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1M chlHC/DIODIanS-p.“

ENVIRONMENTAL PERSPECTIVES: NEAR CRITICAL ADSORPTION IN
MODEL POROUS CARBONS AND CHEMOTACTIC TRANSPORT OF
PSEUDOMONAS SP. STRAIN KC

By

Caroline Jeanne Roush

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1 999

ABSTRACT

ENVIRONMENTAL PERSPECTIVES: NEAR CRITICAL ABSORPTION IN MODEL
POROUS CARBONS AND CHEMOTACTIC TRANSPORT OF PSEUDOMONAS SP.
STRAIN KC

By

Caroline Roush

Remediation technologies that involve the decontamination of soil via supercritical
oxidation require extensive knowledge the interaction between solid and vapor-liquid
equilibrium. This study examines the behavior of a carbon-ethylene system in a slit-pore
environment at near-critical and supercritical temperatures. The purpose of the study was
to compare calculations of the Elliot Suresh Donohue (ESD) simplified local density
(SLD) model to those of a Monte Carlo (MC) computer algorithm to investigate the
strengths and weaknesses of the SLD approach. The SLD approach was used to predict
the adsorption isotherm using energy profiles, and these profiles were in turn compared to
those generated by the MC method.

Chemotaxis refers to 'the biased migration of cells in the direction of a chemical
concentration gradient. Motile chemotactic bacteria can be characterized through two
transport coefficients: the random motility coefficient and the chemotactic sensitivity
coefficient. The focus of this study was to calculate these coefficients for acetate and

nitrate in homogeneous solution and to then investigate the impact of porous media.

ACKNOWLEDGEMENTS

I express my deepest appreciation to Dr. Christian Lastoskie, my graduate advisor. Upon
my arrival at graduate school, I could barely master the fine art of composing e-mail, but
through his help and guidance I no longer consider UNIX to be a four-letter word. He is
highly intelligent, and able to grasp obtuse and highly inarticulate subject material and
explain it in such a way that even those whom are not immersed in the field are able to

comprehend on an intermediate level.

I wish to thank Dr. Carl Lira and Dr. Mark Worden for their contributions and patience
with me. I would not have been able to complete my work here successfully without

their conciSe assessment of my day-to-day work and their pearls of wisdom to help guide

both my fruitful and frustrated efforts.

I also wish to thank Bob, my fellow classmate, neighbor, and lab partner, who has helped
me in so many ways. You are an excellent listener and teacher, and I have fond

memories of those lazy afternoons reading in the soothing green interior of 3258.

I thank my parents for always supporting me, no matter what I do. Their only desire is
for me to be happy, and even if that meant that I were to not utilize all of my potential,
they would still stand behind me. That is the most important kind of support and love

that you can give to a child.

TABLE OF CONTENTS

Page
LIST OF TABLES ..................................................................... v
LIST OF FIGURES .................................................................... vi
LIST OF SYMBOLS .................................................................. iv
CHAPTER 1
1.1 Introduction to Equations of State ....................................... 1
1.2 Computer Modeling ....................................................... 7
1.3 The Simplified Local Density Model (SLD) .......................... 15
1.4 The ESD Equation of State .............................................. 19
1.5 Present Study .............................................................. 22
1.5.1 L=9.1943 ...................................................... 26
1.5.2 L = 4.5 ........................................................... 30
1.5.3 L = 2.37 ......................................................... 34
1.5.4 L = 1.0 ........................................................... 34
1.6 Conclusrons41
CHAPTER 2
2.1 Introduction ............................................................... 43
2.1.1 The Random Motility Coefficient (u) and the
Chemotactic Sensitivity Coefficient(Xo) ........................... 53
2.2 Materials and Methods ................................................... 55
2.2.1 Preparation of Media and Grth Conditions ............. 56
2.2.2 Photography and Image Anaylysis .......................... 60
2.3 Mathematical Model ...................................................... 60
2.3.1 Cell Balance .................................................... 60
2.3.2 Nutrient Balance ................................................ 65
2.3.3 Chemoattractant Balance ...................................... 65
2.3.4 Boundary Conditions .......................................... 65
2.3.5 Initial Conditions ............................................... 66
2.3.6 Computer Simulations ......................................... 67
2.3.7 System Parameters ............................................. 68
2.4 Results and Discussion ................................................... 7O
BIBLIOGRAPHY ........................................................................ 79

iv

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

LIST OF TABLES

Page
Parameters used in calculations for BSD-SLD and MC ................ 23
Common parameters used in calculations for the two models... ......24
Equilibrium bulk pressures calculated by SLD for each reduced
density ......................................................................... 26
Parameter values used in mathematical model ........................... 69
Values found for X05 and XOQ using mathematical model .............. 73

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 6A.

Figure 7.

Figure 7A.

LIST OF FIGURES

Page
(A) A P-T diagram for a one-component system. At the
intersection of the three curves, all three phases are in
equilibrium. This point is known as the triple point.
(B) Saturation line of a one-component system which
separates the two phase region. The region under the
dome shaped curve is the two phase region; liquid and
vapor. When the pressure and temperature are raised
above the critical point, the phenomenon of boiling does
not exist and the phase is referred to as a supercritical
fluid ....................................................................... 2

Representation of L=3 (pore size of 3 ethylene molecules)
and zlen = 3.8057 (see Equation 13) ............................... 14

MC simulation of liquid-vapor equilibrium in a carbon slit
pore of width 19 Angstroms at a low density ...................... 16

MC simulation of liquid-vapor equilibrium in a carbon slit
pore of width 19 Angstroms at a high density .................... 17

Ethylene adsorption on BPL carbon as measured by Reich, et.
al., (1980) and modeled by the BSD-SLD method using efs =
93 K and a slit width of 13.4 Angstroms ........................... 20

MC simulation results (top) and SLD (bottom) for slit width
of L=9.1943 at a subcritical temperature ........................... 27

Density profiles with MC simulation results (top) and SLD
(bottom) for slit width of L = 9.1943 at a subcritical
temperature ............................................................. 28

MC simulation results (top) and SLD (bottom) for slit width
of L=9. 1943 at a supercritical temperature ......................... 29

Density profiles with MC simulation results (top) and SLD (bottom) for
slit width of L = 9.1943 at a subcritical temperature .............. 31

vi

Figure 8.

Figure 9.

Figure 10.

Figure 10A.

Figure 11.

Figure 11A.

Figure 12.

Figure 13.

Figure 14.

Figure 15.

Figure 16.

MC simulation results (top) and SLD (bottom) for slit width
of L=4.5 at a subcritical temperature ................................ 32

MC simulation results (top) and SLD (bottom) for slit width
of L=4.5 at a supercritical temperature .............................. 33

MC simulation results (top) and SLD (bottom) for a slit width
of L=2.37 at a subcritical temperature .............................. 35

Density profiles with MC simulation results (top) and SLD
(bottom) for slit width of L = 2.37 at a subcritical
temperature .............................................................. 36

MC simulation results (top) and SLD (bottom) for slit width
of L=2.37 at a supercritical temperature ............................. 37

Density profiles with MC simulation results (top) and SLD
(bottom) for slit width of L = 2.37 at a supercritical
temperature .............................................................. 38

MC simulation results (top) and SLD (bottom) for slit width
of L=1 at a subcritical temperature ................................... 39

MC simulation results (top) and SLD (bottom) for slit width
of L=1 at a supercritical temperature ................................. 40

Experimental photos of motile chemotactic bacteria.

Evidence of a chemotactic ring is exhibited by the circular

rings of higher cell density. Bottom figure displays evidence

of two chemotactic rings, which may have formed from each

of the two chemoattractants present ................................. 46

Depiction of single cell with flagella. The flagella are used

for motility purposes. The direction that the flagella rotate
dictates which course of movement the cell will take. If the
flagella rotate in a clockwise manner, then they form a small
bundle and the cell swims in a smooth manner (a). If the

flagella rotate counter-clockwise, the flagellar bundle unravels
and the cell tumbles in a new direction (b). [Figure adapted

from Macnab, 1987] ................................................... 47

Observed single-cell behavior in an isotropic medium (left)
resembles a random walk with basal mean run length <To>
and swimming speed v. In the presence of an attractant
gradient (right) run lengths are increased when moving

vii

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

toward increasing concentrations of attractant, yielding a
mean run length <r>greater than <To>. [Figure adapted from

Ford, 1992] ............................................................. 50
Physical representation of motility plate for porous media

study .................................................... ‘ ................. 58
Top view of porous motility plate studies .......................... 59

Typical nutrient profile. In this instance, the nutrient is
Acetate .................................................................. 63

Simulation compared to experiment for 0.1 g/L Acetate
and 0.42 g/L Nitrate ................................................... 71

Simulation compared to experiment for 0.5 g/L Acetate
and 2.5 g/L Nitrate .................................................... 72

Simulation compared to experiment for 0.1 g/L Acetate
and 0.083 g/L Nitrate ................................................ 74

Simulation compared to experiment for 0.5 g/L Acetate
and 0.42 g/L Nitrate .................................................. 74

Simulation compared to experiment for 0.5 g/L Acetate
and 0.083 g/L Nitrate ................................................ 77

Simulation compared to experiment for 0.1 g/L Acetate
and 0.083 g/L Nitrate ................................................ 77

viii

Cs
X00
XOS
XOchf

XOSeff

Err
em’k
ass/k
est/k

LIST OF SYMBOLS

attractive parameter for Van der Waal EOS
repuslive parameter for Van der Waal EOS
shape factor for repulsive term in ESD EOS
half-saturation constant for nitrate, chm'3
half-saturation constant for acetate, gscm'3
chemotactic sensitivity coefficient for nitrate, cmzhr'1

chemotactic sensitivity coefficient for acetate, cmzhr'l

effective chemotactic sensitivity coefficient for nitrate, cmzhr'l
effective chemotactic sensitivity coefficient for acetate, cmzhr'l
uniform separation between carbon walls, Angstroms

difiusion coefficient for nitrate, cmzhr'l

diffusion coefficient for acetate, cmzhr'l

soil porosity

Lennard-Jones well depth of ethylene-carbon atom site interaction.
fitted parameter for ethylene well depth

ethylene interaction parameter, K

carbon interaction parameter, K

ethylene-carbon interaction parameter, K

kl

Kos

NT

Pc
Pt)

[’5

reduced number density

nutrient concentration, g“ cm'3

cell flux

flux of cells due to random motility

flux of cells due to chemotaxis to nitrate

flux of cells due to chemotaxis to acetate

Boltzmann constant, J K'1

constant in ESD EOS

dissociation constant for receptor-attractant complex for nitrate, chm'3
dissociation constant for receptor-attractant complex for acetate, gscm'3
pore size corresponding to number of ethylene molecules in center of pore
random motility coefficient, cmzhr‘l

effective random motility coefficient, cmzhr'I

number of moles

number of molecules

total number of receptors for a chemoattractant

pressure, mass time'2 length”1

shape parameter for attractive term in ESD EOS

nitrate concentration, gQ cm'3
molar gas constant, mass length2 time’2 mole'l 1('1
chemical density, dimensionless

physical density, dimensionless

carbon density, Angstroms'3

VQ

Vs

<<<C

.<

cell tumbling frequency

ethylene molecular diameter, Angstroms
carbon molecular diameter, Angstroms
effective ethylene-carbon intermolecular diamter, Angstroms
acetate concentration, gs cm'3

temperature, K

dimensionless temperature

experimental temperature

soil tortuosity

cell concentration

differential tumbling frequency

maximum specific growth rate on nitrate, hr'l
maximum specific growth rate on acetate, hr'l
one-dimensional cell swimming speed
volume, length3

molar volume, length3mole'l

chemotactic velocity

attractive energy parameter in ESD EOS
yield coefficient for growth on nitrate, gu/gQ
yield coefficient for growth on acetate, gu/gs
distance from carbon surface, Angstroms
carbon center-to-carbon center distance perpendicular to wall, dimensionless

constant in ESD EOS

xi

Z‘m attractive contribution to the compressibility factor

Zrep repulsive contribution to the compressibility factor

xii

Chapter 1

1.1 Introduction to Equations of State

The main goal of the modern thermodynamicist is to accurately predict the
behavior of substances at certain state conditions. Most substances have only three
phases: solid, liquid, and gas, and a great deal of interest lies in what phase a substance
will be in for a specified temperature and pressure. A “phase diagram” is used to
illustrate such behavior. Two examples of phase diagrams are pictured in Figure 1
(Bromberg, 1984, Morrill, 1972). The phase diagram gives ranges of pressure and
temperature over which each phase is stable. These diagrams have two important
features: the triple point and the critical point. The triple point is defined as the point at
which all three phases coexist in equilibrium. Three phases do not coexist at pressures
higher than the triple point pressure. Below the triple point pressure, only the solid and
gas phases exist. In addition, the condensation curve (the region between the liquid and
gas phases) has a parameter known as the critical point. At temperatures and pressures
higher than this critical point there is no distinct phase transition from gas-like densities
to liquid-like densities.

