydisperse multiphase materials. Growth and dissolution occur because fluxes of solute, driven by chemical potential gradients at the interfaces of the dispersed phase material, depend on particle size. The kinetics of these processes are "competitive," dictating that larger particles grow at the expense of smaller ones, overall leading to an increase of the average particle size. The classical treatment of phase coarsening was done by Todes, Lifshitz, and Slyozov, (TLS) in the limit of zero volume fraction, V(sub v), of the dispersed phase. Since the publication of TLS theory there have been numerous investigations, many of which sought to describe the kinetic scaling behavior over a range of volume fractions. Some studies in the literature report that the relative increase in coarsening rate at low (but not zero) volume fractions compared to that / 2 1/ 3 predicted by TLS is proportional to V(sub v)(exp 1/2), whereas others suggest V(sub v)(exp 1/3). This issue has been resolved recently by simulation studies at low volume fractions in three dimensions by members of the Rensselaer/MSFC team

Downey, J. Patton

Facemire, Barbara R.

Frazier, Donald O.

Glicksman, Martin E.

Rogers, Jan R.

Witherow, William K.

English

NASA Technical Reports Server

Title:Evolution of Local Microstructures (ELMS): Spatial Instabilities of Coarsening
Clusters
Principal Investigator: Martin E. Glicksman, Rensselaer Polytechnic Institute
S +'5-
Co-Investigators: Donald O. Frazier*, Jail R. Rogers*, William K. Witherow*,
J. Patton Downey*, Barbara R. Facemire*
*NASA Marshall Space Flight Center, Space Sciences Laboratory
Introduction
This work examines the diffusional growth of discrete phase particles dispersed within a
matrix. Engineering materials are microstructurally heterogeneous, and the details of the
microstructure determine how well that material performs in a given application. Critical
to the development of designing multiphase microstructures with long-term stability is
the process of Ostwald ripening. Ripening, or phase coarsening, is a diffusion-limited
process which arises in polydisperse multiphase materials. Growth and dissolution occur
because fluxes of solute, driven by chemical potential gradients at the interfaces of the
dispersed phase material, depend on particle size. The kinetics of these processes are
"competitive," dictating that larger particles grow at the expense of smaller ones, overall
leading to an increase of the average particle size. The classical treatment of phase
coarsening was done by Todes, Lifshitz, and Slyozov, (TLS) 1'2in the limit of zero volume
fraction, Vv, of the dispersed phase. Since the publication of TLS theory there have been
numerous investigations, many of which sought to describe the kinetic scaling behavior
over a range of volume fractions. Some studies in the literature report that the relative
increase in coarsening rate at low (but not zero) volume fractions compared to that
predicted by TLS is proportional to V_/ 2, whereas others suggest V) / 3. This issue has
been resolved recently by simulation studies at low volume fractions in three dimensions
by members of the Rensselaer/MSFC team) -5
Background and Objectives
Our studies of ripening behavior using large-scale numerical simulations suggest that
although there are different circumstances which can lead to either scaling law, the most
important length scale at low volume fractions is the diffusional analog of the Debye
screening length. This screening length places limits on the extent and applicability of a
mean-field description and can result in local divergences from mean-field behavior. The
numerical simulations we employed exploit the use of a recently developed "snapshot"
technique and identify the nature of the coarsening dynamics at various volume fractions.
Preliminary results of numerical and experimental investigations, focused on the growth
261
https://ntrs.nasa.gov/search.jsp?R=19990040286 2020-06-18T00:56:40+00:00Z
of finite particle clusters, provide important insight into the nature of the transition
between the two scaling regimes. The companion microgravity experiment centers on
particle growth within finite clusters and follows the temporal dynamics driving
