i

EXPANDABLE MONOLAYER CAPACITY FOR

MODELING ADSORPTION COMPRESSION SYSTEMS

By

Daniel Bier

A master’s essay submitted to Johns Hopkins University in conformity with the

requirements for the degree of Master of Chemical and Biomolecular Engineering

Baltimore, Maryland

September, 2016

© 2016 Daniel Bier

All Rights Reserved

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Abstract Adsorption is the process by which gas and liquid molecules reversibly bind to a

surface. The phenomenon is driven by attractive forces between molecules and a surface

but can also be affected by inter-molecular interactions. Adsorption occurs all around us:

whether you are reading this thesis on a paper or screen, molecules of oxygen and

nitrogen are constantly adsorbing and desorbing from the page or screen. Adsorption

also plays a major role in a variety of industrial applications, including catalysis,

chemical and biological separations, and more recently, self-assembly (1).

Traditionally, adsorption was considered an entirely attractive phenomenon.

From this perspective, molecules adsorb onto a surface because they are attracted to both

the surface and each other. However, Donohue and Aranovich showed that molecules

can continue to adsorb onto a surface even when the molecules are repelling each other, a

phenomenon they called “adsorption compression” (2).

This paper explores the idea of an expandable monolayer capacity and developed

a model that provides a theoretical framework to describe adsorption compression. The

motivation for this is twofold. First, although there is abundant experimental evidence of

adsorption compression, there is no theoretical model that accurately describes the

physics of the phenomena. Second, even equations which account for adsorption

compression do not have a way to deal with a changing monolayer capacity. This makes

them fundamentally and practically inaccurate at high densities. The expandable surface

capacity presented here makes it uniquely able to explain and predict adsorbate behavior

during adsorption compression. This could be especially useful for catalysis, a $16

billion per year industry which relies specifically on adsorption (3).

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Acknowledgements

I would like to thank my research advisor, Professor Marc Donohue, not only for

his invaluable assistance throughout this project but also for his mentorship the past three

years. Besides teaching me most of what I know about modeling, he has shown me the

importance of questioning conventional explanations and how to think about problems at

a molecular level. I would also like to thank Professor Gregory Aranovich who I had the

fortune of working with this past year. He provided me with a great deal of help on

adsorption physics as well as understanding experimental data. This project certainly

would not have been possible without the long hours these two people spent with me

postulating and re-postulating ideas on adsorption compression mechanisms and surface

capacity.

I would also like to thank my parents, Jeri and Steve, for always believing in me

the past twenty three years. From the word box to listening to my thesis presentation, the

two of you have been my best teachers.

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Table of Contents Abstract ....................................................................................................................................... ii

Acknowledgements .................................................................................................................... iii

Table of Figures ........................................................................................................................... v

Literature Review ........................................................................................................................ 1

Adsorption vs. Absorption ....................................................................................................... 1

States of Matter – Discrete and Continuous ............................................................................ 2

Adsorption Background........................................................................................................... 5

Historical and IUPAC Isotherms ............................................................................................. 6

Classical Ono-Kondo Equation ............................................................................................. 13

Lennard-Jones Potential Function ......................................................................................... 15

Variable Capacity Corrections .............................................................................................. 18

Modeling and Data Analysis ..................................................................................................... 20

Effect of Varying Parameters on Different Isotherm Types .................................................. 20

Evidence of Adsorption Compression in Past Experiments .................................................. 28

Expandable Capacity Model .................................................................................................. 34

Conclusions ............................................................................................................................... 47

Literature Cited .............................................................................................................................. 48

Appendix ................................................................................................................................... 50

2 Site Model (Mathematica) .................................................................................................. 50

Varying Parameters for soft molecule model (Mathematica)................................................ 51

Variable Capacity Code (Matlab) .......................................................................................... 52

Curriculum Vitae ....................................................................................................................... 57

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Table of Figures

Figure 1: Microscale adsorption (right) and absorption (left). In adsorption, molecules

adhere to a surface. In absorption, molecules are taken into a volume (27) ...................... 1

Figure 2: Macroscale adsorption (left) and absorption (right) (25) ................................... 1

Figure 3: Hexagonal packing lattice. This is generally the most energetically favorable

packing structure. 91% of surface area filled...................................................................... 2

Figure 4: Square packing lattice. 78% of surface area filled. Also referred to as cubic

packing in three dimensions................................................................................................ 2

Figure 5: Depicts the transition of the first column from square to hexagonal packing. In

a large system, these types of transitions represent a nearly continuous change in packing

density rather than a series of discrete transitions. The system is likely going from a

square commensurate packing to a hexagonal incommensurate packing. .......................... 3

Figure 6: Depicts a single column compression to fit a 9th molecule in the first column.

This introduces different magnitudes of inter-molecule forces since molecules line up at

inconsistent distances. ......................................................................................................... 4

Figure 7: Commensurate system with a striped surface. Adsorbate molecules sit in the

sites created by the surface molecules (27). Although adsorbed density is lower than in a

hexagonal lattice, favorable adsorbate-adsorbent attractions are maximized ..................... 5

Figure 8: Incommensurate system with a striped surface. Adsorbate molecules do not sit

in the sites created by the surface molecules due to larger molecule size. Favorable

adsorbate-adsorbent attractions are given up for favorable adsorbate-adsorbate

attractions. ........................................................................................................................... 5

Figure 9: Langmuir isotherm with different values of Єs for each curve. The x-axis of

these graphs is often given as xb, which is calculated from pressure through the use of an

equation of state. The black line on the bottom curve helps show the linear region which

occurs at low pressure. Systems with a higher surface energy have a greater level of

adsorption, but all curves approach x1 = 1 at high pressure. .............................................. 8

Figure 10: BET Isotherm from original 1938 paper by Brunaeur, Emmett, and Teller.