One of the fundamental physical property correlations in the field of

thermodynamics is the equation of state (EOS). The EOS attempts to correlate the phase

 

 

 

 
  
   
  

(A)
Liquid
Solid
P
/ Triple Point
Gas
T
(13)
Liquid
Critical point
P

L-V region

'\

Saturation liquid line

   

 

 

V

Figure 1. (A) A P-T diagram for a one-component system. At the intersection
of the three curves, all three phases are in equilibrium. This point is known as
the triple point. (B) Saturation line of a one-component system which
separates the two-phase region. The region under the dome shaped curve is
the two-phase region; liquid and vapor. When the pressure and temperature
are raised above the critical point, the phenomenon of boiling does not exist
and the phase is referred to as a supercritical fluid.

diagram to an actual equation that gives a representation between the pressure P, the
molar volume V, the absolute thermodynamic temperature T, and is of the form:
f(P,V,T) = o (1)
The simplest equation of state is the ideal gas law. This law concerns itself with a perfect
(ideal) gas and is represented by:
PV=nRT (2)

where P = pressure

V = volume

R = gas constant per mole

T = absolute temperature

n = moles of gas

One of the main weaknesses of the ideal gas law is that it fails to take into account
interaction between the individual gas molecules (and thus the term “perfect/ ideal gas”).
Specifically, it assumes that (1) the interaction between the molecules is so weak that it
can be deemed as nonexistent; and (2) the particles have no size. The engineer must deal
with real (i.e. nonideal) gases. Modeling the phase behavior of a real gas by means of an
equation of state has proven to be a challenge. Ideally, this equation would be simple,
with as few independent parameters as possible, but yet still display the same trends and
values found from experimental measurements. Besides the interaction between
molecules, it would be desirable to predict the structure and motion of the system in

question.

The van der Waals (V DW) equation, formulated in 1873, is the earliest empirical
EOS which predicts liquid-vapor phase equilibria. The VDW equation is cubic in nature
in terms of molar volume or density. It takes the following form:

P = RT/(V-b) - a/v2 ‘ (3)

The representation is made up of two terms, the first term comprised of repulsive
contributions and the second term comprised of attractive contributions. For simple, non-
polar liquids, it has been found that in the neighborhood of the triple point, the structure
of the fluid is determined by repulsive forces (and thus the second term does not
contribute much to the behavior of the fluid, other than to determine the density)
(Widom, 1967). However, around the critical point for liquids, it has been found that the
attractive forces play a significant role in the structure of the fluid (Widom, 1967).
Macroscopically, this means that the fluid has a high compressibility near the critical
point. At low temperatures, liquids have low energy but high entropy, whereas gases
have a higher energy but lower entropy. At temperatures near the critical point, energy
and entropy values are more closely balanced, and thus the fluid becomes much more
sensitive to pressure.

It has been established that the structure of a fluid is mostly determined by short-
range repulsive forces with long-range attractive forces modeled as perturbations to the
former (Henderson, 1979). In addition, it has generally been found that the shape of a
molecule is what determines the intermolecular correlation for dense fluids (Abbott,
1973). The reasons for this lie in the fact that for a dense fluid outside the critical region
of the phase diagram, large energy requirements are needed in order to displace a

molecule even a minute amount. This is because the molecules are tightly packed

together. The “b” parameter is known as the repulsive coefficient and for accurate
calculations is a function of density. In essence, it is “the “volume” taken up by a
molecule when ‘excluding’ other molecules from it, thereby decreasing the space
available for the motions of the other molecules (Lebowitz, 1980).
The “a” parameter is a measure of the intermolecular attraction and is a function

of temperature. Consider the following form of the VDW equation:

P(P,T) = RTP/(U-Pb) - 892 (4)
The attractive term is a representation of the energy per unit volume of the attractive part
of the potential. This energy acts to hold the system together and thus decreases the
external pressure needed to maintain the fluid in a given volume (Lebowitz, 1980). The
VDW equation can also take the following form:

Z = 1/(1-bp) - ap/RT (5)
where Z can be defined as:

z = PV/RT (6)
The VDW equation is obtained from first-order perturbation theory with a hard-sphere
reference system (Henderson, 1979). The leading term can be interpreted as resulting
from the volume available to a collection of molecules of finite size (Abbott, 1989). Note
that the limit as the density approaches zero is the ideal gas law. It can be generally
stated that most EOS’s are compatible with the ideal gas law at low densities. Departure
from the ideal gas law is more prevalent at low temperatures and high pressures where the

intermolecular repulsions are of greater significance.

To explain perturbation theory, consider the fact that there are basically three
methods of obtaining thermodynamic properties: computer simulation, integral equations,
and perturbation theory (Henderson, 1979). Perturbation theory makes use of a reference
system for which the properties are known. Usually, this reference system is taken to be
a collection of hard-sphere molecules whose interactions are purely repulsive.

One of the characteristics of the VDW equation is that it requires linear
isometrics. This has proven to be a source of weakness to the equation, since most fluids
do not lend themselves to linear behavior. Linear behavior implies that there is no
intermolecular interaction. Other EOS’s predict different types of behavior in the
isotherms. For instance, the Dieterici EOS requires isotherms of positive curvature and is
written as:

P = RT/(V-b)exp(-a/V RT) (7)
The Berthelot EOS, requires isotherms of negative curvature and is written as:
P= RT/(V-b) - a/T V (8)

Each of these equations has its strengths in certain respects, but neither of them
simultaneously predicts the observed behavior of gases which indicates negative
curvature at low densities, positive curvature at intermediate densities and negative
curvature at high densities. It should be noted that all three equations; the VDW, the
Dieterici, and the Berthelot, predict a critical point and a vapor-liquid envelope in their
phase behavior. These useful polynomial equations are valid at sufficiently low
temperatures and they also recognize the relative incompressibility of liquids.

The VDW equation has been found to be faulty for fluids consisting of not only

certain spherical and non-spherical molecules but also for single-component hard-sphere

fluids. A large number of subsequent equations of state have thus been produced to
amend the VDW equation in some way have been produced. Another weakness of the
VDW equation is that the attractive and repulsive parameters a and b cannot both be
chosen to fit the reduced temperature, volume and pressure simultaneously. Ofien, these
parameters are dependent on each other. There are a variety of methods used to find
values for these parameters. One is the “brute-force” method, in which nonlinear
regression techniques are employed in order to determine values derived from
experiments (Henderson, 1979). Another method is to clarify some constraints on the
E08 and to then generate values for the parameters by solving a system of equations
(Henderson, 1979).

1.2 Computer Modeling

With the advent of the modern computer, the capabilities of modeling complex
systems have greatly expanded. For example, modeling methods can be used to predict
the properties of pure fluids and fluid mixtures absorbed in porous sorbents. The major
interest in the area of research considered for this paper was the interaction between a
confined fluid in a porous with a slit-pore geometry. The adsorbent and fluid studied
were carbon and ethylene, respectively.

A computer simulation is capable of calculating certain properties in a specified
system, such as a set of molecules in a “box” or microscopic volume at fixed temperature
and density. The box is designed to have certain characteristics, such as periodic
boundary conditions (which negate surface effects). The molecules in the box are
contained in a certain state which has energy that is proportional to the Boltzmann factor

(Henderson, 1979). The molecules are also considered to possess certain internal degrees

of freedom, which in turn make a contribution to the EOS. Computer simulations do not
give thermodynamic properties directly. In order to obtain these properties, integration
must occur over a series of individual states, known as microstates.

There are basically two kinds of molecular simulations: Monte Carlo (MC) and
molecular dynamic. The Monte Carlo method has no time dependence and is based on
stochastic principles. This means that it is probabilistic in nature and is driven by random
number generators (hence: the name Monte Carlo; referring to the random element of the
computation method). The molecular dynamics method is based on a more deterministic
approach derived directly from solving the motions of molecules using Newton’s Second
Law. The Monte Carlo method involves a methodology that evaluates the
configurational integral by sampling the phase space of the ensemble. From this,
thermodynamic properties are made available. Molecular dynamics generates a trajectory
through phase space by direct calculation of intermolecular forces. Thus, molecular
dynamics is capable of producing both thermodynamic and transport information.
Although Monte Carlo simulations are conceptually more abstract than molecular
dynamics simulations, they are generally much easier to program and thus are the
simulation of choice for calculation of thermodynamic properties and equations of state.

As mentioned previously, the goal of a computer simulation is to predict the
proper molecular state or states of a system in order to compute the average that
corresponds to a specific observable property. The proper molecular states of a system is
based on quantum theory. Quantum mechanics predicts the quantum energy states, better
known as microscopic states, of a model system. Statistical mechanics is then employed

to calculate the bulk properties of the system. To better comprehend this overall

procedure, consider the following scenario: a system has bulk macroscopic properties
such as energy, pressure, heat capacity and viscosity. Systems also have microscopic
properties, such as the radial distribution function g(r), which gives insight to the nature
of molecular fluid structure. One of the major goals of molecular thermodynamics is to
bridge the gap between the macroscopic and microscopic levels. In essence, this means
establishing a relationship between intermolecular forces and bulk macrosc0pic
properties.

Some of the microscopic behavior of a fluid can be described by making use of
the radial distribution function, which determines the average value of all pair functions
in a uniform fluid, including, in particular, the energy and pressure of a fluid with central
pair potentials (Lebowitz, 1980). Consider AN (r), which represents the average number
of molecular centers within a spherical shell of thickness Ar at distance r from the center
of a reference molecule. The radial distribution fimction takes on the form:

g(r) = AN (r)/(41tr2Ar) as Ar —) O (9)
where g(r) = 1 corresponds to a state in which no correlation between the molecules
exists (ideal gas). In the realm of real fluids, the radial distribution function is a fimction
of t, which, when multiplied by the average density p of the fluid, yields the average local
density at a distance r from the center of an arbitrarily chosen molecule in the fluid
(Widom, 1967). The deviation of this function from I is also important, as it the extent to
which the average density at a distance r from the center of any molecule deviates from
the mean density of the fluid. At small values of r the deviation is quite large, because

the strong intermolecular repulsion forces make it difficult for other molecules to place

IO

themselves near the reference molecule. At long distances, the deviation goes to zero
(Widom, 1967).

Traditionally, quantum mechanics has been used to find macroscopic properties.
However, quantum mechanics usually concerns itself with individual states, called
microstates, which fluctuate rapidly from one state to another so that certain dynamic
properties, such as energy and pressure, are impossible to calculate (Reed, et. al., 1973).
Thus, a system (known as statistical mechanics) has been devised in which an average of
these fluctuations is taken among the quantum states and the subsequent dynamic
properties can be calculated. The techniques of statistical mechanics offer a reliable basis
for the development of equations of state because they are based on sound physical
considerations. Thermodynamic properties of systems can then be derived in terms of
partition functions, which depend upon the temperature, the volume, and the nature and
ntunber of the particles in the system. An ensemble is a collection of the quantum states
of the system, and the partition function describes the distribution of the system among
the quantum states. A canonical ensemble is one in which the system is constrained at
constant N, V and T (number of molecules, volume and temperature).

