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Small-Angle Neutron Scattering Study of Fumed Silica Powder Compaction
F. Rodriguez-Reinosoet al. (Editors),Characterization of Porous Solids I1 0 1991 Elsevier Science Publishers B.V., Amsterdam
267
SMALL-ANGLE NEUTRON SCATTERING STUDY OF FUMED SILICA POWDER COMPACTION ALAN J. HURD1v2, GREGORY P. JOHNSTONZ, and DOUGLAS M. SMITHZ lSandia National Laboratories, Albuquerque, New Mexico 87185-5800 (USA), 2Center for MicroEngineered Ceramics, University of New Mexico, Albuquerque, New Mexico 87131 (USA) ABSTRACT In a previous study of fumed silica by mercury porosimetry (ref. l), we established an inverse dependence of the powder compressibility n on applied pressure P, heralded by power-law differential volume vs pressure curves. Independent small-angle neutron scattering (SANS) measurements, undertaken to establish microstructure, indicated a decreasing zero-angle intensity, I(q+O), with increasing sample compression. Preliminary analysis suggested that I(q+O) is proportional to the compressibility n . Since it is well known that I(q-+O) is proportional to n for systems in thermodynamic equilibrium, our results implied an analogous relation for systems in mechanical equilibrium. However, the preliminary study was incomplete, lacking scattering data for a wide range of compactions. In this paper, we have tested this relationship, in the low pressure regime, by correlating the scattered neutron intensity from samples in various degrees of compaction. We find that, while the intensity does decrease with increasing compaction (after removing the effects due merely to greater sample density) as expected according to the compressibility analogy, the dependence is not the same as that found by mercury porosimetry. Lightly compressed fumed silica has been studied as a percolating system by several groups (refs. 2-4); our results pertain to states of greater compaction. INTRODUCTION Previous studies (refs. 1.2) of fumed silica compaction have established an empirical "equation of state" between powder pore volume and applied pressure,
The compressibility, which is defined as
&
P
-
1 -
dV ~
V
dP
1
dV
V
dP
= -A
(2)
is therefore inversely proportional to pressure. Here V, the total volume of the powder, is comprised of compressible pore volume V, silica volume V,:
V=V,,+V,.
and incompressible
The scattered intensity at zero angle I(q+O)
might be expected to be proportional to
n
by analogy to thermalized systems.
In fact, preliminary scattering studies on compacted fumed silica powders demonstrated that I(0) decreases with increasing compression. we attempt to test the relationship
In this paper
268
which, if true, would imply a straightforward statistical mechanics for powder behavior. Thus, it should be possible to understand the empirical equation of state in Eq. (1) on a statistical basis.
Our study was motivated by interest
in the pore space surrounding the fractal powders as they are forced together. To illustrate what we mean by a statistical understanding of the state of the powder, we refer the reader to Edwards and Oakeshott (ref. 5).
If the
particles are sufficiently numerous and the local construction rules welldefined, then the macroscopic properties of the powder should be predictable and interesting.
The essential idea is that all configurations consistent
with mechanical stability are equally probable, but that for the overwhelming majority of these states the measurable properties are essentially the same. Thus, we should be able to predict, for example, the volume of a heap of sand. Unfortunately, it is necessary to understand the role of energy in our powder system in order to develop a calculus for the pressure-volume equation of state and the suppression of density fluctuations; we hope to provide these insights with future experiments.
The present study is a first step toward
that goal. Fumed silica is "glass soot" made by burning silicates in flames.
It is
known to be mass-fractal particles, i.e. submicron-sizeaggregates composed of random, weakly branched strings of 100 A silica spheres, with a great deal of internal porosity. Since the limiting small-angle intensity can be related quite generally to the fluctuations in the sample, I(0)
a
2
4 N > = <(N
- UC-)2>
(4)
the relationship being tested here is the proportionality
I
v
'a
? ci
2
<6N >
ap
(5)
The probability W(N,V,A,. . . ) of having N particles in a given total volume V must be sharply peaked at for large N.