Shows multilayer adsorption with each line representing a different thickness of the

adsorbed layer. From Adsorption of Gases in Multimolecular Layers . Brunaeur, Emmett

and Teller ............................................................................................................................ 9

Figure 11: Freundlich Isotherm varying K. As K increases, the strength of the

adsorption increases as evidenced by the higher value of x1 at an equivalent xb. This

could denote a greater attractive force between adsorbed molecules and themselves or the

surface. .............................................................................................................................. 10

Figure 12: Freundlich Isotherm varying n. As n increases, the effect of a change is

pressure is more pronounced for low bulk densities. It does not change the final adsorbed

amount. Could represent attractive strength between molecules in the bulk are, with a

lower n denoting stronger attractions in the bulk, and hence lower levels of adsorption at

low bulk densities. ............................................................................................................ 11

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Figure 13: IUPAC Isotherm Classifications. Comparing experiental data to the IUPAC

isotherms can give information on the possible adsorption mechanism. Type I represents

a microporous surface. Types II and II represent a macroporous surface with strong (II)

and weak (III) adsorbate-adsorbent interactions. Types IV and V represent multilayer

adsorption with hysteresis, which leads to a desorption curve higher than the adsorption

curve. Type VI represents either multilayer adsorption or adsorption with phase

transitions. ......................................................................................................................... 12

Figure 14: Nearest neighbor diagrams for square (left) and hexagonal (right) lattices. In

a square lattice, molecules have four nearest neighbors. In a hexagonal lattice, molecules

have six nearest neighbors. For molecules of the same size, the nearest neighbor distance

is the same for either type of lattice. ................................................................................. 14

Figure 15: Lennard-Jones Potential Function for Argon. The energy between molecules

is shown by the blue line. The force between molecules is shown by the green line.

Force is the derivative of energy. The vertical dashed line goes through the minimum of

the energy function. To the left of this curve, force is repulsive or negative. To the right

of this curve, force is attractive or positive. ...................................................................... 16

Figure 16: Geometric illustration of equation 10. Density of 1/9 corresponds with

molecule distance = 3. This relation does not necessarily hold if the molecules are on a

lattice, but works to find the average distance for off lattice molecules. .......................... 17

Figure 17: Adsorption Isotherm at different temperatures. At low levels of adsorption,

all isotherms overlap (Langmuir region) because there are insufficient molecules for

inter-molecule attractions to occur. At lower temperatures, the isotherm is S-shaped due

to the relationship between the high surface energy and high inter-molecule attractions

which start being generated around xb = 1. Around xb = 0.7, the isotherms start crossing.

This is due to the Lennard-Jones potential, which, at low temperatures, is both more

attractive at low density and more repulsive at high density compared to higher

temperatures. ..................................................................................................................... 21

Figure 18: Adsorption Isotherm with different Єs values. X1 is higher when the surface

attraction is stronger. The final adsorbed density is also higher with a stronger surface

energy. ............................................................................................................................... 22

Figure 19: Adsorption Isotherm at different Єa values. The isotherms overlap initially

(Langmuir region) because inter-molecule forces are negligible at low surface density.

The isotherms intersect around xb = 0.8 because the system with a large Єa value has

both stronger attractions at low density and stronger repulsions at high density compared

to systems with a smaller Єa value. ................................................................................... 23

Figure 20: Two-site Model. On Surface A, sites are far apart and no compression

occurs. On Surface B, sites are moderately close, the second molecule is more difficult to

add, and compression occurs. On Surface C, sites are so close that only one molecule can

adsorb regardless of bulk density. (1) ............................................................................... 24

Figure 21: Two-site model for different values of molecule size to site distance. The x-

axis shows chemical potential and the y-axis shows the average number of molecules

adsorbed (out of two potential adsorption sites). When the molecule size is only slightly

larger than the space between sites, the average number of sites filled approaches two at

high chemical potential. As molecule size increases or site distance decreases, the

average number of molecules adsorbed decreases until only one molecule can fit. ......... 26

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Figure 22: Two-site model for different values of molecule size to site distance. This is

the same data as the previous plot but in xb vs. x1 isotherm form. When the molecule

size is only slightly larger than the space between sites, the average number of sites filled

approaches two at high chemical potential. As molecule size increases or site distance

decreases, the average number of molecules adsorbed decreases until only one molecule

can fit. ............................................................................................................................... 26

Figure 23: Two-site model, average molecules adsorbed at different Єs values. With a

higher surface energy, the adsorbed fraction is higher. The average number of filled sites

is also higher at maximum chemical potential. ................................................................. 27

Figure 24: Two-site model, isotherms for different Єs values. This is the same data as

the previous plot. With a higher surface energy, the adsorbed fraction is higher............ 28

Figure 25: Xenon on MgO isotherm from Adsorption by Powders and Porous Solids:

Principles, Methodology and Applications by Rouquerol, Françoise, Rouquerol, Jean and

Sing, 1999 (21). Each curve represents a different temperature, shown in the original

caption. .............................................................................................................................. 29

Figure 26: To scale diagram of Xenon on MgO adsorption system. The left system

represents a likely low density configuration where the xenon molecules are sitting at the

optimal distance from the surface. On the right are several possible configurations at

high density. On the top right is a commensurate system with high inter-molecule

repulsions. The bottom right diagram shows an incommensurate system. In between

these are various configurations where inter-molecule and molecule-surface interactions

are balanced in different ways. (23) (22) .......................................................................... 31

Figure 27: Xenon on MgO data plotted in New Ono-Kondo Coordinates. In this

coordinate system, positive slopes denote molecule attractions and negative slopes denote

molecule repulsions. Many of these systems shift from attractive to repulsive and back to

attractive to repulsive and back to attractive forces between molecules. (24) ................. 32

Figure 28: New Ono-Kondo Coordinates for curve “e” of Xe on MgO data. This

isotherm shows three regions of attractive forces followed by regions of repulsive forces

for the first two.................................................................................................................. 33

Figure 29: Isotherm for plot e of Xe on MgO data. The circled points represent regions

where the molecules are likely attractive even though adsorbed density is already high.

This could be due to a change in packing. ........................................................................ 34

Figure 30: Isotherm from Abaza thesis simulations. Number of molecules adsorbed is

plotted against chemical potential. This graph shows evidence that rather than adding an

entire row or column at once, the system adds molecules one row at a time. (6) .......... 36

Figure 31: Monte Carlo from Abaza thesis simulations. The surface is large enough to

hold 16x16 molecules with hexagonal packing. It is notable that even though the inter-

molecule repulsions increase at higher chemical potential, even at 18x18 (bottom right),

there are still vacancies denoting inter-molecule attractions. (6) .................................... 37

Figure 32: Isotherm from variable capacity model with different Єsvalues. As expected,

the isotherms with a higher Єs show a higher degree of adsorption compression. .......... 39

Figure 33: Final dimensions with different Єs values. With a larger surface energy, there

is a higher degree of adsorption compression. .................................................................. 39

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Figure 34: Change in vacancies over adsorption from variable capacity model. Initially

one vacancy is lost for each additional molecule adsorbed. Eventually, it is more

favorable to add a site when a molecule is adsorbed. This manifests itself in the graph

with a horizontal jump. ..................................................................................................... 40

Figure 35: Adsorption isotherms from variable capacity model with different Єa values.