microstructural evolution using holography.
This research effort will extend our preliminary ground-based results and develop a
critical microgravity experiment to observe these phenomena in a suitably quiescent
environment. Work accomplished to date shows that the critical cross-over between the
two scaling regimes ( Vv_/2 versus Vv_/_) is a function of both the volume fraction, which,
intuitively, affects the screening length and the cluster size, n, where n is the number of
particles comprising the cluster. Specifically, the cross-over occurs at a critical volume
fraction for which V_ 1/(3n2). One also expects that the volume fraction itself fluctuates
from region-to-region, because the local size and distribution of particles vary. Thus,
spatial variances should arise in the local kinetics. One of the scientific objectives of this
work is to identify the nature of these microstructural fluctuations. This added level of
understanding will be of importance in the development of more predictive models for
microstructural evolution in heterogeneous materials.
Theor_
We begin with a discussion of infinitely dilute coarsening systems based on the TLS
mean-field solution to coarsening kinetics which is restricted to zero volume fraction of
droplet phase. The continuity equation requires that neither nucleation nor coalescence
occur, only smooth growth and dissolution:
OF O
-- +-- [v(R).X]= 0
3t OR
°
Our approach in this work is to seek to measure and predict v(R), the time rate of change
of the radius, R, of a particle, in a given cluster. To obtain v(R), certain simplifying
assumptions are required:
1. Kinetics are limited by volume diffusion in the matrix, and
2. Diffusion remains quasi-static.
Therefore, the diffusion equation describing the concentration field, c(r), in the matrix
phase reduces to the Laplace equation:
V2q_ = 0, .
where q_ =(c- Co)/Co, and Cois the equilibrium solubility.
262
Theboundaryconditionsatthesurfaceof the i-th particle are specified through the
Gibbs-Kelvin local equilibrium relation
q) i, _-_--B R'
.
where k is the capillary length given by, L = 27f)/(kBT), through which all quantities of
length are scaled in this mean-field treatment; y is the droplet-matrix interfacial energy; f_
is the atomic volume of the dispersed phase; and kB is Boltzmann's constant.
Solution to the Laplace equation, subject to the Gibbs-Kelvin boundary condition, may
be written in terms of the Coulomb potential
.
where q0i is the coulomb potential at a distance r from an ion of charge zie.
The diffusion potential is, therefore,
_B TLS
= ,'-,..I+ ,
The diffusion analog of electrostatic charge is source/sink strength (volume flux), B rLsi '
given by
,
where R* is the critical radius given by, R ° = )_ / q_.o, and qp_is the background matrix
potential, assumed uniform throughout the matrix phase. The zero volume fraction
restriction in TLS theory results in the total neglect of direct interactions among particles.
A more realistic approach which introduces the influence of volume fraction on
coarsening kinetics was employed by Marqusee and Ross. 6 Their approach applies a
spatially coarse-grained background diffusion potential, which is the diffusion analog of
the electric potential arising from a given charge density, p. An expression for electric
potential influenced by P is,
V2q) = _P 7.
_O
The diffusion analog is, therefore,
V2_ = -4_, 8.
263
where or, a source/sink density in the matrix space, is analogous to p from electrostatics.
introduces a coarse-grained local background potential, tp(r), due to droplets
surrounding a coarsening droplet centered at point r in the matrix,
V exp(-K.lr- r[)+ .
q0(r) =/_/. [r - r[ .
The volume flux, Bi, is now,
10.
where K represents the natural cut-off distance for direct particle interactions via the
diffusion field, beyond which the particles are isolated from each other by the intervening
two-phase medium.
For droplet clusters comprising finite coarsening systems, one may define physically
distinct coarse-grained length scale parameters; among them are:
the Debye screening radius, AD-- K_.
the average droplet radius in the cluster, A R _ (R),
therefore, AD = [4r_Nv<R>] -_j2 As in electrostatics, for large A D (sparse particle
density), the screened potential approaches the unscreened potential.