W depends on some number of
extensive parameters A , . . . , which remain unspecified in our discussion of powders; for now we assume that a single function A suffices to encapsulate
269 our ignorance.
By expanding W in the neighborhood of its peak at N 4 > , we
can express the width, or fluctuation in N, in terms of the curvature in A.
W(N,V,A,. . . ) = W()
+
1 2 a2A (N-) - I 2 aN2 N-+>+
' '
(By stability arguments, the linear term in the expansion must be zero and the sign of the A-curvature must be negative.)
Thus, the Gaussian width-squared
is 2/(a2A/aN2) and is equal to 2<6N2>. But adding particles to a constant volume is equivalent to compressing a sample of constant mass; hence, it is actually the variation in A with volume that matters. -(aA/aV),
In thermodynamic systems, A is the (Helmholtz) free energy and
is the pressure, so
a2A
2
----..- 2 aN2 <6N >
v2 N~
ap -
av
(thermodynamic systems). (7)
Thus, in thermodynamic systems, Eq. ( 5 )
follows directly from the key
relation between free energy and pressure.
For powder compaction, it must
again be an energy argument, but we have yet to identify the fate of the mechanical energy put into the system. EXPERIMENT Eight samples of Cab-0-Sil (grade EH-5, Cabot Corporation) fumed silica were prepared in closed aluminum cells with 3 mm path length.
The cell was an
aluminum cylinder, 1 . 2 6 cm inside diameter, with 1 mm thick aluminum windows pressed into each end.
Densities ranged from 0,038 g/cc to 1 . 2 g/cc; the
loose powder density (see below) is even less than that of sn aerogel. Small-angle neutron scattering was performed at the Manuel Lujan Los Alamos Neutron Scattering Center (LANSCE). 0.05
The useful wave vector range was
< q < 0.16 A-1 after correction for scattering from an empty cell. The
high sample densities (1.2 g/cc) were done at the Missouri University Research Reactor (MUM). ANALYSIS Figure 1 shows the measured sample density as a function of l/a, where a is a compression factor defined by a-V/V, with V the compressed sample volume and V, the volume o f loose powder that was compacted. By least-squares fitting to p=po/a, we found po=0.0296 g/cc (loose powder density).
270
Q
9
0
2.0
0.0
4.0
1 /.
80
60
Figure 1.- Densities of compressed fumed silica samples. In the very low pressure regime, we noticed that the scattering curves all had the same form but different amplitudes.
Each curve was divided by
p
and
by sample thickness to correct for the effects of scattering mass; this brought the very low pressure curves into coincidence, proving that very little structural difference existed between these samples; the higher density samples did, however, exhibit deviations at low q as seen in Figure 2 .
We
compared the total scattered intensity between curves by dividing each curve by a "reference curve" f(q)
in order to leave only amplitude information.
f(q) was formed by simply averaging the six lowest density curves and is shown in Figure 2 with two other representative curves. Finally, the lowest-q intensity datum from each normalized curve was taken as our approximate I(0)
.
(rather than attempt a dangerous extrapolation to
0) and plotted in Figure 3 . The abscissae are VP., where V, is the pore q volume (calculated from the density) and V, is the silica volume (calculated from the mass). For low densities (V,/V, > lo), the intensity was found to be constant when normalized in the above manner indicating no significant interparticle interference accessible to the
SANS.
At higher densities,
however, I ( 0 ) was found to drop as particles packed closer together and scattered more coherently.
271
0.001
0.01
0.1
9 Figure 2.- Scattering curves of compressed fumed silica. scattering curve f(q) p-0.306
for very low density
g/cc; (c) p=1.00 g/cc.
samples p C 0 . 0 9
(a) Average g/cc;
The denser samples scatter less intensity at
small angles.
-
0
1
(b)
10
100
VdVS
Figure 3.- Approximate zero-angle intensity vs. normalized pore volume.