At low adsorbed density, adsorption is higher when Єa is large because of higher

attractive forces. At higher densities, the repulsive force is higher for a large Єa, and

adsorption is lower. ........................................................................................................... 41

Figure 36: Change in vacancies over adsorption from variable capacity model.

Depending on the value of Єa, new sites are added at different adsorbed densities. ....... 41

Figure 37: Change in sites over adsorption from variable capacity model. Depending on

the value of Єa, sites are added at different points of the adsorption. Ultimately, more

sites are added when Єa is smaller because of the lower inter-molecule repulsions at high

adsorbed density................................................................................................................ 41

Figure 38: Final dimensions of adsorbed layer and degree of adsorption compression

with different Єa values. The degree of adsorption compression is greater for a system

with a lower Єa value due to lower inter-molecule repulsions at high adsorbed density..42

Figure 39: Isotherms in New Ono-Kondo coordinates for varying Єs. The initial slopes

are all equal because inter-molecule forces are equal. The different y-intercepts

correspond precisely with the different surface energies. The shift from positive to

negative slope denotes the change from inter-molecule attractions to repulsions. ........... 44

Figure 40: Isotherms in New Ono-Kondo coordinates for varying Єa. The y-intercepts

are all equal because surface energies are equal in each systen. The different slopes

correspond precisely with the different Єa values. The shift from positive to negative

slope denotes the change from inter-molecule attractions to repulsions. ......................... 45

1

Literature Review This literature review begins with a discussion on adsorption background, including

states of matter, lattice theory, historical and experimental isotherms, and the Ono-Kondo

equation of adsorption. Once this is established, the paper discusses a set of equations that

attempt to account for adsorption compression with an expanded but constant monolayer

capacity. The equations can be manipulated to produce the New Ono-Kondo coordinates (4), a

useful way to interpret experimental data. These equations and coordinates become the basis for

the adsorption model presented.

Adsorption vs. Absorption Before continuing, it may be helpful to briefly distinguish adsorption from its sister

sorption process, absorption. While adsorption is the binding of molecules to a surface,

absorption is the uptake of molecules into a volume. This can be humorously depicted as a cake

being eaten: “absorption”, and a cake which has been thrown at somebody’s face: “adsorption”.

[cake pics]

With this distinction made, the remainder of the paper will focus on adsorption.

Figure 1: Macroscale adsorption (left) and absorption (right) (25).

Figure 2: Microscale adsorption (right) and absorption (left). In adsorption, molecules adhere to a surface. In absorption, molecules are taken into a volume (27).

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States of Matter – Discrete and Continuous The three common states of matter – gas, liquid, and solid – are generally described in

terms of inter-molecular distance and orderliness. For a decrease in temperature or an increase in

pressure, molecular density generally increases. The common exception to this is the liquid to

solid transition of H2O, where density decreases due to a change in molecule packing. For a

simple theoretical system, this increase in molecule density manifests itself in the form of two

discrete phase transitions: gas to liquid and liquid to solid. This behavior is usually what

happens to molecules in bulk. However, this three state classification is insufficient for

describing most adsorption systems since the adsorbate molecules are strongly influenced by

surface interactions.

Aside from the three states of matter, a second type of transition that can occur as

molecule density increases is a rearrangement in packing shape. The two most common types of

lattices are cubic (or square in 2D) and hexagonal, with hexagonal generally being the lower

energy, or more favorable, state.

Figure 3: Hexagonal packing lattice. This is generally the most energetically favorable packing structure. 91% of surface area filled.

Figure 4: Square packing lattice. 78% of surface area filled. Also referred to as cubic packing in three dimensions.

3

Although both of these systems would be classified as solids due to the orderly arrangement of

the molecules, the cubic lattice has 22% void space and the hexagonal lattice has 9% void space

making it the densest packing arrangement (5). At a certain molecule density, the system could

change from cubic to hexagonal packing. This would represent a discrete transition separate

from the gas-liquid or liquid-solid transitions.

However, in Monte Carlo simulations (6), there is evidence that rather than a single

transition of this sort, packing transitions might occur one row at a time.

Figure 5: Depicts the transition of the first column from square to hexagonal packing. In a large system, these types of transitions represent a nearly continuous change in packing density rather than a series of discrete transitions. The system is likely going from a square commensurate packing to a hexagonal incommensurate packing.

When considering that a real-life system might consist of many moles of the adsorbate,

these by-row changes in packing represent a near continuous transition rather than a series of

discrete transitions.

In the case of adsorption compression, a third type of rearrangement can occur when a

single row gets compressed.

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As in the previous case, this can happen row by row, and then column by column,

producing a nearly continuous increase in the molecule density.

A fourth type of transition that an adsorption system can undergo is when the

monolayer’s distance from the surface changes. At an atomic level, the surface is always

corrugated to some degree due to the structure and placement of molecules at the surface of the

adsorbent. At lower densities, molecules might sit in the valleys of the surface’s electron cloud.

But if the molecules are larger than the sites created by the valleys, an increase in density might

make the inter-molecule interactions highly repulsive. In this case, the entire monolayer can lift

off the surface with a corresponding rearrangement in adsorbate packing. This is referred to as a

commensurate to incommensurate transition. In a commensurate system, the adsorbate

molecules fit in the sites created by the surface molecules. In an incommensurate system, the

adsorbate molecules are larger than the surface sites, and therefore must lift off the surface to

avoid repulsions with each other.

Figure 6: Depicts a single column compression to fit a 9th molecule in the first column. This introduces different magnitudes of inter-molecule forces since molecules line up at inconsistent distances.

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The modeling portion of this thesis focuses on the third type of rearrangement: density

increase by single row compression.

Adsorption Background Many processes rely either directly or indirectly on adsorption. In heterogeneous

catalysis, recent studies have shown that adsorption compression is important in lowering the

energy barrier for a chemical reaction to occur (7). In gas adsorption, a common separation

technique, adsorption is used to purify a contaminated gas by having the gas pass over a surface

that the contaminant can adsorb to. Chromatography, another separation technique, involves

passing a gas or liquid through a column where certain components get selectively adsorbed.

