Inhibited Nucleation and Growth Reactions
J. Phys. Chem. B, Vol. 108, No. 50, 2004 19435
where x is the fraction remaining, n is a reaction order, m is
more correctly defined as a nucleation-growth parameter, and
q is an initiation parameter. When far from equilibrium and
when m ) 0 and n ) 1, this reduces to a first-order reaction.
The conventional Prout-Tompkins model has n ) m ) 1. The
conventional Prout-Tompkins model can be integrated analyti-
cally only for constant temperature. The parameter q is related
to the initial condition (integration constant) that enables the
reaction to get started, since dx/dt is identically 0 at x ) 1 for
the Prout-Tompkins model as normally written. Equation 15
is included in the most recent version of Lawrence Livermore
National Laboratory (LLNL) kinetics analysis program (Kinet-
ics05).4 Equation 15 is numerically integrated over the relevant
experimental conditions, typically (but not restricted to) iso-
thermal and constant heating, and multiple data sets are fitted
simultaneously by nonlinear regression. An analogous equation
could be written for the Avrami-Erofeyev approach by replac-
ing (1 - qx)m by (-ln (qx)p), where p is the growth dimen-
sionality in the Avrami formalism.15
Defense Corporation (HOL 81H030-033) for LLNL using the
Bachmann synthesis process and was determined to be >99.90%
pure as analyzed by HPLC for RDX impurities. Particle-size
analysis indicated that >90% of the material was between 30
and 500 µm in diameter. Other DSC experiments used crystal
fragments derived from a large single crystal of â-HMX grown
by H. H. Cady and provided to LLNL by the University of
Delaware. Fragments averaging 1-mm diameter were used. A
few experiments used the HMX formulation PBX 9501, a blend
of 95% HMX, 2.5% Estane, and 2.5% BDNPA-F. For the
optical studies, pure â-HMX crystals were prepared by the
method of Siele et al.22 Octahydro-1,5-diacetyl-3,7-dinitro-
1,3,5,7-tetrazocine (DADN) was treated with 100% HNO3 and
P2O5 at 50°C for 50 min, followed by quenching in ice water.
Slow recrystallization from acetone yielded HMX as colorless
microcrystals.
B. Reaction Measurements. Optical movies were recorded
to help understand earlier AFM experiments.11 A Leica optical
microscope (total magnification 800×) was used for the dynamic
movies to determine the velocity of the phase transition within
individual crystals. The size of each crystal was determined with
a calibrated reticle. Movies were recorded in real time using a
color CCD camera and a standard VCR. Sample heating was
accomplished using a Veeco temperature controller. The samples
were heated from ambient temperature to 300 °C with a ramp
rate of 20 °C/min. The resolution of the heater stage is 0.2 °C,
and the small size of both the sample and the heating stage
ensures uniform heating over the entire crystal. Movies are
available as Supporting Information.
Now consider that eq 15 is not reversible for solid-state
reactions in the way it would be for homogeneous reactions.
One could write a net reaction rate having nucleation-growth
rate laws in both directions.
dx/dt ) -kfxn(1 - x)m - kr(1 - x)nxm
(16)
In this case, x indicates a mass fraction of component A, so the
exponents are switched in the forward and reverse terms. For
simplicity, the q factor is not shown explicitly. Note that in the
limit of n ) m, one can factor out the k’s to form an equation
equivalent to eq 15, since Keq ) kf/kr. However, the physical
rationale for the form of the Prout-Tompkins model is not valid
for complete reversibility along the reaction coordinate.
As pointed out by Avrami2 more than 60 years ago, expanding
growth regions consume unnucleated defects and eventually
coalesce with neighboring growth regions. While the reaction
interface could be reversed over some infinitesimal distance, it
can neither retrace its steps once coalescence has occurred nor
reconstruct the original defect distribution. Consequently, we
consider eq 15 to be unidirectional. Once the reaction is
complete, one could use the same equation in the reverse
direction, albeit with a different defect energy distribution
characteristic of the product material, but eq 15 is not a valid
way to treat extensive reversibility during the course of the
reaction.
Equations 15 and 16 have both similarities and differences
from that of Henson et al.12 They construct a kinetic equation
with both forward and reverse first-order reactions more along
the lines of eq 16. Their additional first-order term relates more
explicitly to nucleation than the traditional Prout-Tompkins
model, which we modify in a different manner. We previously
showed20 that the first-order term in the nucleation-growth
formalism of Nam and Seferis21 is equivalent to our q factor if
the activation energies are the same for the first- and second-
order processes. In addition, a nucleation-growth reaction is not
reversible to any substantial extent along the same reaction
pathway, as stated earlier. Consequently, we consider eq 15 a
better approach for incorporating thermodynamic inhibition
phenomenologically than either eq 16 or the equations of Henson
et al.12
A differential scanning calorimeter (DSC), TA Instrument
model 2920, and its associated software, Universal Analysis,
were used for additional analyses. All data were collected at
0.2 s-1. DSC23 measures the difference in the heat flow between
a sample and an inert reference as a function of time, where
both the sample and the reference are subjected to a controlled
temperature-pressure environment during that time. The instru-
ment design used here is commonly called the heat flux design.
Indium, tin, lead, and zinc from TA Instruments were used to
calibrate the temperature and enthalpy response of the instrument
at a heating rate of 10 °C/min. Onset temperatures at other
heating rates were corrected using measurements of indium and
tin melting points at 0.5, 5, 25, and 100 °C/min.
All samples were weighed in a Sartorius MC 5 Electronic
balance accurate to e0.005 mg. In all cases, the pan with sample
was matched to a reference pan within 100 µg to balance heat
flows due to heat capacity.
5. Calibration of the Detailed Model for HMX. The
nucleation and growth aspects of the detailed model are
calibrated separately. We start with general observations,
followed by a calibration of the growth kinetic parameters,
because they are simpler to extract. Nucleation kinetic param-
eters, which are more difficult to extract, are then estimated.
The reaction rate of the phase transition is assumed to be
proportional to heat flow. Representative calorimetry traces are
shown in Figure 2 for 0.4-mg samples of HMX batch B-844.
At slow heating rates for small samples, one can discern
individual particles undergoing the phase transition. At higher
heating rates, the individual grain resolution is lost, because
the lower activation energy for growth than for nucleation makes
the individual peaks broader in temperature. The distribution
of nucleation times reflects the probabilistic nature of nucleation.
The envelope outlined by many crystals reflects both the basic
rate law (e.g., first-order) for the nucleation process and any
distribution effects related to particle size, defect energy, or both.
4. Experimental Methods
A. Samples. Three sources of pure â-HMX were used in this
study. One material (batch B-844) was manufactured by Holston