272 [We note that the large-q data in the raw scattering curves do not quite approach a Porod asymptote (slope of -4), as noted previously (ref. 6), indicating a somewhat rough primary particle surface.
Since the curves
coincide at large q (after correcting for scattering mass),
the interfacial
area is unchanged with compaction except at the highest densities, when it decreases.
We infer that chains in the aggregate do not break to form new
surface area (or, if they do, an equal amount of surface area is annihilated simultaneously).] Using n
a
1/P from Eqs. (1) and (2). we would expect
I(O)
a
P-'
a
v3
P
Instead we observe (for V,/V,
a
vY P
v
=
0.75 f 0.25
(9)
We conclude that Eq. ( 3 ) does not hold for our powder samples. The failure of Eq. (3) does not mean that it has no ergodic analog for fumed silica powder, but merely that the large scale fluctuations measured by Eq. (4) do not control the compressibility in the same way as in thermal systems. ACKNOWLEDGMENTS The scattering experiments were done at LANSCE (Los Alamos Neutron Scattering Center) and at MURR (Missouri University Research Reactor).
We are
indebted to Phil Seeger (LANSCE) and David Mildner (MURR) for their expert help, to Peter Pfeifer for helpful discussions about pores, and to Don Stuart for technical assistance. This work was supported by Sandia National Laboratories under DOE Contract DE-AC04-76-DP00789and the UNM/NSF Center for MicroEngineered Ceramics. REFERENCES
1. 2.
3.
4. 5. 6.
D. M. Smith, G . P. Johnston, and A . J. Hurd, J . Colloid Interface Sci., in press. F. Ehrburger and J. Lahaye, J. Phys. France 5 0 , (1989) 1349. J. Forsman, J. P. Harrison, and A. Rutenberg, Can. J. Phys. 65, (1987) 767. J. M. Heintz, F. Weill, and J. C. Bernier, Mater. Sci. and Engin. m, (1989) 271. S. F. Edwards and R.B.S. Oakeshott, Physica D 3 8 , (1989) 8 8 . A. J. Hurd, D. W. Schaefer, D. M. Smith, S . B. Ross, A. Le Mehaute, and S . Spooner, Phys. Rev. B 2 (1989) 9742.
267
SMALL-ANGLE NEUTRON SCATTERING STUDY OF FUMED SILICA POWDER COMPACTION ALAN J. HURD1v2, GREGORY P. JOHNSTONZ, and DOUGLAS M. SMITHZ lSandia National Laboratories, Albuquerque, New Mexico 87185-5800 (USA), 2Center for MicroEngineered Ceramics, University of New Mexico, Albuquerque, New Mexico 87131 (USA) ABSTRACT In a previous study of fumed silica by mercury porosimetry (ref. l), we established an inverse dependence of the powder compressibility n on applied pressure P, heralded by power-law differential volume vs pressure curves. Independent small-angle neutron scattering (SANS) measurements, undertaken to establish microstructure, indicated a decreasing zero-angle intensity, I(q+O), with increasing sample compression. Preliminary analysis suggested that I(q+O) is proportional to the compressibility n . Since it is well known that I(q-+O) is proportional to n for systems in thermodynamic equilibrium, our results implied an analogous relation for systems in mechanical equilibrium. However, the preliminary study was incomplete, lacking scattering data for a wide range of compactions. In this paper, we have tested this relationship, in the low pressure regime, by correlating the scattered neutron intensity from samples in various degrees of compaction. We find that, while the intensity does decrease with increasing compaction (after removing the effects due merely to greater sample density) as expected according to the compressibility analogy, the dependence is not the same as that found by mercury porosimetry. Lightly compressed fumed silica has been studied as a percolating system by several groups (refs. 2-4); our results pertain to states of greater compaction. INTRODUCTION Previous studies (refs. 1.2) of fumed silica compaction have established an empirical "equation of state" between powder pore volume and applied pressure,
The compressibility, which is defined as
&
P
-
1 -
dV ~
V
dP
1
dV
V
dP
= -A
(2)
is therefore inversely proportional to pressure. Here V, the total volume of the powder, is comprised of compressible pore volume V, silica volume V,:
V=V,,+V,.
and incompressible
The scattered intensity at zero angle I(q+O)
might be expected to be proportional to
n
by analogy to thermalized systems.