When the canonical partition function is calculated, the separate energy
contributions from the various molecular degrees of freedom are examined. For an
isolated system, total energy is conserved, and this total energy is made up of kinetic
energy and potential energy. The potential energy is the sum of both repulsive forces and
attractive forces, much like the VDW E08, and is the result of intermolecular
interactions. To calculate this potential energy, a few assumptions are made. One is that

the intermolecular pair-potential energies are those for an isolated pair of molecules. A

11

second assumption is that the configurational energy of a system of many molecules is
the sum of all possible isolated pair energies. This is known as the assumption of
pairwise additivity. A third assumption is that the pair potential depends only upon the
distance between the molecular centers of mass. Two types of interactions are relevant to
this modeling study between adsorbing fluid molecules (which we designate the fluid-
fluid interaction potential) and those between fluid molecules and the solid surface of the
carbon adsorbent.

One of the popular models for soft-sphere pair potential is the Lennard-Jones
equation. This equation is of the form:

(Ml) = 46 “1(0n/ Z)12 - (Ga/Z)6] (10)
where z is the separation distance between molecules and <5ff and 0'ff are fitted parameters
for the bulk ethylene well depth and molecular diameter, respectively. These can be
obtained through either virial coefficients or from viscosity measurements. This equation
consists of two terms, the first being the repulsive term and the second being the
attractive term. The equation implies that repulsive interactions in a fluid can be
represented by a “soft-sphere” pair potential at short distances, with weak dispersion-type
attractions at longer distances. It is sometimes useful to separate these two concepts. The
short-range part keeps the particles apart and is responsible for the local correlation while
the long-range part sees only the gross (macroscopic) density profile of the fluid and
provides an attractive potential well (mean field) for the fluid particles (Lebowitz, 1980).
Mean-field theory implies other molecules have an influence on each particle that moves

in an average field. Other types of intermolecular potentials are available to describe pair

l2

interactions of molecules with attractions and repulsions; e.g. the square-well potential.
This potential has been found to adequately describe the macroscopic properties through
a wide range of PVT values in the phase diagram, including the critical and triple points
(Widom, 1967).

A separate model for solid-fluid interaction that is similar in form to the Lennard

Jones equation is described by the Steele 10-4-3 potential:

¢S,(z) = ew[(2/5)(os,/z)'° - (cg/z)4 - G‘s/(3A(z+0.6lA)3’] (11)
where 6“, is the wall potential parameter, 2 is the direction from the surface of the carbon
and odis the effective ethylene-carbon intermolecular diameter calculated by taking the
average of the fluid and solid diameters. The A represents a uniform separation between
slab walls. The wall potential parameter 6w is given by

e w = 2n 6 sfpgtrssz (12)
where ps is the carbon density and 6,, is the Lennard-Jones well depth of the ethylene-
carbon atom site interaction. The “10” and “4” terms represent the repulsive and
attractive interactions of the fluid molecule with the wall plane, while the “3” term results
from the summation of the attractive part of the potential over the remaining layers of the
solid.

Each pore is bounded by two parallel slabs infinite in the x and y directions. The
walls are considered smooth, and separated by a fixed width in the z-direction. The
comparisons between theory and simulation in this study were made at identical values of
L, the ratio (slit width from carbon surface to surface)/(molecular diameter of ethylene).

This corresponds to the number of ethylene molecules that can fit side by side in the free

volume of the pore (in the z-direction). Thus, L=l represents a pore width of one
ethylene molecule. Another dimensionless measure of slit width is the carbon center-to-
carbon center distance perpendicular to the wall, zlen. Figure 2 illustrates the physical
distinction between zlen and L. The parameters 0',s and 0'Ff represent the carbon and
ethylene molecular diameters respectively. Thus:

zlen = L + 0'3/0ff (13)
For carbon and ethylene 0”: 3.4 and on: 4.22 Angstroms.

The local composition of a fluid confined in a slit can be obtained by making use
of computer simulations. One advantage of these simulations is that calculations can be
done for a large range of density, temperature, bulk composition, and energies of
interaction. Ofien times, experimental data for such broad ranges of temperatures and
pressures are difficult, if not impossible, to obtain. The Monte Carlo program randomly
generates a large number of trial configurations for a system consisting of (a fixed number
of molecules in a volume of space appropriate to give the desired density. The program
begins with an initial configuration of 108 to 864 molecules in a fee lattice at a specified
temperature and density. Subsequent configurations are generated by displacing
molecules to new positions. After each attempted move, the new position of the
molecule is checked against the positions of the other molecules and the change in total
energy of the system is calculated. Probability rules from canonical Monte Carlo
simulation govern whether this new move is accepted or rejected. Moves which result in
a decrease in the total energy are accepted. If a displacement increases the total energy,

then the move is rejected or accepted based on a randomly generated number. The

 

zlen

 

Figure 2. Representation of L=3 (pore size of 3 ethylene molecules)
and zlen = 3.8057 (see Equation 13).

15

probability of the move being accepted decreases exponentially as the value of the total
energy change increases. Afier the system has been allowed to relax from its initial
lattice configuration, the properties of each configuration are recorded and at the end of
the simulation these values are averaged to yield equilibrium thermodynamic properties.
Equilibrium was determined to be established when the energy and density profiles
generated smooth curves, with minimal fluctuation.

1.3 The Simplified Local Density Model (SLD)

Equations of state describe the relationship between the pressure P, the molar
volume V, and the absolute thermodynamic temperature T of substances. The cubic EOS
takes into consideration fluid-fluid interactions that induce phase changes. In an
adsorbing fluid, only a fraction of the volume of the porous solid is actually occupied
bythe fluid phase. An adsorption model at the surface of the wall must therefore also be
included for modeling the EOS of a confined fluid.

For this research, a slit pore type of geometry was explored, which does not lend
itself to bulk fluid calculations. The slit geometry has been used to accurately describe
the behavior of certain gases sorbing onto selected materials, such as carbon. Adsorption
refers to the sorption of a substance onto a solid surface. Adsorption isotherms describe
the uptake of a gas (or liquid) onto a solid surface (or wall) at constant temperature and as
a function of pressure. The adsorbing molecules fill the pore initially by forming a
monolayer on the carbon surfaces (the walls). At higher densities, the space in between
the slit walls fills up with additional adsorbate molecules, as shown in Figure 3 and
Figure 4. Note that in these figures the wall is removed, and only the fluid behavior is

depicted. The walls are located just beyond the outer edge of the furthest layer of fluid.

l6

 

Figure 3. MC simulation of liquid-vapor equilibrium in a carbon slit
pore of width 19 Angstroms at a low density.

 

Figure 4. MC simulation of liquid-vapor equilibrium in a carbon slit pore
of width 19 Angstroms at a high density.

18

Adsorption models, much like equations of state, can take on many forms.
Empirical methods (such as those of Freundlich and Langmuir) can be employed to
describe monolayer adsorption. For instance, the Langmuir two-parameter states that the
surface of the solid contains only adsorption sites and that the adsorbed species does not
interact with other adsorbates, only with the adsorption site. Although mathematically
simple, this type of method fails to take into account adsorbate interactions in the lateral
direction. At the other end of the modeling spectrum are molecular simulation methods
(e.g. Monte Carlo). The simplified local density model (SLD) tries to build a bridge
between these two extremes (Chen, et. al., 1997, Rangarajan, er. al., 1995 and
Subramanian er. al., 1995).

SLD includes interactions between the adsorbed molecules at various distances
from the wall. The method involves adapting an EOS to describe adsorption behavior.
SLD has been used in the study of isotherms for adsorption on a flat wall system with
fluids and solids. The total molar chemical potential of a bulk fluid is made up of two
parts: the chemical potential due to the fluid-fluid interactions and an attractive potential
due to the fluid-solid interactions. At all points from the wall, the molar chemical
potential is a constant. The individual contributions may change, but the total remains
the same. Near the wall, the adsorbent exerts an attractive potential on the adosorbate.
SLD also makes use of the local density approximation (LDA) to calculate the adsorbate
chemical potential. The LDA states that at a certain distance from the wall, all the
thermodynamic properties my be calculated using the same density value, which is itself
an average. The density values change with respect to position, and therefore, SLD fails

to predict any fluid structure found throughout the slit of near the wall.

1.4 The ESD Equation of State

The most popular equations of state are cubic in volume or density. These
equations take on the following form:

P = RT/(V - b) - a/(v2 + ubV + wb) ' (14)

Certain values can be assigned to the u and w parameters to obtain different EOS’s. For
instance, to obtain the original VDW equation, set u = w = 0. The Redlich-Kwong (RK)
equation has values of u = l and w = 0. The Peng-Robinson (PR) equation has values of
u = 2 and w = -l. Ideally, a cubic EOS would give an accurate representation for both
vapor-liquid equilibrium and volumetric properties at sufficiently low temperatures.
There is evidence in the literature that practically identical vapor-liquid equilibrium
(VLE) values (T-P-composition) can be obtained from cubic equations of state containing
two to four parameters, and these results are frequently comparable to those obtained
from more complex equations of state (Yu, et. al., 1986). However, the equations vary
widely when it comes to representing volumetric properties, especially liquid volumes
(Abbott, 1989). Three ethylene adsorption isotherms that are modeled by the ESD-SLD
method are depicted in Figure 5. Good agreement between the model and experiment are
shown. The three isotherms correspond to subcritical, supercritical and near-critical
conditions.

The Peng-Robinson (PR) equation has perhaps had the most widespread use
among cubic equations. The PR parameters are calculated using vapor pressure
information over the range from the normal boiling point to the critical temperature. This

well-known equation has been demonstrated to give both good liquid-vapor equilibrium

1“” (mmollg)

 

20

212.7K y

301.4 K

 

1L
0 L——— ~+- ,A_ ~ . + -— + , i

 

 

0 0.5 1 1.5 2 2.5

P (MPa)

Figure 5. Ethylene adsorption on BPL carbon as measured by Reich,
et. a1, (1980) and rmdeled by the BSD-SLD method using efs = 93 K
and a slit width ofl3.4 Angstroms.

21

information and volumetric properties for pure substances and mixtures. The PR EOS
takes the following form:

P = RT/(V-b) -a(T)/(V(V+ b) + b(V- b)) (15)

A more recent equation of state that is gaining in popularity is the ESD equation
of state (Elliot, Suresh and Donohue, 1990). This model has been used to characterize
nonspherical and associating molecules. The authors split the problem up into three main
parts. They address the issues of (l) the effect of nonsphericity on repulsive forces, (2)
the effects of attractive dispersion forces, and (3) the effects of molecular association.
They chose to take the approach of modifying available theories in each of these three
realms and then combining them into one universal application. The repulsive term to the
equation takes the following form:

Zrep = 4cn/(l-1 .911) (16)
where c is the shape factor and n=bp, which is the reduced density. 2"” is the repulsive
contribution to the compressibility factor according to:

PV/RT = 1 + 2"" + Z“ (17)
where R is the gas constant, and V is the molar volume.

The attractive term to the equation takes the following form:

Z“ = -(9.5an)/(1 + 1.7745Y) (18)
where Y is the attractive energy parameter and q is given by:

q=l +1.90476(c - l) (19)

22

1.5 Present Study

With the basic framework for this work laid, the specifics of the project can be
outlined. The system for comparison is the adsorption of ethylene on carbon. The ESD
equation of state is used with the SLD assumption to predict the thermodynamic
properties of the fluid. One of the goals of this study was to compare calculations of the
ESD SLD model with the Monte Carlo (MC) calculations to investigate the strengths and
weaknesses of the SLD approach. A slit pore geometry was chosen to represent the
adsorption surface for both models. Activated carbon has a rather complicated geometry,
and it has been found that a simple slit pore model realistically duplicates experimental
results within certain boundaries by making some assumptions.