This is similar to the process used in most drinking water and wastewater purification systems

(8). More recently, scientists are exploring adsorption and its role in self-assembly by creating

ultrathin films with special optical and physical properties (9). In many of these cases, the

Figure 7: Commensurate system with a striped surface. Adsorbate molecules sit in the sites created by the surface molecules (27). Although adsorbed density is lower than in a hexagonal lattice, favorable adsorbate-adsorbent attractions are maximized.

Figure 8: Incommensurate system with a striped surface. Adsorbate molecules do not sit in the sites created by the surface molecules due to larger molecule size. Favorable adsorbate-adsorbent attractions are given up for favorable adsorbate-adsorbate attractions.

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process was developed without a complete understanding of the exact forces at play in the

system.

…many areas in which technological innovation has covered adsorption

phenomena have been expanded more through art and craft than through science.

A basic understanding of the scientific principles is far behind, in part because the

study of interfaces requires extremely careful experimentation if meaningful and

reproducible results are to be obtained (10).

However, a better understanding and a more robust model of adsorption could be used to make

these processes more effective and efficient.

In the context of adsorption, the surface is named the adsorbent and the molecules are

referred to as the adsorbate, or adsorbate molecules. Adsorbate molecules can be in one of two

states: freely moving in the bulk or adsorbed onto the adsorbent. To distinguish between these

two states, two variables are defined: bulk concentration xb and adsorbed concentration x1. By

plotting x1 as a function of xb, certain properties of an adsorption system can be seen. Generally

a plot of xb versus x1 is made at a constant temperature, making the curve an isotherm. Isotherms

vary greatly depending on the mode of adsorption and the molecules involved. Therefore,

correctly interpreting experimental isotherms can give scientists a lot of information about the

system they are studying. This has led to a “renaissance of isotherm modeling (8)”, as one paper

describes it.

Historical and IUPAC Isotherms When molecules adsorb onto a surface, they can start forming additional layers even

before the first layer is complete. However, as a simplifying assumption, the adsorbed layer is

often considered to be a monolayer (11). With this assumption in place, we can generate the

widely used Langmuir isotherm, which utilizes the additional simplifying assumptions that there

are no inter-molecule interactions and all molecule-surface interactions are equal (10).

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The mathematics behind Langmuir adsorption starts with an equilibrium relationship

between the adsorbed and bulk state (12) (13) (14).

Where [A] is the concentration of adsorbate molecules in the bulk, [S] is the

concentration of surface sites, and [AS] is the concentration of filled sites. If the rate constant

for the forward reaction is ka and the rate constant of the reverse reaction is kd, K can be defined

as ka/kd. Ɵ is then defined as the fraction of sites that are occupied, making (1- Ɵ) the

unoccupied fraction. The rate of the forward reaction/adsorption is then proportional to [A]* (1-

Ɵ)* ka, which is equal to the rate of desorption at equilibrium, Ɵ*kd. [A] is directly related to the

pressure, so making this substitution and solving for the fraction occupied gives the Langmuir

equation:

Langmuir isotherm gives 2 distinct regions. At low pressure, the equation reduces to

and adsorption is a nearly linear function of pressure. As pressure increases, there is a second

region where the adsorbed fraction approaches one asymptotically.

[Note: In a later section, the Langmuir equation is derived through an alternate path by using the

Classical Ono-Kondo Theory. In that section, K is shown to be a function of several other

variables, including the surface energy, Єs.]

The following plot shows the effect of changing the surface energy on the Langmuir

isotherm.

8

Figure 9: Langmuir isotherm with different values of Єs for each curve. The x-axis of these graphs is often given as xb, which is calculated from pressure through the use of an equation of state. The black line on the bottom curve helps show the linear region which occurs at low pressure. Systems with a higher surface energy have a greater level of adsorption, but all curves approach x1 = 1 at high pressure.

The expandable capacity model proposed by this paper utilizes Langmuir’s first

assumption - that adsorption occurs in a monolayer - both because monolayers do occur over a

wide range of conditions and because even in the case of multilayer adsorption, the first layer is

the key determinant of adsorption behavior (15). All molecule-surface interactions within a

given system are also considered equal, an assumption which breaks down under certain size

scales and geometries but which our model could easily be adapted to account for. However,

inter-molecule interactions are not set to zero, as this assumption is only valid at very low

surface densities and would require us to entirely ignore the region that can give adsorption

compression.

Aside from the 1916 Langmuir isotherm, there are several other notable isotherms. In

1938, Brunaeur, Emmett, and Teller published their famous paper on multi-layer isotherms. The

BET isotherm, as it was eventually called, essentially utilizes the same assumptions as the

Langmuir isotherm except that it allows for multiple layers to form. Specifically, an adsorbed

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Langmuir Isotherms at Different Surface Energies

xb

x1

Es = -10

Es = -5

Es = -3

Es = -1

xb

x1

9

molecule in one layer serves as the site for an adsorbate molecule in the next layer, with the

topmost layer in equilibrium with the bulk

phase adsorbate. The following diagram is from their original paper, and is an experimental

adsorption isotherm of Nitrogen onto an iron catalyst (16).

Each line represents a different thickness of the adsorbed layer.

Brunaeur, Emmett and Teller found that their equations predicted all the experimental data to

within 10%. The equation still does not take into account lateral inter-molecule interactions,

meaning it would not be able to predict or categorize adsorption compression. However, the

isotherm has proven very useful in characterizing the surface area of porous catalysts (15).

Another isotherm which takes a completely different approach than either Langmuir or

BET is the Freundlich isotherm, which utilizes empirical data rather than a theoretical

Figure 10: BET Isotherm from original 1938 paper by Brunaeur, Emmett, and Teller. Shows multilayer adsorption with each line representing a different thickness of the adsorbed layer. From Adsorption of Gases in Multimolecular Layers . Brunaeur, Emmett and Teller.

10

mechanism. Freundlich found that for many adsorption systems, the adsorbed fraction increases

with pressure by the equation

Where x is the mass of the adsorbate and m is the mass of the adsorbent (hence x/m = x1) (17). P

is bulk pressure and K and n are empirical constants based on the system at a given temperature.

The following two graphs show how an isotherm changes when varying K and n respectively.

Increasing K increases the strength of the adsorption. Physically, an increase in K would

represent a surface which more strongly attracts molecules or molecules which more strongly

attract each other while adsorbed. As a result, the final degree of adsorption is also higher with a

higher K value.