In fact, preliminary scattering studies on compacted fumed silica powders demonstrated that I(0) decreases with increasing compression. we attempt to test the relationship
In this paper
268
which, if true, would imply a straightforward statistical mechanics for powder behavior. Thus, it should be possible to understand the empirical equation of state in Eq. (1) on a statistical basis.
Our study was motivated by interest
in the pore space surrounding the fractal powders as they are forced together. To illustrate what we mean by a statistical understanding of the state of the powder, we refer the reader to Edwards and Oakeshott (ref. 5).
If the
particles are sufficiently numerous and the local construction rules welldefined, then the macroscopic properties of the powder should be predictable and interesting.
The essential idea is that all configurations consistent
with mechanical stability are equally probable, but that for the overwhelming majority of these states the measurable properties are essentially the same. Thus, we should be able to predict, for example, the volume of a heap of sand. Unfortunately, it is necessary to understand the role of energy in our powder system in order to develop a calculus for the pressure-volume equation of state and the suppression of density fluctuations; we hope to provide these insights with future experiments.
The present study is a first step toward
that goal. Fumed silica is "glass soot" made by burning silicates in flames.
It is
known to be mass-fractal particles, i.e. submicron-sizeaggregates composed of random, weakly branched strings of 100 A silica spheres, with a great deal of internal porosity. Since the limiting small-angle intensity can be related quite generally to the fluctuations in the sample, I(0)
a
2
4 N > = <(N
- UC-)2>
(4)
the relationship being tested here is the proportionality
I
v
'a
? ci
2
<6N >
ap
(5)
The probability W(N,V,A,. . . ) of having N particles in a given total volume V must be sharply peaked at
W depends on some number of
extensive parameters A , . . . , which remain unspecified in our discussion of powders; for now we assume that a single function A suffices to encapsulate
269 our ignorance.
By expanding W in the neighborhood of its peak at N 4 > , we
can express the width, or fluctuation in N, in terms of the curvature in A.
W(N,V,A,. . . ) = W(
+
1 2 a2A (N-
' '
(By stability arguments, the linear term in the expansion must be zero and the sign of the A-curvature must be negative.)
Thus, the Gaussian width-squared
is 2/(a2A/aN2) and is equal to 2<6N2>. But adding particles to a constant volume is equivalent to compressing a sample of constant mass; hence, it is actually the variation in A with volume that matters. -(aA/aV),
In thermodynamic systems, A is the (Helmholtz) free energy and
is the pressure, so
a2A
2
----..- 2 aN2 <6N >
v2 N~
ap -
av
(thermodynamic systems). (7)
Thus, in thermodynamic systems, Eq. ( 5 )
follows directly from the key
relation between free energy and pressure.
For powder compaction, it must
again be an energy argument, but we have yet to identify the fate of the mechanical energy put into the system. EXPERIMENT Eight samples of Cab-0-Sil (grade EH-5, Cabot Corporation) fumed silica were prepared in closed aluminum cells with 3 mm path length.
The cell was an
aluminum cylinder, 1 . 2 6 cm inside diameter, with 1 mm thick aluminum windows pressed into each end.
Densities ranged from 0,038 g/cc to 1 . 2 g/cc; the
loose powder density (see below) is even less than that of sn aerogel. Small-angle neutron scattering was performed at the Manuel Lujan Los Alamos Neutron Scattering Center (LANSCE). 0.05
The useful wave vector range was
< q < 0.16 A-1 after correction for scattering from an empty cell. The
high sample densities (1.2 g/cc) were done at the Missouri University Research Reactor (MUM). ANALYSIS Figure 1 shows the measured sample density as a function of l/a, where a is a compression factor defined by a-V/V, with V the compressed sample volume and V, the volume o f loose powder that was compacted. By least-squares fitting to p=po/a, we found po=0.0296 g/cc (loose powder density).