the extent of the total coarsening system, Atot, is defined for a "spherical cluster" by
/3 n 1/3
Atot = , 11.
where n is the total number of particles remaining at a given time, and for AD >
Ato t, screening is not a factor.
One can now relate the time-rate of change of radius,
v(Ri) - 22Dc°f) B,,
R
i
whereby Bi is now representative of the volume flux of a droplet in a finite coarsening
cluster. The experimental parameters for determining the time-rate of change of each
12.
264
droplet within a given cluster are accessible through physicochemical measurement and
holography.
Experiment
_Experimental objectives of ELMS are:
• to measure v(Ri) of each droplet cluster by using holography
to determine, in a mixed-dimensional system (3-D droplets undergoing 2-D
diffusion), the existence, if any, of spatial instabilities in microstructural homogeneity
due to a heretofore undefined "screening length"
to verify in a fully 3-D system in microgravity the TLS coarsening rate deviation
dependence on local volume fractions for global volume fractions < 1%, where the
volume fraction for a finite coarsening cluster is,
>-_(4/3)_/_ 3
W _ i=l
(4/3)13,o,
13.
The experimental approach is comprised of ground-based work to extend prior mixed-
dimensional studies and a flight experiment to study finite coarsening kinetics in three
dimensions.
A. Ground-based experiment
The ground-based experimental study ofdiffusional coarsening will proceed in a liquid-
liquid two-phase system. Ideally, one performs the experiment at an isopycnic point to
maximize reduction of gravity-driven convective disturbances. This would allow
observations over the long times required to investigate diffusional coarsening. A
holographic technique has been instrumental in prior work on this subject and will
continue to be the primary method to study both the influence of local environmental
conditions on individual droplet size histories and measurements of global averages.
Most of the ground-based coarsening data will be collected during two-dimensional
diffusion of three-dimensional droplets attached to a glass surface (mixed-dimensional
case). 4's
Ground-based studies will proceed as follows:
• Prepare droplet distributions for sparse and densely populated clusters for finite
coarsening systems undergoing 2-/) diffusion;
265
Configure a pre-programmed x-y translator with predetermined droplet positions in
tandem with a sharp tool for scoring pinpoint defects onto high-quality, pristine cell
walls. This should induce heterogeneous nucleation at desired positions;
• Fabricate an optimal cell for achieving predictable cluster size and number
distributions, and perform experiments;
• Study temporal correlation of droplet size and number distribution for mixed-
dimensional coarsening.
B. Flight Experiment
The space-based coarsening study requires long-duration, quality microgravity to
prevent phase separation via Stokes settling. In this instance, there is no wall attachment
to prevent sedimentation, so establishing microgravity conditions is an imperative. As the
estimated time of the experiment is almost four months, a Space Station facility will
ultimately be required.
In order to understand the need for quality microgravity, it is helpful to examine the
velocity of a 100gm diameter droplet (the largest size anticipated) at a temperature near
the isopycnic point. The Hadamard-Rybczynski relation provides the velocity of droplets,
2Er'o alal
V_=-
3 Drh + 21"11]
, 14.
where P2-P_ is the mass density difference between the droplet and matrix phases, q_ is
the viscosity of the matrix fluid, and q2 is the viscosity of the droplet phase fluid. The
Hadamard-Rybczynski relation is valid provided that the droplet surface is clean, and the
Reynolds number, Re-=plVsIr/rl_ , is small compared to unity. Droplets are assumed to
remain spherical. Assuming the use of a succinonitrile-water system, and using a
succinonitrile-rich droplet phase with a 0.1K deviation from the isopycnic temperature,
Ap = 3.85 x 10 s g/cm 3. Such a deviation could occur from either experimental
uncertainties in the isopycnic set point and the temperature measurement system, as well
as from cycling or drift caused by the temperature control system. The magnitude of the