Figure 11: Freundlich Isotherm varying K. As K increases, the strength of the adsorption increases as evidenced by the higher value of x1 at an equivalent xb. This could denote a greater attractive force between adsorbed molecules and themselves or the surface.

11

Figure 12: Freundlich Isotherm varying n. As n increases, the effect of a change is pressure is more pronounced for low bulk densities. It does not change the final adsorbed amount. Could represent attractive strength between molecules in the bulk are, with a lower n denoting stronger attractions in the bulk, and hence lower levels of adsorption at low bulk densities.

Increasing n changes the shape of the isotherm, specifically by increasing adsorption at low

pressures. However, the final adsorbed capacity is unchanged.

Brunauer, and later IUPAC, have developed classification schemes in an attempt to

match the shape of the adsorption isotherm to the adsorption mechanism (18).

12

Figure 13: IUPAC Isotherm Classifications. Comparing experiental data to the IUPAC isotherms can give information on the possible adsorption mechanism. Type I represents a microporous surface. Types II and III represent a macroporous surface with strong (II) and weak (III) adsorbate-adsorbent interactions. Types IV and V represent multilayer adsorption with hysteresis, which leads to a desorption curve higher than the adsorption curve. Type VI represents either multilayer adsorption or adsorption with phase transitions.

If an isotherm resembles the type I isotherm, then the system likely has a microporous

surface. The next two isotherms (II and III) are for macroporous surfaces with strong (II) and

weak (III) adsorbate-adsorbent interactions. Types IV and V represent multilayer adsorption

with adsorption hysteresis. This leads to a desorption curve that is higher than the adsorption

curve. Specifically, type IV is multilayer with strong inter-molecule interactions, and V is

multilayer with weak interactions. Finally, isotherm VI is a stepped isotherm, which can occur

due to a phase change, commensurate-incommensurate transition, or a rearrangement (such as a

shift from a cubic to hexagonal lattice). As Donohue and Aranovich note, although the IUPAC

isotherms cover a broad range of adsorption systems, they do not cover all systems and assume

that all isotherms are monotonically increasing functions of pressure (18). They found that the

Ono-Kondo theory is able to predict all known types of adsorbent behavior.

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Classical Ono-Kondo Equation This section will go through a derivation of the Ono-Kondo equation based on Donohue

and Aranovich’s paper Intermolecular Repulsions in Adsorption Systems (2).

For a system in equilibrium, a molecule may adsorb onto the surface or desorb back into

the bulk with no net change of free energy, i.e.

where H is the enthalpy term which consists of molecule-surface interactions and inter-molecule

interactions in the bulk and on the surface9. S is the entropy term, which is based on the number

of configurations the system can be in for a given xb and x1.

To obtain ΔH as a function of x1 and xb, it can be written as:

where the three Є values are related to the energy of interactions for molecule-surface, inter-

molecule in the adsorbed layer, and inter-molecule in the bulk layer respectively. Usually all

three of these energies are treated as constants, but in treating adsorption compression Єa is

varied with density based on the average distance between molecules. The zx constants are

coordination numbers which represent the average number of interactions that each molecule has

experiencing the corresponding energy. For example, on a square lattice monolayer, a molecule

has four neighbors. Therefore z1 would be four (effectively multiplying Єa by four), whereas z1

would be six in a hexagonal lattice (Figure 14)

14

Figure 14: Nearest neighbor diagrams for square (left) and hexagonal (right) lattices. In a square lattice, molecules have four nearest neighbors. In a hexagonal lattice, molecules have six nearest neighbors. For molecules of the same size, the nearest neighbor distance is the same for either type of lattice.

These coordination numbers are just an approximation; for the example above the red circle in

the cubic lattice is considered to have four neighbors (blue), with all other circles providing no

interaction. In reality, each of the other circles would provide varying degrees of interaction

based on their proximity. This is referred to as the “nearest neighbor approximation.”

To obtain ΔS as a function of x1 and xb, it can be broken into

Where W1 is the number of system configurations in which an arbitrary site on the surface

contains an adsorbate molecule and a site in the bulk is empty. Conversely, W2 is the number of

configurations where the site on the surface is empty and the site in the bulk contains the

molecule. K is the Boltzmann constant. If W0 equals the total number of system configurations

W1 + W2, then

( ) ( )

ΔS can then be written as

(( ) (( ) )

Combining the expanded equations for entropy and enthalpy gives

( )

( )

15

This is the Ono-Kondo equation of adsorption. The full equation is nonlinear and has no analytic

solution. However when Єa and Єb are zero it reduces to the Langmuir isotherm. To generate the

Langmuir, Єa and Єb are set to zero and x1 is solved for, giving

( ) ( )

This equation is used to plot the Langmuir isotherm. As Єs increases, the initial linear region

gets steeper, meaning the value of x1 is higher for a given value of xb. This is expected: a higher

surface energy should increase the degree of adsorption.

Lennard-Jones Potential Function Although the Ono-Kondo equation above is derived based on lattice theory, it can be

adapted to account for off-lattice molecules by replacing the constant Єa with a function that

changes with average molecule distance. The most commonly used potential function is the

Lennard-Jones Potential. The function gives potential energy, ϕ, as a function of molecule

distance, r.

The Lennard-Jones equation takes the form

( ) (

)

(

)

where Є is in units of energy and describes the interaction potential between molecules, and σ is

in units of length and describes the molecule size or van der Waals radius (19).

16

Mathematically, when r is large, the (σ/r)6 term dominates. With a negative value for ,

this produces a negative value of . Physically, this is representative of attractive van der Waals

forces that occur between the electron clouds of molecules. As the distance decreases, the (σ/r)12

terms becomes more important until the function hits a local minimum (assuming Є is negative).

This r represents the distance that the molecules would sit from each other at equilibrium in the

absence of any other forces. At smaller distances than this, the potential starts increasing steeply

as the (σ/r)12 term dominates. Physically, this represents the molecules repelling each other due

to overlapping electron clouds. Replacing Єa with this function allows the adsorbate molecules

to be “soft”, meaning they can become compressed.

0 1 2 3 4 5 6 7-5

0

5Lennard Jones Potential Energy and Force

Distance Between Molecules

Energ

y, F

orc

e

Energy

Force

Figure 15: Lennard-Jones Potential Function for Argon. The energy between particles is shown by the blue line. The force between molecules is shown by the green line. Force is the derivative of energy. The vertical dashed line goes through the minimum of the energy function. To the left of this curve, force is repulsive or negative. To the right of this curve, force is attractive or positive.