270
Q
9
0
2.0
0.0
4.0
1 /.
80
60
Figure 1.- Densities of compressed fumed silica samples. In the very low pressure regime, we noticed that the scattering curves all had the same form but different amplitudes.
Each curve was divided by
p
and
by sample thickness to correct for the effects of scattering mass; this brought the very low pressure curves into coincidence, proving that very little structural difference existed between these samples; the higher density samples did, however, exhibit deviations at low q as seen in Figure 2 .
We
compared the total scattered intensity between curves by dividing each curve by a "reference curve" f(q)
in order to leave only amplitude information.
f(q) was formed by simply averaging the six lowest density curves and is shown in Figure 2 with two other representative curves. Finally, the lowest-q intensity datum from each normalized curve was taken as our approximate I(0)
.
(rather than attempt a dangerous extrapolation to
0) and plotted in Figure 3 . The abscissae are VP., where V, is the pore q volume (calculated from the density) and V, is the silica volume (calculated from the mass). For low densities (V,/V, > lo), the intensity was found to be constant when normalized in the above manner indicating no significant interparticle interference accessible to the
SANS.
At higher densities,
however, I ( 0 ) was found to drop as particles packed closer together and scattered more coherently.
271
0.001
0.01
0.1
9 Figure 2.- Scattering curves of compressed fumed silica. scattering curve f(q) p-0.306
for very low density
g/cc; (c) p=1.00 g/cc.
samples p C 0 . 0 9
(a) Average g/cc;
The denser samples scatter less intensity at
small angles.
-
0
1
(b)
10
100
VdVS
Figure 3.- Approximate zero-angle intensity vs. normalized pore volume.
272 [We note that the large-q data in the raw scattering curves do not quite approach a Porod asymptote (slope of -4), as noted previously (ref. 6), indicating a somewhat rough primary particle surface.
Since the curves
coincide at large q (after correcting for scattering mass),
the interfacial
area is unchanged with compaction except at the highest densities, when it decreases.
We infer that chains in the aggregate do not break to form new
surface area (or, if they do, an equal amount of surface area is annihilated simultaneously).] Using n
a
1/P from Eqs. (1) and (2). we would expect
I(O)
a
P-'
a
v3
P
Instead we observe (for V,/V,
a
vY P
v
=
0.75 f 0.25
(9)
We conclude that Eq. ( 3 ) does not hold for our powder samples. The failure of Eq. (3) does not mean that it has no ergodic analog for fumed silica powder, but merely that the large scale fluctuations measured by Eq. (4) do not control the compressibility in the same way as in thermal systems. ACKNOWLEDGMENTS The scattering experiments were done at LANSCE (Los Alamos Neutron Scattering Center) and at MURR (Missouri University Research Reactor).
We are
indebted to Phil Seeger (LANSCE) and David Mildner (MURR) for their expert help, to Peter Pfeifer for helpful discussions about pores, and to Don Stuart for technical assistance. This work was supported by Sandia National Laboratories under DOE Contract DE-AC04-76-DP00789and the UNM/NSF Center for MicroEngineered Ceramics. REFERENCES
1. 2.
3.
4. 5. 6.
D. M. Smith, G . P. Johnston, and A . J. Hurd, J . Colloid Interface Sci., in press. F. Ehrburger and J. Lahaye, J. Phys. France 5 0 , (1989) 1349. J. Forsman, J. P. Harrison, and A. Rutenberg, Can. J. Phys. 65, (1987) 767. J. M. Heintz, F. Weill, and J. C. Bernier, Mater. Sci. and Engin. m, (1989) 271. S. F. Edwards and R.B.S. Oakeshott, Physica D 3 8 , (1989) 8 8 . A. J. Hurd, D. W. Schaefer, D. M. Smith, S . B. Ross, A. Le Mehaute, and S . Spooner, Phys. Rev. B 2 (1989) 9742.