velocity of such a droplet at 104g is 1 X 10"4mm/hr. In l-g, the same droplet would fall
1mm in 1hr, and would reach the bottom of the test cell before noticeable ripening had
occurred. The dispersion would become rapidly depleted of larger ripening droplets. The
relative position of the droplets would also vary significantly in time. It would not be
possible to explore local microstructural effects from diffusion, because the droplet
population would be altered more rapidly by sedimentation effects. This and other
factors, such as distortion around the diffusion field, would obscure the kinetics. These
factors impose the imperative for microgravity investigation of three-dimensional
coarsening kinetics and observation of evolution of microstructure.
266
Preparation for microgravity studies will begin by assessing two approaches for droplet
dispersion:
1. Jet break-up in microgravity to control the distribution of droplets in each cluster.
From Tomitika, the Rayleigh stability criterion, q, is expressed as,
q= 2q2( -
15.
where rl_ and rh are viscosities of the fluids; y is the interracial tension; R is radius of
liquid cylinder; and Xd -- 27r_/kd. 2d is the wavelength of disturbance caused by, e.g.,
vibrations or nozzle roughness. For appropriate jet velocity and nozzle wetting
characteristics, the average droplet size in a cluster is approximately twice the nozzle
diameter. For q < 0, disturbances damp out; for q = 0, disturbances remain stable,
resulting in oscillating waves; for q > 0, disturbances become capillary waves which lead
to liquid cylinder breakup [pressure limits drop size].
Table 1. Results from prototype: distribution of butyl-benzoate droplets in
water 7
dE.(g__ <d>(pm) _ <d>/d_
77 149 ±66 1.9
153 284 ±130 1.9
203 348 ±190 1.7
dN is the nozzle diameter; <d> is the average cluster droplet diameter; cy is the standard
deviation within the cluster from average cluster droplet diameter.
2. Site saturation method [Nucleation of Crystals from Solution (NCS)] s for
forming droplet clusters by quench.
Finally, holographic imaging is the optical tool selected for observing these phenomena.
Figure 1 contains a sequence of holograms during the mixed-dimensional coarsening of
succinonitrile-water. Figure 2 represents maps of local diffusion fields derived from
spatial and size distributions of all droplets observed in the corresponding holograms of
Figure 1 using monopole approximations. Clearly, holography provides more data than
does conventional optical imaging techniques. A reconstructed hologram is amenable to
various optical characterization techniques, e.g., in situ microscopy, and interferometry.
An especially important aspect of the experimental study includes determining the extent
of modifications required to allow incorporation of holography within the operating
envelope of the mill±kelvin thermostat (MITH) hardware.
267
References
1. O.M. Todes, J. Phys. Chem (Soy) 20, 629 (1946).
2. I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961).
3. V.E. Fradkov, M.E. Glicksman, S.P. Marsh, Phys. Rev. E. 53(3), 3925 (1996).
4. J.R. Rogers, J.P. Downey, W.K. Witherow, B.R. Facemire, D.O. Frazier, V.E.
Fradkov, S.S. Mani, and M.E. Glicksman, J. Electronic Mat. 23(10), 999 (1994).
5. V.E. Fradkov, S.S. Mani, M.E. Glicksman, J.R. Rogers, J.P. Downey, W.K.
Witherow, B.R. Facemire, and D.O. Frazier, J. Electronic Mat. 23(10), 1007 (1994).
6. J.A. Marqusee and J. Ross, J. Chem. Phys. 80, 536 (1984).
7. J.R. Rogers, Ph.D. Dissertation, University of Colorado, 1992
8. R.L. Kroes, D.A. Reiss, and S.L. Lehoczky, Microgravity Sci. Technol. VIII/1 52-55,
(1995).
268
Holograms of mixed-dimensional coarsening of succiononitrile-water
system in pyrex cells: discrete phase is succinonitrile rich.
Hologram 100 Hologram 129
Hologram 151
Figure 1.
269
Holographic data for mixed-dimensional coarsening, with mapping
of local diffusion fields using monopole approximations. Black
circles show particle sizes, shades of blue indicate areas of solute
depletion, whereas shades of red show enriehment_
Hologram 100 Hologram 129
Hologram 151
Figure 2.
270


Evolution of Local Microstructures (ELMS): Spatial Instabilities of Coarsening

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server