17

Physically, the minimum value of the potential energy occurs when the potential is -Є, or

the force is zero. The other parameter σ is the radius at which the potential energy goes from

attractive to repulsive. You can also find the value of r associated with the minimum in the

potential by taking the derivative of the Lennard-Jones potential function. The result of this

gives

The variable r is a function of the density of molecules on a surface. r is obtained from adsorbed

density, or x1, through the relation equation

where D is the molecule diameter. This relationship can be seen from a diagram

Figure 16: Geometric illustration of equation 14. Density of 1/9 corresponds with molecule distance = 3. This relation does not necessarily hold if the molecules are on a lattice, but works to find the average distance for off lattice molecules.

where

and the average molecule distance is shown as 3*D, or D*(1/9)

-0.5

It might be worth noting that even though many other potential functions exist, the Lennard-

Jones function is the most easily integrated, greatly simplifying the math. However, our model

could just as easily replace the LJ function with any alternative potential function.

18

Variable Capacity Corrections One difficulty that arises when using the Ono-Kondo equation combined with Lennard-

Jones is that it mixes lattice theory and the corresponding assumptions with an off-lattice model.

To illustrate the conundrum, a system in which , then according to lattice theory, the

surface is filled and x1 = 1. However, the Lennard-Jones equation allows for more molecules to

be added to the surface until the system is at energetic equilibrium. But x1 cannot increase above

unity because the Ono-Kondo equations break down in this regime. To see this, one can look at

Equation 9, which is the entropy term of the Ono-Kondo equation. By plotting entropy as a

function of x1 with a constant xb, x1 is seen to approach unity, entropy decreases as expected.

However, after passing unity, entropy becomes an imaginary number, which has no physical

significance.

The other issue that arises is the definition of capacity. Although the left side of the

Lennard-Jones potential gets increasingly steep, there is no minimum distance, and therefore no

maximum system capacity. And a capacity that occurs at high bulk density does not necessarily

exist yet when bulk density is low or moderate. This dilemma will be addressed in full in the

modeling section of the paper.

In order to accommodate a system that can “overfill” the surface, there needs to be a

method to determine the high density monolayer capacity, which will be a function of the

molecule-surface and inter-molecule interactions of the adsorbed molecules. This will be done

using the method developed by Aranovich and Donohue (20).

We can solve for this adjusted capacity by writing an equilibrium relationship of ΔH = 0

in the following manner

(

)

19

Where g0 is a constant based on the radial distribution function and K is a packing factor. This is

the equilibrium at low temperatures where the TΔS entropy term is negligible, since the

maximum capacity will occur when the molecules have less energy, i.e. a low temperature.

Plugging in the Lennard-Jones potential equation for gives

(

)

Solving for am, the adjusted capacity of the system, gives

(

)

The K-1*σ-2 term is the capacity of a lattice without compression, which is solely a function of

the molecule size σ and the packing factor K. Now rather than using x1 as the term for adsorbed

density, it is replaced with a/am, with “a” being the adsorbed density in molecules per area and

“am” as the maximum adsorbed density.

Defining this new surface capacity allows continued use of the Ono-Kondo equations

because x1 is prevented from exceeding unity. However, when plotting isotherms, it is often

useful to use a/amh as adsorbed fraction, where amh is the hard molecule capacity, so that the

extent of adsorption compression can be easily seen. This is a mathematical solution to the

problem of adsorption compression, but it still has some physical issues when trying to

approximate a real system. For example, when bulk density is low, using the high bulk density

capacity for calculating entropy overestimates the number of potential system configurations.

These limitations are accounted for in the expandable capacity model.

20

Modeling and Data Analysis The modeling and data analysis section is broken into three parts. In Part 1, two

isotherms are generated by different methods: a lattice theory for soft molecules and a two-site

model. For each model, the effect of varying different parameters on adsorption is shown. In

Part 2, Xenon on Magnesium Oxide adsorption data from the highly cited text Adsorption by

Powders & Porous Solids (21) is used to show that when plotted in New Ono-Kondo

coordinates, there is clear evidence of adsorption compression. This text makes no mention of

adsorption compression, suggesting the phenomenon is often missed when looking at

experimental data. In Part 3, I use the expandable capacity model I developed to generate

isotherms and show how one can use New Ono-Kondo coordinates to calculate the surface

energy, initial inter-molecule energy, or the initial number of sites if one of the three is known.

Often the surface energy can be obtained from Henry’s constant, making this an effective

method.

Effect of Varying Parameters on Different Isotherm Types

Adsorption Compression Model with a Constant Soft Molecule Capacity

The soft molecule model determines a surface capacity am which allows compressions

and is based on the Єs and Єa at xb equals unity (20). This capacity serves as the denominator of

x1.

Three basic system parameters that can be varied are temperature, Єs, and Єa . To see the

effects of changing these parameters on the system, isotherms can be plotted of xb versus x1,

where x1 is equivalent to a/amh. Using amh rather than am allows us to see how “compressed” the

system is.

21

The first parameter varied is temperature, T. Temperature affects both the surface and

molecule energy, evident from the term Єx/kT where k is the Boltzmann constant. From a

mathematical standpoint, it is known that increasing temperature will decrease x1 for a given xb

since T is in the denominator. This also makes sense physically. Temperature is related to the

average kinetic energy of molecules, and molecules that are moving more quickly are less likely

to stick together. This phenomenon is taken into account in many equations of state for gases,

where the attractive force is negligible at extremely high temperatures. The following plot

shows a series of isotherms at different temperatures.

Figure 17: Adsorption Isotherm at different temperatures. At low levels of adsorption, all isotherms overlap (Langmuir region) because there are insufficient molecules for inter-molecule attractions to occur. At lower temperatures, the isotherm is S-shaped due to the relationship between the high surface energy and high inter-molecule attractions which start being generated around xb = 1. Around xb = 0.7, the isotherms start crossing. This is due to the Lennard-Jones potential, which, at low temperatures, is both more attractive at low density and more repulsive at high density compared to higher temperatures.

22

As expected, at lower temperatures the molecules will adsorb more quickly because they

have less kinetic energy. As T increases, a/amh decreases for a given xb. At low xb however,

each curve has the same slope. This can be considered the Langmuir region, because there are

not yet sufficient molecules for inter-molecule interactions, and therefore x1 is mainly a function

of xb and Єs. At T = 500 K, the S shaped behavior of the curve is lost. The S shape represents a

phase change; it indicates two different x1 values at the same bulk density. Above 500 K, it

becomes too hot for the molecules to start behaving like liquid, and they remain in a gaseous

state throughout the adsorption.

