Capillary condensation, invasion percolation, hysteresis, and discrete memory
R. A. Guyer
Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts 01003
K. R. McCall
Department of Physics, University of Nevada, Reno, Nevada 89557
Received 18 December 1995
A model of the capillary condensation process, i.e., of adsorption-desorption isotherms, having only pore-
pore interactions is constructed. The model yields 1 hysteretic isotherms, 2 invasion percolation on desorp-
tion, and 3 hysteresis with discrete memory for interior chemical potential loops. All of these features are
seen in experiment. The model is compared to a model with no pore-pore interactions the Preisach model and
to a related model of interacting pore systems the random field Ising model. The capillary condensation
model differs from both. S0163-18299606725-2
I. INTRODUCTION
Hysteresis is often seen in a measured quantity’s response
to a probe. For example, the displacement u ( t ) response of
an oscillator to a periodic driving force F ( t ) is hysteretic. In
this case the hysteresis in u versus F is due to a damping
induced phase difference between u ( t ) and F ( t ). Hysteresis
with discrete memory H/DM is less familiar although it is
seen in a wide variety of physical and model systems.
1–5
Discrete memory or end point memory refers to the memory
and memory erasure that occur at the extrema of the probe
protocol. In magnetism the probe is the external field; in
capillary condensation the probe is the chemical potential. A
very restricted set of hysteresis models also have discrete
memory. An elegantly simple explanation of H/DM is given
by the Preisach model,
6–8
a model for the behavior of a large
number of noninteracting, individually hysteretic units.
In many instances where the Preisach model is used the
basic condition for its application, noninteraction or indepen-
dence, is absent. For example, the Preisach model has been
extensively used to describe the capillary condensation pro-
cess, i.e., adsorption-desorption isotherms.
9
However, the in-
vasion percolation feature seen in many isotherms
4
provides
compelling evidence for pore-pore interactions and suggests
that a Preisach model has circumscribed validity.
10,11
More sophisticated models have been developed for
adsorption-desorption isotherms
12,13
and for systems that
possess H/DM.
14
The purpose of this note is to describe a
capillary condensation model CCM that, in marked con-
trast to the Preisach model, owes all of its behavior to the
interaction of individually nonhysteretic units, and to show
that this model possesses H/DM. The CCM possesses the
essential features necessary to give the desired qualitative
behavior of adsorption-desorption isotherms although it has
been stripped of much of the detail that would make it a
practical model for inverting data to learn the pore space
geometry. In Sec. II the capillary condensation model is de-
scribed. In Sec. III results of numerical simulation of the
CCM are presented. The essential content of the CCM and
its relationship to other models, e.g., the random field Ising
model, are discussed in Sec. IV.
II. MODEL
The capillary condensation process is characterized by the
geometry of the pore space, the rules for the evolution of
fluid configurations in the pore space, and by the chemical
potential protocol to which the fluid is subjected. For defi-
niteness and simplicity we adopt the following model.
a The pores, cylinders of length b , are the bonds of a
square lattice L L . There are N =2 ( L / b )
2
pores.
b The pores are in the presence of a fluid at chemical
potential , i.e., the pore space is connected to a chemical
potential reservoir.
c A pore is either empty, in state =0, or filled with
fluid, in state =1. A pore in state 0 has a thin liquid film on
its surface and vapor in its interior. A pore in state 1 is filled
with liquid and may have a meniscus at its ends depending
on the state of its near neighbors the six pores with which it
shares a node.
d To the i th pore we assign a radius r
i
from the prob-
ability density ( r ) and a critical value of the chemical po-
tential
i
from the probability density ( ). There may be a
correspondence between
i
and r
i
so that
r dr =
d . 1
To avoid details that depend on ( r ) and ( ) but are ines-
sential to our discussion we map the spectrum of critical pore
chemical potentials onto the interval 0,1 by ordering
1
,...,
N
from smallest to largest and using p
i
= ordinal
number of
i
)/ N . We map the reservoir chemical potential
to the appropriate point in the sequence p
1
,..., p
N
and con-
tinue to denote it as .
e The fluid in the system is carried through a chemical
potential protocol ( t ). Here t simply indicates that the
chemical potential evolves in time. This time evolution oc-
curs so slowly that the fluid configurations in the pore space
are taken to be a sequence of stable or metastable fluid con-
figurations C ( t ) =C ( t
m
) =C
m
consistent with the chemical
potential protocol, ( t ) = ( t
m
) =
m
.
f On chemical potential increase
m+1
m
. If
m
p
i
m+1
, then
i
=0 →1.
PHYSICAL REVIEW B 1 JULY 1996-I VOLUME 54, NUMBER 1
54 0163-1829/96/541/184/$10.00 18 © 1996 The American Physical Society