The goal of the SLD approach was to predict the adsorption isotherm using
density and energy profiles. The profiles were then compared to those generated using the
MC method. The fluid chemical potential is itself made up of two parts: a repulsive term
and an attractive term. The assumption was made that both of these terms were
determined by the local density approximation (Rangarajan, et. al., 1995). In previous
SLD papers, fluid within of5 = (6s + (5)/2 was ignored. In this study, the authors allowed
the cutoff distance to be the point where the local fugacity of the fluid was approximately
one percent of the value of the bulk fugacity. This cutoff point has a different value for
each slit size.

One of the major features found in the MC method is that distinct layers of fluid
form throughout the slit. However, where these layers form depends on the size of the

slit. For instance, in a small slit, (L=l) a very distinct monolayer forms in the center of

23

the slit. However, at larger slits, layers first form at the walls of the slit, and then
gradually fill in to form monolayers in the middle of the slit (see Figures 3 and 4).

To implement the MC simulation, a FORTRAN code was amended from a pre-
existing version (Lastoskie, 1994). The program was executed using the values of L and
pc (the chemical density) desired to make the comparisons with the SLD model (see
Table 2).

For the SLD model, the parameters used were chosen based on experimental data
for ethylene adsorption onto activated carbon (Reich, et. al., 1980). The fluid-solid
interaction parameter 8ka and the slit size were varied in order to obtain trends seen
experimentally at temperatures of 21 1 .22 K and 339.42 K. A value for 8ka of 93 K and a
slit width of 13.4 Angstroms from carbon center-to-carbon center were selected to fit the
data reasonably, as was shown in Figure 5. The Salk parameter is not required for the
SLD model. Table 1 outlines the parameters used in the calculations for the theory and
simulation.

Table 1. Parameters used in calculations for BSD-SLD and MC

 

PPM—VERT—

A W
p. 0.114 Angstroms'3
0,, 4.22 Angstroms
“as 3.40 Angstroms
0., 3.81 Angstroms

Salk 25.0 K

8...!k 125 K

err/k 225 K

 

 

 

 

For the MC calculations, a value of err/k = 224.7 K was selected for ethylene. The

SLD and MC methods predict different state densities for non-ideal gas conditions,

24

making direct comparisons impossible. Therefore, in order to conduct meaningful
comparisons between the two models, a common basis between the two methods was
established. Comparison at two temperatures are provided, one subcritical and one
supercritical. To discuss the trends observed, several slit widths, reduced densities and
temperatures were chosen. Table 2 outlines the various parameters utilized for the two

models.

Table 2. Common parameters used in calculations for the two models.

 

 

 

 

 

 

 

 

 

pa. pp.

L = 1 zlen = 1.8057
0.2 0.1107604
0.3 0.1661406
0.4 0.2215207

0.45 0.2492108

L = 2.37 zlen = 3.1757
0.1 0.0746292
0.3 0.2238877

0.539 0.4022515

0.85 0.6343483

L = 4.5 zlen = 5.3057

0.1 0.0848144
0.3 0.2544433

0.579 0.4910756

1 0.8481445
L=9.1943 zlen = 10
0.1 0.091943
0.3 0.275829
0.579 0.53235
0.9 0.827487

 

 

 

The parameters pc“ and pp“ are defined by the following equations:
poi.I = poo-113 (20)

pp* = ppri'3 (21)

25

For the MC simulation, dimensionless parameters are used as inputs for the
program. The dimensionless temperature, T*, is defined as:

T* = T,,k/t:flv (22)
where the experimental temperature is TC, and k is Boltzmann’s constant. SLD and MC
were compared at a common chemical density pc. As stated earlier, to run the MC
program, a physical density pp was desired. The chemical density pc and the physical
density (employed by the program) are related by the following equation:

p. = p.(L/zlen) (23)
The comparison of energy values was obtained in terms of the dimensionless energy per
molecule for each case. It is interesting to note that the repulsive energy does not
contribute to the departure function for the ESD equation of state. For the MC method,
reduced energy E* was calculated as:

E* = E/efi. (24)

At the higher temperature, calculations extend as high as 188 MPa, as calculated
with the SLD model. For some cases in the smallest slit, there are also some very low
pressures. Table 3 illustrates the values of the reduced densities and corresponding
pressures using the ESD equation of state. Each pair of plots outlined in Figures 6-13
may be used to compare the reduced energy profiles calculated by the two methods. In
addition to the energy profiles, density profiles were generated as well. These were
utilized for general comparisons and in most cases, proved to be rather limited in their
insight. These profiles proved to be most beneficial in the instances where the energy

profiles between the two models differed the most.

Table 3. Equilibrium bulk pressures calculated by ESD-SLD for each reduced

density.
T = 211.22 T = 339.42
chemical density P (MPa) P (MPa)
pc I. = 1.0
0.2 1.83E-06 6.96E-03
0.3 1.37E-04 0.219
0.4 5.13E-02 40
0.45 46.8 188
L = 2.37
0.1 1.88E-04 4.15E-02
0.3 3.42E-03 0.388
0.539 3.52E-02 2.73
0.85 6.25 149
L = 4.5
0.1 1.00E-03 0.167
0.3 6.72E-02 2.25
0.579 0.337 9.45
1 33.7 170
L = 9.1943
0.1 1.42E-02 0.758
0.3 0.392 5.76
0.579 0.613 15.6
0.9 0.662 75.6

1.5.1 L = 9.1943

26

 

 

 

 

 

 

 

 

 

 

 

For a slit width corresponding to L=9.1943 (zlen = 10) two distinctive behaviors
were noted for each of the two temperatures as illustrated in Figure 6 and Figure 7. It
should be noted that the energy profiles pertaining to the MC method have been averaged
at each position to result in the symmetrical profiles shown. At the lower temperature of
211.22 K and reduced density of 0.1, a thicker surface “energy” layer of fluid and a flatter

energy profile are observed for the MC method and both models indicate a lack of fluid in

27

 

I Energy vs. Position 1:211 .22, L=9.1943

 

 

 

_..|_.d=.3
d=.579
.34, ,d= .9

 

 

 

 

 

 

 

 

 

l ErrerwvsPoeitionT=211zz L=9.1943
l

 

Sid—=7—
_n_d=.3

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6. MC simulation results (top) and SLD (bottom) for slit width
L = 9.1943 at a subcritical temperature.

28

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DemityvsPoaitionT=3aa42L=a1943
9.00
8.001
7.00.
>.6001 . ! i”_._d=.1
. 3" 5.0). T 11 l+d=3
l d=.579
l d=.9 _
Densityvs. PosifionT=339.42 L=9.1943
0.03
0.03 4 -—‘
v l+d=.1
> 0.02 —4 \ l
9.: «‘5 . . ’_|_.d=.3
l g 0.02- '. , l
l q, ; l d=.579
: o 0.01 - j ‘
.--,.,.--d=.9
0.01 - l
l 0.00 - _ -.
0 10 20 30 40
fistamefiunwrbonoemermngsuum)

 

 

Figure 6A. Density profiles with MC simulation results (top) and SLD (bottom)
for slit width of L = 9.1943 at a supercritical temperature.

29

rw____#__._._,z- 22 #__.._LL

Energy vs. Position T=339.42, L=9.1943

   

-2 ,
41f
‘3 +

“:1 '8 .1:—

-1o 1

-12 l

l

l

-14 L___.._,,,____. . ., 7 L —+ 4 l l
o l

l

1

 

 

 

10 20 30 40
Distance from carbon center (Angstroms)

 

l_____a_ A A L 7 , LL

 

 

Energyvs. Position T = 339.42. L= 9.1943

 

 

 

 

 

0 10 20 30 40
Distance from carbon carter (Angstroms)

 

 

Figure 7. MC simulation results (top) and SLD (bottom) for slit width of
L = 9.1943 at a supercritical temperature.

30

the center of the slit. At a reduced density of 0.3, the difference in the thickness of the
contact layers decreases and both models predict low fluid density in the slit’s center. At
a pc = 0.579, the SLD method does not predict fluid in the center of the slit, whereas the
MC method predicts a more uniform density profile. This obServation is indicated by the
density behavior profiled in Figure 6A. At the highest reduced density, corresponding to
a value of 0.9, “layering” of the fluid is observed in the MC profile, but is absent in the
SLD profile. In both models, fluid is present in the center of the pore, but the fluid is
much more structured in the MC method (See Figure 6A). For the MC method, the
energy near the wall is found to be more density (pressure) dependent.

At the supercritical temperature of 339.42 K, the same trends in behavior were
observed for the density of 0.1. At densities of .3 and .579, the SLD predicts a more
uniform energy profile. The density behavior for the two methods is outlined in Figure
7A. At the highest density, corresponding to a value of 0.9, the layering behavior is again
observed for the MC simulations. As before, the MC energy is more density dependent at
locations near the wall.

1.5.2 L = 4.5

For a slit width corresponding to L=4.5 (zlen = 5.3057) the profiles at the lower
temperature show similar behavioral patterns as those observed in the larger slit at all
densities. This behavior is illustrated in Figure 8 and Figure 9. At the supercritical
temperature, Figure 9 shows similar trends to those observed in the larger slit. An
additional observation is that contact layers in the MC method are hard to distinguish due

to the more uniform density profiles at this point, whereas the SLD gives consistent

31

 

Demityvs. PaidonT=211.22.L=9.1943 l

 

 

 

 

 

 

 

 

 

 

|
l

 

ll
liog'go;

 

in

 

 

 

 

 

 

l ‘ ___._-__ .z.___*

E
l
‘
l
l
l

 

Figure 7A. Density profiles with MC simulation results (top) and SLD (bottom)
for slit width of L = 9.1943 at a subcritical temperature.

0 u
-2 +
‘ -4 -l =
z E ‘6? =
3 '8 i . I l d = .579
“4°r “‘ l d-10
-14 i '
“'16 .1————-__—#_— L. __._ L— __-_,,_.-__ __r._m _ . _ __--Li
0 5 10 15 20
Distance from carbon center (Angstroms)
L L __ w”. L , L._,-__-fi,___j
Energy vs. Position T = 211.22, L = 4.5
0 -.
-1 l f
-2 l ' , 1+ T 7é‘T—l
3 -3 1 l l j_n_d= .3 l
3 -4 .1 '
= .5 l 2 I. l d=.579
Ill 7 ,1 I
'61 4; , l ,. ..d=1.0
-7 j *7 * -—--
«8 +_ ,. __ -. -. ; -_,
0 5 10 15 20
ustance from carbon center (Angstroms)
_- L r. - _.._ ._”U

32

 

Energy vs. Position T=21 1 .22, L=4.5

 

 

 

 

 

 

 

 

 

Figure 8. MC simulation results (top) and SLD (bottom) for slit width of
L = 4.5 at a subcritical temperature.

33

7 7777. ~———— p—F-M 7‘ 5 7777‘ iqn._fi___,

Energy vs. Position T=339.42, L=4.5

 

 

 

 

 

 

 

 

y---"m~'"'.x 13:13
i 4 . ._ .: arr—
l 5 o : l_._d= .3
l 5 .3 . , d= .579
' -10. ‘ ., , “=10
-12 , ’W’” *4 ~" I
-14. .. _--—+ + ———+# A 7 «A 5
0 5 10 15 20
Distance from carbon center (Angstroms)
2 n, ,2 L, fi__U
* Energy vs. Position T = 39.42, L = 4.5
l
l o .
-1 r:
l -2
I § -3 .
; 5 4 .l
-5 -j‘
l -6 l
i -7 l T‘" T""'_' T I " Ti "—7
l 0 5 10 15 20
1 Distance from carbon center (Angstroms)
l
l

 

 

-_~. __ . _ _____1 , .. _. .

Figure 9. MC results (top) and SLD (bottom) for slit width of L = 4.5
at a supercritical temperature.