Another parameter to vary is Єs.

Figure 18: Adsorption Isotherm with different Єs values. X1 is higher when the surface attraction is stronger. The final adsorbed density is also higher with a stronger surface energy.

23

At lower values of Єs (more negative values), the surface provides a larger attractive

force, and x1 is higher. In the case of Єs = -10, x1 reaches 0.9 while xb is less than 0.01. This

looks like a vertical line unless the x-axis range is adjusted.

Finally, inter-molecule energy Єa can be varied.

As the inter-molecule energy becomes more attractive, the extent of adsorption increases for a

given xb until the molecules start repelling. At this point, an increase in xb will lead to less

adsorption in systems with a more negative Єa. This is why the isotherms cross each other

around xb = 0.8.

Ea 4

Ea 2

Ea 1

Ea 0.5

Figure 19: Adsorption Isotherm at different Єa values. The isotherms overlap initially (Langmuir region) because inter-molecule forces are negligible at low surface density. The isotherms intersect around xb = 0.8 because the system with a large Єa value has both stronger attractions at low density and stronger repulsions at high density compared to systems with a smaller Єa value.

0.0 0.2 0.4 0.6 0.8 1.0xb0.0

0.2

0.4

0.6

0.8

1.0

1.2x1

xb

x1

24

Two Site Model

The previous methods discussed for analyzing adsorption systems involve the

independent variable xb and the dependent variable x1, where x1 represents the fraction of

adsorbed molecules. For simulations, the number of molecules is often in the hundreds for

simplicity purposes, but in reality this fraction consists of a very large number of molecules.

However, an alternative method involves looking at just two molecules.

Rather than considering an entire adsorbate-adsorbent system, the two-site model only

considers the relative size and spacing of two active sites on the surface and the two molecules

that can adsorb onto them (1). If the spacing of the active sites is such that both can be occupied

without significant inter-molecule interactions, then as bulk density increases, the likelihood of

each site being occupied increases fairly linearly. However, in a system with closer together

sites, the average number of adsorbed molecules plateaus at 1, and then does not go up again

until xb increases more. Essentially, the first site is filled as expected, but the second site is not

filled until the considerable inter-molecule repulsions are overcome.

Figure 20: Two-site Model. On Surface A, sites are far apart and no compression occurs. On Surface B, sites are moderately close, the second molecule is more difficult to add, and compression occurs. On Surface C, sites are so close that only one molecule will adsorb regardless of bulk density. (1)

25

To examine the two-site model mathematically, the grand canonical partition function, Ξ,

is calculated for a system in which the bulk phase is in equilibrium with two surface sites. There

are four possible states for this system: State 1 where neither site is filled, State 2 and State 3

where one or the other site is filled, and State 4 where both sites are filled. If both sites are filled,

the Lennard-Jones potential is used for the energy between the two adsorbed molecules.

The grand canonical partition function then becomes

( ) ( ) ( )

With the first term representing State 1, the second term representing States 2 and State 3, and

the fourth term representing State 4. is the chemical potential, which is directly related to xb

by the equation

( )

The average number of molecules on the surface (which can be between 0 and 2) then

becomes:

( ( ) ( ))

This gives another important parameter to vary: the ratio of sigma to the distance between sites.

This parameter determines if there will be repulsions or attractions when both sites are filled, and

how strong either force will be. Assigning the variable σ/r to this, the LJ function can be written

as

( ) ( ) ( )

With all other parameters held constant, the number of sites filled goes up as the ratio of

sigma to well size increases.

26

Figure 21: Two-site model for different values of molecule size to site distance. The x-axis shows chemical potential and the y-axis shows the average number of molecules adsorbed (out of two potential adsorption sites). When the molecule size is only slightly larger than the space between sites, the average number of sites filled approaches two at high chemical potential. As molecule size increases or site distance decreases, the average number of molecules adsorbed decreases until only one molecule can fit.

N

Figure 22: Two-site model for different values of particle size to site distance. This is the same data as the previous plot but in xb vs. x1 isotherm form. When the molecule size is only slightly larger than the space between sites, the average number of sites filled approaches two at high chemical potential. As particle size increases or site distance decreases, the average number of molecules adsorbed decreases until only one molecule can fit.

xb

x1

<N> (average number of molecules adsorbed) at different σ/r values

27

There are several distinct ranges of σ/r to note. If σ is much larger than r, the molecules do not

have a significant interaction and two molecules adsorb onto the surface fairly easily. As the

ratio decreases, it takes a higher bulk density to reach the same x1 values. For example, in the

above figure, σ/r values of 1, 1.02, and 1.04 all eventually approach x1=1, but it takes longer for

the curves with higher values of sigma. As σ/r continues to increase, the curve ends up with a

lower final value of x1.

Another parameter to vary is the surface energy. For a constant σ/r ratio and Єa,

decreasing the surface energy decreases the expected number of molecules.

Figure 23: Two-site model, average molecules adsorbed at different Єs values. With a higher surface energy, the adsorbed fraction is higher. The average number of filled sites is also higher at maximum chemical potential.

28

Figure 24: Two-site model, isotherms for different Єs values. This is the same data as the previous plot. With a higher surface energy, the adsorbed fraction is higher.

All of these isotherms have the same shape, they only differ in their steepness.

Evidence of Adsorption Compression in Past Experiments

As stated previously, scientists previously believed that the forces between adsorbed

molecules were either attractive or neutral. This is partly because when looking at an isotherm

plotted x1 as a function of xb, it is generally not possible to see the effects of adsorption

compression. However, Aranovich and Donohue showed that when you plot ln[x1*(1-

xb)/(xb*(1-x1)] as a function of x1, the natural log term has a positive slope for molecule

attractions, and a negative slope for molecule repulsions (4). They call these “New Ono-Kondo

Coordinates”.

We used these coordinates to look at Xenon on Magnesium Oxide data (21). This system

was expected to show adsorption compression because of the van der Waals radii of xenon and

magnesium oxide.

xb

x1

29

Below is data from the text Adsorption by Powders and Porous Solids. The author notes

that the isotherms shows clear evidence of either molecule rearrangements or complete phase

transitions. These are visible in the form of steps, marked by dashed lines.