34

patterning of these layers with large mean density decreases in the center.
1.5.3 L = 2.37

For a slit width corresponding to L=2.37 (zlen = 3.1757), the greatest discrepancy
between the two methods is observed at the reduced densities 0f 0.1 and 0.3 and the
subcritical temperature as seen in Figure 10. There is a layer of fluid observed at the
walls with an absence in the center of the slit and an accompanying lack of energy for the
SLD method. This is not the case for the MC method, and this is indicated in the density
profile in Figure 10A. The behavior patterns between the two theories for the higher
reduced density of 0.539 indicate similar behavior for the SLD method as before. The
MC method predicts a sharp increase in energy at the center of the pore at a density of
0.85, whereas the SLD method doesn’t exhibit this increase. However, this sharp
increase in energy does not have similar corresponding behavior in the density profile for
the MC method. Also, the MC density profile does not exhibit the presence of fluid in
the center of the slit at this density, whereas SLD does. Figures 11 and 11A depict the
energy and density behavior found at the supercritical temperature. Both models are
qualitatively similar.
1.5.4 L = 1

For a slit width corresponding to L=1 (zlen = 1.8057), the behavior is illustrated
in Figure 12 and Figure 13. At both the lower and supercritical temperatures, quite
similar behavior is observed for the designated densities. This is the opposite of what
was expected at the outset of the comparison when it was anticipated that the smallest

pores might show the greatest differences between the two models. It should be noted

35

 

,.____—.____._..__._...___.__ — __ _ — —.L____.—_

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Energyvs.PositionT=211.22,L=237 l

l

0 {WV-"€13,“ may». qugygyyynv“ :

‘2 i fi—"— ——-1'
49 l '+J-1
S; —I-—d = 3

26‘ ‘ 05%)

11.1 -8 i . . d = 85

40. "v “T" '—<

JLLLLL .r LLL . l

o 5 10 1

Distance from Carbon Center (Angstroms) 1

L . , LL L- L- LL -7

Energyvs. PositionT=211.22, L=2.37 i

1

0 ., l

41 g s

'2 ‘ :0 = .1 '1!

§ :3 —I—d = .3 ii

5 -5 d= .539 l

6 L.-. - d = .85 l

.7 ___,}

-8 l _ LL . _ !

o 5 10 15 l

Ustance from carbon center (Angstroms) 5

 

 

 

 

Figure 10. MC simulation results (top) and SLD (bottom) for a slit width
of L = 2.37 at a subcritical temperature.

36

 

T T TAT—W

DersityvsPositionT=211.212,L=2.37 l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.00
. 7.00. .5 .
6.00. l
as?”
d=.3 g!
d=.539l:
-0185};
l
o 2 4 H 5 a“ 10 12 14 l
Mmefrorncarbonoerleflmm) l
0931va warm" T=211.22,L=237 ;
QCB .
000. ' L :
>002- +4“ 11
z _I_d=.3
3012
o d=.5rfi
0001.
L. d=.85
(101-
000- l
0 14 1
l
l
l LLL L L L 1

 

 

Figure 10A. Density profiles with MC simulation results (top) and SLD (bottom)
for slit width of L = 2.37 at a subcritical temperature.

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Energy vs. Position T=339.42, L=2.37
l 0 frrrtsrrreemm unuwvw-r-ver-rr
l -1 j,
l .
l j l i § 1 +9 =
E .4 , l _n_d = .3
5 _5 l j | d = .539
.6 = , ‘ LL” = '85
-7 l "1,, ,- ,
-8 . :th L L ,, L L. L LL
0 5 10
Distance from carbon center (Angstroms)
‘L
1
l
l 0
I -1 l _ ___
L -2 _ l+d=.1
E -3 - [ _|_d =.3
5 4 ~ ,. d=.539
1 '5 — ! -~,./—--«d= 85
l '6 A L
l .7
l
0

 

 

Figure 11. MC simulation results (top) and SLD (bottom) for slit width
of L = 2.37 at a supercritical temperature.

38

 

DensityvsPositimT=339AZL=237 T

 

 

 

 

 

 

L A _. LL L L L.__—_“ ,L‘J

 

 

 

 

 

 

 

 

”T . .. T" ' '1 "PM”?
Densrtyvs. PosrbonT=339Az L=2.37
0.03
0.03. _ ,LLL
+d-Wi
> 0.02 . .
.2 d=.3
g 0.02 :4—
.» d=.539
D 0.01 -
__ .d=.85
0.01 ,. LLLL.
0.00 , g .
0 2 4 6 8 10 12 14
Dstanoefmmcarbonoemermrgsu'm)

 

 

Figure 11A. Density profiles with MC simulation results (top) and SLD (bottom)
for slit width of L = 2.37 at a supercritical temperature.

39

EnagyvsFMsfimnTfihtZZlfito

 

0 mewwwmm

  

 

 

 

 

 

 

1 m - - -

a -1 i _|_d=.3 i
~45‘ l
g l i d: .4 l
in .2 ,_ ' __ i;

' ‘ 304%,?

l -3 4 - . 4 —-—- 1 t —a

j 0 2 4 6

1 Distance from carbon centermrgsb'oms)

l

1 Energyvs. PositionT = 211.22. L= 1.0

. 0

-1 .‘ ___M__ p

1 § :3 :4_d=.3 "

l 3 l ‘ d= 4

l w '5 l l - A.

.6 - 1 d=.45

l -71 .LL -—«~

l ‘8 i —— r 1

l 0 2 4 6 8

; Distancefromcarbon center (Angstroms) 1

 

Figure 12. MC simulation results (top) and SLD (bottom) for slit width of
L = 1 at a subcritical temperature.

40

 

  

 

 

 

 

 

 

 

 

 

 

 

 

Distance from carbon center (Angstroms)

ITTT’TT TTTT T T‘ T TTTTTT T T T T T T T T TTTT TTTTT'TT TTI
Energy vs. Position T=339.42, L=1.0 I
o " -MWVW‘lrm-WWW5IWWW\V\ mwwmeWe-Wr-ms I
‘0-5 I :0 = 2T
>5 _ =
g 1 I .m..d- .3
":1 _1.5 I d - .4
l
-2.5 .L_—_-_-_.- - - - -9 _-_ I .1 I
0 2 4 6 8
Distance from carbon center (Angstroms)
L.L,LLL . L L L - L LL LLLI
lTTTTTTTTTTTTT T T. .T TTTT T TT'
Energy vs. Positron T = 339.42, L = 1.0 I
l
0 ‘1 l
-1 1 ———- 51
-2 I +d = .2 II
E '3 I _|_d=.3 II
2 -4 I d=.4 ’I
In 1 I:
'5 TI L, Md: 45II
'6 1 ‘1
T7 Ti— —T—TT—TT T T T ‘- 7 f l I
0 2 4 6 8 I

 

 

 

Figure 13. MC simulation results (top) and SLD (bottom) for slit width of
L = 1 at a supercritical temperature.

41

that the reduced densities at this slit width cover a much smaller range than at the other
slit sizes. The pressure ranged from values of 1.83x10” MP3 to 46.8 MPa, for the
subcritical temperature, and values of 6.96x10'3 MPa to 188 MPa, as noted in Table 3. In
both methods, similar energy profiles were observed.
1.6 Conclusions

Several trends are common for the slit widths considered. For lower densities, the
SLD energy profile is “thinner” and more pronounced near the wall than the MC profile.
At higher densities, the MC method predicts a more structured fluid. The SLD energy
profile doesn’t depend as strongly on density near the wall. This work has not directly
uncovered the reason for the smaller density dependence of the SLD energy profile at the
wall. Comparisons of the MC and SLD density profiles are difficult due to the layered
structure of the MC results. The energy profile predicted by SLD is directly related to the
local density, whereas in MC, nonlocal effects arise. If this observation is indeed due to
nonlocal effects, then this is a shortcoming of the SLD method, and the deficiencies are
more severe for larger pores. Another possible shortcoming is the lack of repulsive fluid-
fluid interaction in the ESD model. Currently, the repulsive contribution to the ESD
equation is modeled using the bulk fluid mean field term which is known to be deficient
even in bulk fluids. An increased fluid-fluid repulsion near the wall would result in a
flatter density profile at a fixed mean density, improving agreement between the models.
However, rather than modifying the ESD equation, it would be preferable to adapt the
SLD to a different equation that has a repulsive contribution to energy, and thus develop a

greater understanding of the repulsive energies near the wall.

42

There is a significant cancellation of attractive and repulsive contributions in the
MC energy profile, which is quite structureless relative to the density profile. This
cancellation evidently extends to the local entropy, since the local chemical potential is
invariant with respect to position, and is given by a combination of energy and entropy.

Mention should be made as to the quantitative values between the two sets of
energy values used to make the comparisons. For the slit widths of 9.1943 and 4.5, the
SLD energy values were exactly half those of the MC method. At a slit width of 2.37 this
trend changes in that both methods are on about the same scale. At a slit width of 1.0, the
trend differs yet again, in that the SLD method values are now about 2.5 times those of

MC. No explanation has yet been determined as to the cause of this behavior.

Chapter 2
2.1 Introduction

One of the areas of significant growth in experimental research in the last two
decades is in environmental technology. This is due in part to the contamination of the
environment through chemical spills and leaks that took place in earlier decades. These
contaminants reside in the soil or in water aquifers and reservoirs for years and
continually pose a threat to human water supplies and farming properties. It has only
been in more recent years that environmental research has taken off on a large scale,
resulting in a number of elegant design systems and methods whose focus is to
extensively reduce the evidence of past destructive behavior. This has mainly been due
to recent government environmental laws that strive to ensure that new pollutants are not
added to the environment and that old contaminants are reduced to certain minimal levels.
These methods focus on the removal of the contaminants in a way that is both
unobtrusive to the natural environment and also inexpensive. Combining these two
variables has proven to be quite a challenge to today’s environmental engineers.

Most of the substances that are contaminants in the environment are organic in
nature. An elegant method of decontamination is the use of bacteria to naturally reduce
these organic substances into carbon dioxide and water. The goal is to disturb the
environment containing the contaminants as little as possible, but yet ensure that bacteria
are making contact with the toxins. Much research has been done on this subject, and

various strains of bacteria have been used. Perhaps the most widely used has been

43

Escherichia coli. This strain has been proven to degrade a number of toxic substances,
such as PCB’s (polychlorinated biphenyls), and carbon tetrachloride. The two main
methods for degrading hazardous contaminants in situ are biostimulation and
bioaugmentation (also known as in situ bioremediation). The former method involves
stimulating the already existing bacterial population and the latter refers to the addition of
certain bacterial strains to an existing site. Of the two methods, bioaugmentation proves
to be a greater challenge, since cells often have a difficult time adapting to a new
environment where they must compete against already existing cells. However, if the
new cells are able to adapt well to adverse circumstances and have certain advantages
over the existing cells, then they can prove to compete successfirlly and survive in even a
hostile environment.

Bacteria are small in size (1-2 pm) and are rather simple organisms, which lends
some case in the study of their behavior. Typically, they survive by their ability to adapt
to their existing environment. This can be accomplished through two methods. First, if
they sense a more desirable environment, they can move towards it. Secondly, they can
adapt to change their internal metabolic processes. The latter method is typically a slow
process, because it requires the changing of the genetic make-up of the cell. Thus, at least
for the case of motile bacteria, swimming towards a more favorable environment proves
to be the more advantageous of the two methods. Bacteria have been shown to move in
response to gradients of certain substances, such as hydrocarbons, metal ions, alcohols,
sugars, and amino acids. These substances can be grouped into two main categories:

attractants and repellants. Obviously, attractants are the materials that bacteria would

45

respond favorably to, since they contribute to the cell’s growth. Attractants can also be
substances that have structures that are similar to the food sources essential to the
bacteria.

It has been found that some strains of bacteria respond favorably to concentration
gradients of certain nutrients (Adler, 1966). This behavior has been described as
chemotaxis, and the nutrients that cause this behavior are known as chemoattractants.
For bioremediation purposes, cells exhibiting chemotactic behavior may have an
advantage over the existing native cell population. This behavior exhibits itself by the
formation of a band of high cell density that develops ahead of the already existing cell
population. An example of this type of behavior is depicted in Figure 14. In this figure,
the bacteria metabolize the acetate, thereby generating a gradient as they consume the
chemical and move outward towards the edge of the petri dish. Adler performed an
experiment in which a plug of bacteria was placed at the mouth of a tube containing a
potential chemoattractant suspended in agar (Adler, 1966). It was found that as bacteria
consumed the attractant, a gradient was created, and subsequently a band of high cell
density formed that traveled up the path of the tube. There were two main characteristics
Adler noted in his experiment. When galactose was in excess of oxygen Adler found that
the first band of bacteria that traveled along consumed the oxygen to oxidize a part of the
galactose and then the second band used up the remaining galactose anaerobically.
However, when oxygen exceeded galactose, the first band of bacteria aerobically
consumed all the galactose and lefi behind the unused oxygen, which was in turn
consumed by the second band of bacteria. Adler found similar results when he used

glucose and amino acids in place of the galactose. This phenomenon may prove

46

 

Figure 14. Experimental photos of motile chemotactic bacteria. Evidence
of a chemotactic ring is exhibited by the circular rings of higher cell density.
The bottom figure displays evidence of two chemotactic rings, which may
have formed from each of the two chemoattractants present.

47

(a)

(b)

 

Figure 15. Depiction of a motile cell with flagella. The direction that the
flagella rotate dictates which course of movement the cell will take. If the
flagella rotate in a clockwise manner, then they form a small bundle and
the cell swims in a smooth manner (a). If the flagella rotate counter-
clockwise, the flagellar bundle unravels and the cell tumbles in a new
direction (b). [Figure adapted from Macnab, 1987].

48

to be beneficial in the environment if the cells would view certain toxins as
chemoattractants and migrate faster to come in contact and thus consume them. In fact,
some bacteria have been shown to display chemotactic behavior towards
trichloroethylene (TCE), which happens to be the most coMon toxic hydrocarbon in
aquifers located in the United States. (Barton, et. al., 1996).

Motile bacteria possess the ability to propel themselves through the surrounding
medium by making use of flagella, which are in essence small tails that are located on the
exterior wall of the cell with lengths of 5-10 pm. Typically, cells will swim in a
completely random path that consists of a series of runs and tumbles. Runs are
characterized as relatively straight lines of movement that can last for a few seconds.
Tumbles are characterized as random changes in direction, and typically last for only a
fraction of a second. The direction that the flagella rotate dictates which course of
movement the cell will take (see Figure 15). If the flagella rotate in a clockwise manner,
then they form a small bundle and the cell swims in a smooth manner. If the flagella
rotate counterclockwise, the flagellar bundle unravels, and the cell tumbles in a new
direction. The movement of the cells through the medium has been likened to a three-
dirnensional random walk, with changes in direction caused by tumbles (Ford, 1992).
This motion shares some similarities with diffusion, in which Brownian motion dictates
the path of the molecules and changes in direction are caused by molecular collisions. In
the absence of a chemical gradient cells move according to their random motility. There
are a couple of differences between Brownian motion of molecules and bacterial motility.

Bacteria are known to have a small bias to persist in their direction of rotation after

49

tumbling, resulting in a nonuniform random turn angle distribution, whereas in the
Brownian motion of molecules the direction after collisions has a uniform random turn
angle distribution. Secondly, the change in direction made by the cell takes a finite time,
whereas in Brownian motion a change in direction is considered to be instantaneous
(Duffy, et al., 1995). In the presence of a chemical attractant gradient however, the
course of the bacterial cell is altered (see Figure 16). The cells will tumble less often if
they are moving towards a higher concentration of attractant, and thus the net effect is an
increase in cell density migration towards the attractant. This ability to adjust the
tumbling frequency in the presence of chemical gradients is known as chemoklinokinesis
(Ford, 1992), and the overall effect of this phenomenon is known as chemotaxis.
Research involving bacterial motility has for the most part involved plate studies.
These studies make use of a semisolid agar medium that contains all of the nutrients the
cells require to survive. The medium is then inoculated with cells. These methods can be
adapted to include the study of chemotactic behavior of cells in porous media, the
medium they encounter in natural environments. Some study has been done on the
transport of Escherichia coli through porous media (Barton, et. al., 1997), although the
effect of the porous medium on chemotactic behavior was undetermined. This was
reasoned to be partly due to the fact that shallow gradients were present in the
experimental studies. In contrast, very little study has focused on Pseudomonas sp. KC
(PKC), which is an aquifer-derived organism and somewhat similar to E. coli. Some of
PKC’s motility properties have been studied via the diffusion gradient chamber (DGC)
(Emerson et. al., 1994). PKC has proven to be effective in the degradation of carbon

tetrachloride under denitrifying conditions (Criddle 1990, Knoll 1994, Witt 1994).

50

/\

W V .

Increasing attractant
concentration

 

Figure 16. Observed single-cell behavior in an isotropic medium (lefi)
resembles a random walk with basal mean run length <To> and swimming
speed v. In the presence of an attractant gradient (right) run lengths are
increased when moving toward increasing concentrations of attractant,
yielding a mean run length <r> greater than <To>.

[Figure adapted from Ford, 1992].

51

Several substances have been shown to serve as chemoattractants for E. coli, including
oxygen, sugars, and amino acids (Mesibov, et. al., 1972). The species of Pseudomonads
have been shown to exhibit chemotaxis towards high concentrations of a variety of
substances, including fixed nitrogen (Ford, 1992).

Chemoreceptors are special proteins that are located at the cell surface and contain
binding sites to which only chemical substrates that are structurally similar to the site can
bind (Ford, 1992). Certain Chemoreceptors are capable of detecting chemoattractants and
these in turn influence the activity of the cell. There has been some debate in the past as
to whether bacteria respond to either spatial gradients (across the surface of the cell) or
temporal gradients in concentration of attractant (or repellent). The argument for spatial
gradients contends that, since the bacterium is moving through the spatial gradient at a
given velocity, it appears to the bacterium that the concentration is changing with time.
However, in a study done by Macnab and Koshland (1972), it was found that the bacteria
responded to temporal gradients. These studies contributed to the conclusion that the
bacteria possess both long term and short term memories in terms of the bound receptor
sites. The short-term memory is able to detect changes in the number of bound receptor
sites in between tumbles, while the long-term memory is able to detect changes from
several minutes past. The long-term memory then enables the cell to change its course of
movement in order to stay in a more favorable environment (Macnab and Koshland,
1972)

The transport of bacteria through porous media is determined mainly by three
things: bacterial properties, porous-medium properties, and porous-medium

hydrodynamics. The bacterial properties are factors such as size, motility and surface

52

characteristics, and certain parameters can be calculated that account for these properties.
The porous-medium properties are parameters such as porosity, tortuosity, particle
diameter and surface properties. The porous-meditun hydrodynamics include interstitial
pore velocity. Many of these characteristics are interrelated, and the combination of these
effects directly influence the adsorption of cells to porous-medium surfaces (Camper, et.
al., 1993). Research has shown that the spreading rate of bacteria through soils can
depend on the physical or chemical differences among soils, as well as the water content
of the soil (Soby, er. al., 1983). Much has been done on the study of bacterial transport
through soil, but most of this research has focused on passive modes of transport. These
modes do not consider the active role of bacteria making their own transport.
Researchers have found that the transport characteristics of bacteria cannot be predicted
by use of size and motility information, and so considerations such as the adsorption rate
coefficient were used (Camper, et. al., 1993). However, a direct measurement of the
adsorptiOn rate coefficient is nearly impossible to make.

The model used as a basis for this study considers two main transport coefficients
that govern the chemotactic behavior of bacteria: the chemotactic sensitivity coefficient
XOS (to chemoattractant S), and the random motility coefficient. The former is a function
of the specific chemoattractant (such as aspartate or acetate), and both coefficients depend
on the specific organism and the type of medium, whether an isotropic, homogeneous
medium or a porous environment such as soil. The purpose of this study was to find
approximate values for the chemoattractant sensitivity coefficients for PKC toward

acetate and nitrate (X05 and X00, respectively) for the homogeneous medium (agar gel),

53

and to then use an empirical correlation for pet}, the random motility coefficient in a
porous medium, to estimate X0Scflr and Xoqcff for PKC in soil. This was accomplished
through a series of lab experiments and mathematical modeling to describe the cellular
migration. Comparison of motility patters from the model and from experiment yielded

quantitative estimates of the strain PKC motility coefficients.
2.1.1 The Random Motility Coefficient (u) and the Chemotactic Sensitivity

Coefficient (X0)

Transport in a porous meditun is hindered primarily by the volume fraction of
obstructions. The coefficients u and X0 describe the dispersion and directed motion of
bacterial populations, respectively. Both of these coefficients depend on not only the
geometry of the porous media, but also the bacterial species, and are thus related to the
swimming speed, the tumbling frequency, and the turn angle distribution associated with
single cell behavior. One method for determining these coefficients is by utilizing
population assays.

The random motility coefficient can be viewed as analogous to a molecular
diffusion coefficient and is a representation of the dispersion of the bacterial population
in the absence of advective flow. The random motility coefficient in the absence of a
gradient (bulk solution) is given by the term [10. For a porous medium, the effective value
of the random motility coefficient 1.1,,ff can be expressed as a function of the geometry of

the soil matrix itself:

“eff = (aft-”’10 (1)

54

where 8 represents the soil porosity (free volume) and 1: represents the soil tortuosity
(Barton and Ford, 1995; Duffy et al., 1995). Tortuosity increases as the particle diameter
decreases. The random motility coefficient also varies according to bacterial species. A
typical value of no for E. coli is 1.5x10" cmz/s (Ford and Lauffenburger, 1991); 110 for
PKC was found to be 2.0x10” cmz/s (Schmidt, et. al., 1997). Past studies have indicated
that the presence of the porous medium reduces the random motility of the bacterial
population (Barton and Ford, 1997). For high-mesh packing materials, geometrical
spacing of particles is such that the distances between particles are of the same order of
magnitude as an average run length (distance traveled between changes in direction) of a
bacterium (Barton and Ford, 1997). Thus, because the bacteria are colliding with the
porous matrix, their run lengths decrease and a reduction in the random motility is
observed.

The chemotactic sensitivity coefficient is specific to each chemoattractant. Like
the random motility coefficient, this parameter is affected by the presence of porous
media which affect the chemical gradients. In the past population assays have been used
to determine ntunerical values for the chemotactic motility coefficients. The coefficient
can be expressed as:

X0 = vuzNT (2)
where v represents the differential tumbling frequency, 0 is the one-dimensional
swimming speed, and NT is the total number of receptors (Ford, 1992). This relationship
(with units of distanceZ/time) represents the fractional change in dispersal capability for a

bacterial population per unit fractional change in receptor occupancy, which is a result of

55

individual cells increasing their run lengths as they sense increasing concentrations of an
attractant (Ford and Lauffenburger, 1991). One study observed that the mean run times
increased exponentially with the change in the number of receptor-attractant complexes
over mean run times measured in the absence of a chemical gradient (Berg and Brown,
1972). An average value of X0 found for E. coli K12 responding to fucose was 8.1x10‘5
cmz/s (Ford and Lauffenburger, 1991 ).

2.2 Materials and Methods

The bacteria used in these experiments were Pseudomonas sp. strain KC (PKC).
PKC is an anaerobic, denitrifying bacterium. Various chemoattractants for PKC include
oxygen, acetate, nitrate, glycerol, and glucose.

In order to survive, cells need a carbon source and an electron acceptor.
Typically, the cells are inoculated into a medium that contains these two components.
The carbon source for these experiments was acetate (added in the form of sodium
acetate), but other substances such as glucose or glycerol can be used as well. The
electron acceptor was nitrate (added in the form of sodium nitrate), but in the absence of
nitrate oxygen may be used.

During the course of growth, cells first take the nitrate and reduce it to nitrite
during the denitrification process. It was found experimentally that in the excess of
acetate to nitrate, the nitrite intermediate product was further reduced to form nitrogen
gas (Setiawan, et. al., 1997). As a result, nitrogen bubbles formed. Without the excess
acetate the intermediate was reduced partially to nitrous oxide, which is in turn soluble in

water.

56

2.2.1 Preparation of Media and Growth Conditions

Medium D (Criddle, et. al., 1990) contained (per liter of deionized water) 2.0 g of
KHZPO4, 3.5 g of KZHP04, 1.0 g of (NH4)2SO4, 0.5 g of MgSO4' 7HZO, 1 ml of trace
nutrient stock TN2, 1 ml of 0.15 M Ca(NO3)2, 3.0 g of sodium acetate, and 2.0 g of
sodium nitrate. In some experiments, different amounts of sodium acetate and sodium
nitrate were used to study their effects on growth and chemotactic behavior. Medium D l
was prepared with trace nutrient stock solution TN2. Stock solution TN2 contained (per

liter of deionized water) 1.36 g of FeSO,’ 7HZO, 0.24 g of NazMoO4 ' 2HZO, 0.25 g of

 

CuSO4 ' SHZO, 0.58 g of SnSO4 ' 7HZO, 0.29 g of Co(NO3)2 ' 6H20, 0.11 g of NiSO4 ' 1’-
6HZO, 35 mg of NaZSeO3, 62 mg of H3BO3, 0.12 g of NH4VO3, 1.01 g of MnSO4 ' H20,
and 1 ml of H2804 (concentrated). After the components for the medium were
assembled, the pH of the medium was adjusted to 8.2 with 1 M NaOH. The medium was
adjusted to this pH because it has been found that PKC grows well under these conditions
(Tatara, et. al., 1993). Once the medium reaches a pH of about 8.0, the-formation of a
white precipitate is observed. Research has indicated that the removal of this precipitate
from the medium results in a significant decrease in the iron level of the medirun (Tatara,
et. al., 1993). The medium was then autoclaved at 121°C for 20 minutes.

Cells were adapted to Medium D (acetate minimal medium unless otherwise
indicated) and grown at 35°C, with rotary shaking at 200 rev/min (New Brunswick
gyratory shaker), in a 500-mL aluminum foil-covered Erlenmeyer flask containing 100
mL of medium. An inoculum from such an adapted culture was added to fresh mineral

medium and grown as described above.

57

Experiments were conducted with swarm plates, which are sterile petri dishes
with a diameter of 3 inches. These plates were prepared by pouring ~35 mL of hot,
sterile Medium D into the plate. The medium contained enough agarose to produce a
0.28% solution (by weight). The agarose prevents convective liquid movement within
the plate while still allowing the cells to swim. After the agar had solidified, the plates
were inoculated at the center of the plate with 20 uL of PKC liquid culture using a
micropipette to disperse the cells evenly throughout the depth of the agarose. The plates
were then stored in an anaerobic environment. Anaerobic conditions were obtained by
using a GasPak 150 Anaerobic System (V WR Scientific). Typically, the chemotactic
response was identified by the formation of a ring or outer band of cells after a time
period of about 24 hours.

For the case of porous media studies, the above procedure was amended. A
circular shaped screen, with a height of about 0.5 inches and a radius of either 2.5 or 1.25
inches was placed in the center of each plate. Sterile sand, obtained from sand cores at
the Schoolcraft site, was then poured into this circular screen region as shown in Figure
17. The sand was then saturated by pouring agarose liquid into the plates. The center of
the sand region was inoculated afier the agarose liquid had been poured and cooled.

Figure 18 illustrates the top view of the porous motility plate.

58

Sand

 

 

 

Screen mesh 1 I l

ooooooooooooooooooooo
.D‘
"""""
.....
o...
'0
.-
o

............................................
so
.......
TTTTTTTTTTTT
..........
TTTTT

o a .T‘TT,

o o
........
c. .-

u .-
.....
oooooo
.........
ooooooooooooooo
nnnnnnnnnnnnnnnnnnnnnnnnnnnnn

TT'. .a)
C D
. o
o. .o
o. .0
o .0
”””””

o. a.

I. .C
0000000000
""""""""""""""""""""
ooooooooooooooooooooooooo

 

 

 

Figure 17. Physical representation of motility plate for porous media study.

 

Screen edge

Agar

Inoculation point

Sand
I.

Figure 18. Top view of porous motility plate studies.

60

2.2.2 Photography and Image Analysis

When pictures of the swarm plates were desired, the plates were placed onto a
transilluminator box (TB). Inside the TB, two 30 cm fluorescent lights (single 8W, cool
white bulbs) provided diffuse illumination from about a 45 degree angle beneath the
plates. The bottom of the TB was covered with thick black felt which provided a dark
background. A portion of the light was diffracted by the cells toward a camera that was
mounted directly above the plates. Images resulted that made the regions of the plate
containing cells appear white against the dark background of the felt. The image analysis
system used to record cell growth and motility patterns was a Color QuickCam camera
(Connectix, San Mateo, CA) connected to a PC. Images were captured using PhotoFinish
2.02 (Zsofi, Marietta, GA) sofiware. The images were edited and saved in JPG-format.
2.3 Mathematical Model

In order to model a dynamic system correctly, two main types of equations must
be included; balance equations and constitutive equations. The balance equations account
for changes in state variables throughout all time and space. Constitutive equations are
used in order evaluate flux and reaction terms in the balance equations.
2.3.1 Cell Balance

The modeling of chemotactic behavior in cells must first begin with a series of
conservation equations. The balance equations taken into account for this problem
include nutrient, chemoattractant and cell balances.

The cell density u is a function of both time and position. The balance equation

for the cell density is:

61

%=—V-J,+f(H)u (3)
where Ju is the cell flux, and f(H) is a function for cell growth on a nutrient H. Note that
this equation does not take into account cell death or reproduction. A constitutive
equation for the cell flux has been proposed by Keller and Segel (1971). This equation
has two terms, the first to take into account the diffusion-like random motility of the cell,
and a second term that takes into account the convection-like chemotactic motion of the

cell:

J, = —,uVu + V‘gu + Vugu (4)

The above equation is actually a modification of the original Keller-Segel equation,
amended to include a second chemoattractant, Q. The random motility coefficient is Iu,
VuS is the chemotactic velocity in response to chemoattractant S; and VuQ is the
chemotactic velocity in response to chemoattractant Q. These chemotactic velocities are

functions of the chemoattractant concentration. Combining equations (3) and (4) gives:

T3; T aV’u - V - (un)— V - G/uQu)+ f(H)u (5)

Constitutive relations for finding VuS and VuQ were developed by Rivero, and her
method is referred to as the RTBL model (Rivero er. al., 1989). The constitutive relation

given by the RTBL model for the chemotactic velocity is:

62

(6)
Vus =vtanh w—Mfliz—VS
(KDS +S)

where v is the swimming speed of the cell, ais the cell tumbling frequency, N TS is the
total number of receptors for the chemoattractant S on the cell surface, and K DS is the
dissociation constant of the receptor-S complex. An analogous expression can be written
for the chemotactic velocity in response to chemoattractant Q. The ftmdamental driving
force behind VuS is the concentration gradient of the attractant S. As the cell consumes
the attractant around itself, it creates a gradient. This gradient is detected by the cells and
causes them to move in an outward manner. Figure 19 shows a three-dimensional
depiction of a chemical gradient for the plate experiments. Note the sharp decline in the
concentration of acetate at the edge of the cells’ growth ring. The chemotactic velocity is
modeled as an advective flow term, although the driving force is a chemical gradient, not
a hydraulic one (Barton and Ford, 1996). Previously, Segel had predicted a linear
relationship between the chemotactic velocity and the gradient, but afier comparing these
results to experiment, Rivero derived a hyperbolic tangent relationship. In the presence

of shallow gradients, equation (6) reduces to:

[(03 (7)

V = ———VS
us 108 (KDS+S)2

where X05 is the chemotactic sensitivity coefficient to the attractant S. This parameter
represents a fractional change in the dispersal capability per unit fractional change in

receptor occupancy with units of distancez/time (Ford, 1992). Again, equations in terms

63

”T I':
15:01:?

5535‘?“ \“’;;‘1 ‘9’» t": 1;“7797
\‘ “‘ rttl‘Iill’t‘t‘t't ’07,?

‘ii

 

Figure 19. Typical nutrient profile. In this instance, the nutrient is
Acetate.

of chemoattractant Q can be constructed in a similar manner. In the absence of a
chemical gradient, the coefficient reduces to a constant value that can correlated with
individual cell properties such as swimming speed, tumbling probability (in absence of a
gradient), and directional persistence (Ford, 1992). It has been shown that after cells
tumble, their reorientation is not entirely random (Berg and Brown, 1972). Cell motility
coefficients can be determined using experimental techniques such as the capillary assay
(Rivero-Hudec and Lauffenburger, 1986), the laser densitometry assay (Dalquist, et. al.,
1972), and the stopped-flow diffusion chamber (SF DC) assay (Ford, et. al., 1990). For
this study, the random motility coefficient in porous media is of interest. It has been
found that random motility in a porous medium decreases with decreasing particle
diameter (Duffy, et. al., 1995).

Cell growth is another component that needs to be included in the overall cell
balance. Growth has been modeled in the past as a Monod-type saturation process, and
we will follow this convention (Widman, 1997). The cell balance of equation (5), when

combined with constitutive relations for the cell motility cell growth rate (on nutrient H),

takes the form:

 

é—_ 2 K08 . KDQ V” (8)
.3 71V " V'Klit 1..., islIWSl‘ IIWIMQICH

where v stands for the maximum specific growth rate of the cells growing on H and C0 is

the half-saturation constant.

65

2.3.2 Nutrient Balance
The second balance that needs to be taken into consideration is the nutrient

balance. The nutrient balance is given by:

' 9
—§—H—=DHV2H— vH _u_ ()
5t Co-I-HYH

 

where DH is the nutrient diffusion coefficient and YH is the yield coefficient for cell
growth on H. The assumption made for the nutrient diffusion coefficient is that it is a
constant, and that the medium is isotropic.
2.3.3 Chemoattractant Balance

The chemoattractant balance follows a similar form as the nutrient balance. The
balance equation assumes F ickian diffusion and a consumption rate following Monod-

type kinetics:

fizDSVZS-
a: CS +S

vSS u ' (10)

 

where 05 is the chemoattractant diffusion coefficient; V3 is the specific chemoattractant
consumption coefficient; and CS is the saturation constant for consumption of S, which
corresponds to the concentration at half the maximum consumption rate. A similar
equation can be written for chemoattractant Q.
2.3.4 Boundary Conditions

It should be noted that adhesion to the porous surface by either attractant or
bacteria has been neglected. In essence, this means that only the fluid phase of the porous

medium needs to be taken into account. The cell density is modeled in a two-

 

66

dimensional plane, corresponding to the motility plate. A zero total flux boundary
condition is applied at all boundaries (Q) of the plate (Widman, 1997).

For the two-dimensional cell balance, the boundary conditions are given as:

. 11
#V2 u-ZosV'I[—EALT}”VS]-ZOQV' [ KDQ I‘VQ =0 ( )
(K...+s) (n+0)i ..

Across the walls of the motility plate, no flux of chemoattractants or nutrients

occurs. Thus:

(12)
=0 and 6S =0

y=0 y y=5

as.
5y

 

Similar equations can be written for nutrient H and a second chemoattractant Q.
2.3.5 Initial conditions

The center of each motility plate was inoculated with a micropipette. In order to
model this mathematically, the shape of the injected cell peak at t = 0 was approximated

using an exponential function:

u (13)

U(x',y') = °
exp(wIIx'2 +y'2)

 

where u0 is the initial concentration of cells and w is a peak width factor. The variables

x’ and y’ are defined so that x’=0 and y’=0 are at the center of the arena (Widman, 1997).
The initial conditions for the concentrations of chemoattractants S and Q were

S(x,y)=S0 and Q(x,y)=Q0 for all x and y, where S0 and Q0 were the initial concentrations

of acetate and nitrate, respectively, in the medium. The initial condition for the nutrient

 

67

H was that H(x,y)=H0 for all x and y, where H0 is the initial concentration of acetate in
the medium.
2.3.6 Computer Simulations

Solving a system of nonlinear, coupled partial differential equations for a two-
dimensional type of analysis is not an easy task. To solve the system of balance
equations, an Alternating Direction Implicit (ADI) algorithm was utilized (Camahan et
al., 1969). The program was written in FORTRAN 77 (Widman, 1997) and executed in
the UNIX operating system. The ADI computer program was able to solve a system of
four balance equations (Equations 8,9,10 and an additional balance for chemoattractant
Q) and included terms for two chemoattractants. Output from this program was then
imported to Matlab where images of the cell density profiles could be generated. The
ADI method uses two difference equations to solve each two-dimensional unsteady-state
partial differential equation. The first difference equation is implicit only in the x-
direction and the second only in the y-direction. The equations are solved in succession
at time steps of A112. The ADI method is an unconditionally stable method for which
convergence occurs with a discretization error of the order [(At)2+(Ax)2]. For the model
presented here, Ax = Ay (Widman, 1997).

The ADI program was originally written to model DGC environments. DGC
stands for diffusion gradient chamber and is essentially a square motility plate with
porous barriers on two sides. These barriers allow the transfer of solutions used as
sources for concentration gradients. To model experiments using motility plates, certain

parameters of the ADI program were adjusted accordingly, as described below.

 

68

Specifically, acetate served as both nutrient H and as one of the chemoattractants, S.
Since the program was written such that growth of cells due to chemottractant uptake is
assumed to be negligible compared to grth due to nutrient uptake, the potential error of
calculating “double growth” was avoided.
2.3.7 System Parameters

In order to model the motility experiments correctly, all the input parameters
(other than the chemotactic sensitivity coefficients) needed to be determined

independently. The random motility coefficient for PKC in a bulk medium was

 

previously determined from a laser-diffraction capillary assay technique (Schmidt et. al., v
1997). This value was found as a function of the agar concentration used to form the gel
medium.

The diffusion coefficients of acetate (D3) and nitrate (Dq) at a temperature of
25°C were taken from Cussler (1994). Yield coefficients Y s and Yq were obtained from
Knoll (1994). Values for the dissociation constants of the receptor-attractant complex for
chemoattractants S and Q were taken to be those of aspartate and oxygen, respectively
(Widman, 1997). These dissociation constants are extremely difficult to measure in
practice. The saturation constants for S and Q, and the maximum specific growth rates on
H, S and Q were taken from previously published data (Knoll, 1994); (Setiawan and
Worden, 1997). Table 4 lists the parameter values used for modeling in both the bulk
fluid and porous matrix simulations. Computational results for both the random motility

and chemotactic sensitivity coefficients for the bulk and porous regions are also

69

represented in Table 4. The analysis from which these coefficients were obtained is

discussed in the next section.

Table 4. Parameter values used in mathematical model.

 

 

E’aramter Syrrbol Vaiuctorauk ValueforPorous
random motility ccetiicient 11 1.96x10" cm2 hr" 240x10” cm2 I!”
chemotacticsensitivitycoetiicierttornitiate X00 0043an hr" 1.38x10‘20m’ hr"
diemotaoticsensitivityooeflioientforaoetate X105 5.87x10"ar12 hr" 1.88x10‘3anzlr"
hair-sanitationcorstmtforgmmon nitrate Co 1.20x10’5900n" 1.20x10‘gqcm"
hdf-satuaionoonstartforgoufl'ionaoetate Cs 1.00x10‘gscm° 1.00x1o"gscrn‘3
diffusion coefficient for nitrate Do 0.07 cm2 hr“ 0.07 cm2 hr“
truism coefficient for acetate Ds 0.04 on2 hr" 0.04 cm’ hr"
dssociation corstatttorreccptcr-atuactant oorrplexfor nitrate Koo 3.30mi5 goon“ 3.30::10‘5 gqcm"
cissociationoonsmrorreceptor-amactait oonpiexioracerate Kos 200x10‘gscm° 2.00x10‘gsan°
rnaximmspedficg'umi rateon ritrate Vo 0.13 hr" 0.1311r'1
maxinun specificgmntn tateon acetate Vs 0.13 hr" 0.13 hr"

yield ooeflioientforg'mflrmnitrate Ya 01809190 031309190
yieldeoeflidentforgmflronaoetde Ys 0.2219191: 0221919:

 

 

 

 

 

70

2.4 Results and Discussion

As stated before, the goal of this study was to make use of a mathematical model
to estimate the chemotactic sensitivity coefficients X08 and X0Q (for chemoattractants
acetate and nitrate respectively) for both the bulk and porous medium

With a known value of the random motility coefficient in the bulk medium (no)
for PKC, the values for X08 and X0Q were determined by varying these coefficients in the
mathematical model until they matched the experimental results to some level of
satisfaction. To compare the model to the experimental method visually, the relative
density of the cell population and the diameter of the growth ring of cells were used in a
series of plots. The plots were then compared and agreement indicated a result. The
results indicated that X0Q was rather insensitive to the concentration of nitrate in the
surrounding medium, but XOS was very dependent on the relative concentration of acetate.
An average value between these two extremes was used in the model. Figure 20 and
Figure 21 compare the computer simulation of PKC migration in the left-hand column
and experimental photographs in the right-hand column for two different concentrations
of acetate and nitrate at different time points. Table 5 reports the individual values for
X08 and X0Q that provide the best agreement between the simulation and experiment, as
well as the mean values for XOS and X0Q that resulted from averaging the results obtained
for the six different concentrations of acetate and nitrate. Each optimization required a
pattern that was formed by first setting one coefficient equal to zero and then fitting the
remaining coefficient. The first coefficient was then increased until agreement was no

longer met. This process was then reversed, and eventually optimized values were

 

71

Time = 48.5 hours

 

Figure 20. Simulation compared to experiment for 0.1 g/L Acetate
and 0.42 g/L Nitrate.

 

72

I lnlt' 48.531uuru

 

11111: 71131101113

   

'l mic "’5 1111119.

 

Figure 21. Simulation compared to experiment for 0.5 g/L Acetate
and 2.5 g/L Nitrate.

73

obtained. In some of the figures, the plots corresponding to the model take on a shape
resembling more of a diamond, rather than a circle. The direct cause of this phenomenon
has not yet been determined, however, it has been surmised that this is an artifact of the
program. The cell patterns developed in response to the underlying, time-dependent
concentration profiles of the chemoattractants and nutrient. Examples of latter profiles
are shown, along with the corresponding cell profile, in Figure 22 and Figure 23. Both a
graphs have the concentration, g/cm’, on the vertical axis, and the spatial position, in cm, I

on the horizontal axes. The jagged edges are artifacts of the graphing program.

 

 

Table 5. Values found for XOS and X0Q using mathematical model. Li
Medium Concentration cm2 hr‘1 5% cm2 hr‘1 ,

0.1 g/L Acetate 0.083 g/L Nitrate 0.01 0.08

0.1 gIL Acetate 0.42 glL Nitrate 0.0085 0.003

0.1 g/L Acetate 2.50 g/L Nitrate 0.013 0.08

0.5 g/L Acetate 0.083 g/L Nitrate 0.0015 0.027

0.5 gIL Acetate 0.42 gIL Nitrate 0.001 0.008

0.5 gll. Acetate 2.5 g/L Nitrate 0.0012 0.06
Average 5.87x10’3 0.043

 

 

 

 

 

Past research has indicated thath can deviate from the mean value at the lowest
concentration because a different signaling mechanism is utilized at these low attractant
concentrations (Ford and Lauffenburger, 1991). This may explain why XOS differed
widely between 0.1 g/L of acetate and 0.5 g/L of acetate; however this same order of
magnitude was also present in the nitrate concentrations used, and no such significant

deviation occurred.

74

Time = 96 hour

   

Figure 22. Simulation compared to experiment for 0.1 g/L Acetate
and 0.083 g/L Nitrate.

Time = 101.5 hours

\.

      

Figure 23. Simulation compared to experiment for 0.5 g/L Acetate
and 0.42 g/L Nitrate.

 

75

With set values of XI,S and X00 for the bulk medium, a value for um. was
determined using Equation 1. The Schoolcrafi sand used in the experiments had a
porosity (e) of 39.9% (Criddle, er. al., 1997) and an estimated tortuosity (t) of 3.25
(Duffy, et. al., 1995). It should be noted that the tortuosity was estimated using a particle
diameter of 200 um (Criddle, et. al., 1997) and the simulation results of Duffy and his co-

workers. From equation 1, an 88% reduction in the effective value 11,“ relative to the

 

value in the bulk 110 is predicted. Previous research has predicted that the chemotactic

sensitivity coefficient will be reduced by the same factor as the random motility

 

coefficient (Barton and Ford, 1997). In the same manner as before, the values for XOM J
and X00,” were determined by varying these coefficients in the mathematical model until
they matched the experimental results. Experimentally, this was done by determining the
time at which the cells first broke through the porous barrier and then using the model to
match this time. Instead of an 88% reduction for XOS and X00, the model predicted a 68%
reduction in these coefficients. These values have an error of within 10%, in that if both
values are increased or decreased by 10%, the model no longer fits the experiment.
Figure 22 and Figure 23 show the computer simulation in the left-hand column and
experimental photographs in the right-hand column for two different concentrations of
acetate and nitrate at different time points. These figures do not show an exact fit.
However, lower values for the effective parameters would have made the comparison
with the lower concentration of acetate much worse. Likewise, higher values for the

effective parameters would have made the comparison much worse for the higher

76

concentration of acetate. Therefore, the values obtained provide a balance between these
two cases.

Some disagreement is observed in the case of the chemotactic sensitivity
coefficient for acetate and nitrate in the porous medium. In some cases (in results not
shown), the presence of the porous medium seemed to expediate the chemotactic effect

and the cells migrated through the medium at a faster rate. For the above results, the

:1 IQUL,“

migration rate was slowed by the porous medium, but when the diameter of the porous

region was decreased by 3A, an increase in the migration rate was observed. When the

 

Fart-3'1“ '

model was matched to these results, values of .587 cmzhr‘l and 4.3 cmzhr" were found for

I

XOS and XOQ respectively, which is an 100% increase over the effective values. Figures
24 and 25 show the results from these calculations. With such conflicting results, the
validity of the computer model is brought into question. The computer model should be
capable of predicting the experimental results observed at both sizes of the porous region.
Numerous assumptions were made and since the estimated values for the
parameters can not be measured or reported for the specific experimental system, the
particular values for the transport coefficients are subject to some degree of uncertainty.
Also, the chemotactic sensitivity coefficients for acetate and nitrate have not been
previously calculated. However, the bulk values are within a range of values found for
other chemical substances (Ford, 1992). Another consideration is that there may be
surface interactions taking place between the bacteria and the sand, and this could be
expected to significantly reduce the random motility. Currently, the model does not take

these interactions into account. Future work could also include determining u,ff through

77

Time = 41 hours

 

Figure 24. Simulation compared to experiment for 0.5 g/L Acetate
and 0.083 g/L Nitrate

Time = 41 hours

      

Figure 25. Simulation compared to experiment for 0.1 g/L Acetate
and 0.083 g/L Nitrate.

 

78

experimental means. This could be done by somehow “turning off” the chemotactic
contribution and letting the cells migrate out purely through random motility.
Experimentally, this might be accomplished by saturating the receptors with a high level
of attractant. The motility patterns that occur are strictly dUe to random motility and

growth through the medium.

 

 

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'W
V.
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