Figure 25: Xenon on MgO isotherm from Adsorption by Powders and Porous Solids: Principles, Methodology and Applications by Rouquerol, Françoise, Rouquerol, Jean and Sing, 1999 (21). Each curve represents a different temperature, shown in the original caption.

The following table shows the van der Waals radius of the three molecules and the

covalent radius between magnesium and oxygen in the surface lattice:

30

Atom or Ionic Compound Van der Waals Radius (pm)

Xenon 216 (22)

Magnesium 220 (22)

Oxygen 155 (22)

Magnesium-Oxygen 211 (23)

This diagram represents a geometrically accurate depiction of the xenon-MgO system

using the values above. It is meant to provide a potential mechanism for the observed phase

changes.

31

The configurations above were created manually using geometry, not by a simulation of

potential energies.

When the data is plotted in New Ono-Kondo coordinates, the following graphs are

obtained:

Figure 26: To scale diagram of Xenon on MgO adsorption system. The left system represents a likely low density configuration where the xenon molecules are sitting at the optimal distance from the surface. On the right are several possible configurations at high density. On the top right is a commensurate system with high inter-molecule repulsions. The bottom right diagram shows an incommensurate system. In between these are various configurations where inter-molecule and molecule-surface interactions are balanced in different ways. (23) (22)

32

Figure 27: Xenon on MgO data plotted in New Ono-Kondo Coordinates. In this coordinate system, positive slopes denote molecule attractions and negative slopes denote molecule repulsions. Many of these systems shift from attractive to repulsive and back to attractive to repulsive and back to attractive forces between molecules. (24)

In each plot, there is a change from a positive to negative slope, indicating the shift from

attractive to repulsive forces. These correspond with the steps in the graph, which the author

identified as the rearrangements of molecules. This provides strong evidence that the system’s

lattice changed to accommodate more molecules by making the inter-molecule interactions

attractive again.

In several of the isotherms, including isotherm “e” below,

18

19

20

21

22

23

24

25

26

27

0 0.2 0.4 0.6 0.8 1

ln[x

1*(

1-x

b)/

(xb

*(1

-x1

))]

X1

Xenon on MgO Isotherms in New Ono-Kondo Coordinates

a

b

c

d

e

f

g

h

i

j

x1

96.86 K

100.47 K

106.20 K

108.44 K

111.02 K

116.14 K

118.72 K

121.15 K

126.17 K

131.19 K

x1

33

Figure 28: New Ono-Kondo Coordinates for isotherm “e” of Xe on MgO data. This isotherm shows three regions of attractive forces followed by regions of repulsive forces for the first two.

there are a second and third region which have a positive slope. These systems were able to

rearrange multiple times. In isotherm “e”, at x1 = 0.6, 0.7, and 0.8,the system underwent a

configuration change which allowed the adsorbed molecules to attract each other again. When

looking at the original isotherm for this plot,

20.5

21

21.5

22

22.5

23

23.5

24

0 0.2 0.4 0.6 0.8 1

log[

x1*(

1-x

b)/

(xb

*(1

-x1

))]

x1

Xenon on MgO Plot "e", 111.02 K in New Ono-Kondo Coordinates

34

Figure 29: Isotherm for plot e of Xe on MgO data. The circled points represent regions where the molecules are likely attractive even though adsorbed density is already high. This could be due to a change in packing.

one can see the second and third times where the inter-molecule interactions become attractive

again (circled in red). Since the authors did not notice the presence of adsorption compression,

the method of plotting data in New Ono-Kondo coordinates is a useful tool for viewing these

molecule interactions. Other benefits of these coordinates are discussed in the next section.

Expandable Capacity Model

An Algorithm for Accounting for the Changing Capacity of a Surface

As shown previously, it is extremely difficult to determine when a surface is “full” for

two reasons. First, different parts of the lattice can sometimes rearrange to accommodate more

molecules. Second, as pressure increases, so do the allowable repulsions, which increases

capacity. But since all current equations for adsorption assume a constant capacity during

increases in xb, they have a fundamental flaw which causes problems when considering the

entropy and the rest of the system. Basically, even if the adsorbed molecules will eventually

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

x 1

xb

Xenon on MgO Isotherm "e"

35

rearrange or compress to fit more molecules, it is incorrect to use the final capacity “am”

throughout the adsorption because it assumes more configurations for the system then are

possible at that value of xb.

The xenon on magnesium oxide data previously discussed supports this idea that the final

capacity is distinct from the initial capacity, because the system goes through phases of molecule

attractions followed by repulsions. In order to have attractions again at a higher bulk pressure, it

is necessary that the capacity must have increased.

Abaza also found evidence of this phenomenon. She modeled adsorption using Monte

Carlo simulations based on probability of the system configuration. As xb increased in the

simulations, the size of the lattice underwent several jumps: 16x16 to 16x17 to 17x17 (6).

36

Figure 30: Isotherm from Abaza thesis simulations. Number of molecules adsorbed is plotted against chemical potential. This graph shows evidence that rather than adding an entire row or column at once, the system adds molecules one row at a time. (6)

These two sets of data, one based on an experiment and the other based on theory, strongly

support the idea that the capacity of the system is an increasing value, and that the final capacity

is simply the capacity at xb equals unity.

Another feature of note in Abaza’s data is that even at the final capacity of 18x19, with

most of the molecules overlapping (repelling), there are regions in space where some

neighboring molecules have attractions rather than repulsions.

37

Subtracting the number of molecules from the number of sites shows that the number of

vacancies is 342-340 = 2. At high levels of adsorption, the aggregate of all the holes add up to

two vacancies. Physically this makes sense because filling in the final few vacant sites vastly

reduces the number of configurations, and therefore the entropy of the system. This supports the

idea that in an equation of adsorption, the denominator for x1 should be constantly changing to

account for the increase in sites.

Figure 31: Monte Carlo from Abaza thesis simulations. The surface is large enough to hold 16x16 particles with hexagonal packing. It is notable that even though the inter-molecule repulsions increase at higher chemical potential, even at 18x18 (bottom right), there are still vacancies denoting inter-molecule attractions. (6)

38

Changing Capacity Model

We developed an algorithm which allows the system’s capacity or number of sites to

increase each time xb increases. In the code parameters, an initial number of sites is defined

which serves as the initial denominator for x1. The outer loop of the code increases the

numerator of x1 by a constant fraction on each iteration. A series of inner loops then calculate xb

values for two different scenarios. In scenario one, the system keeps the same number of sites,

so x1 equals the new numerator over the old denominator. In scenario two, the denominator of x1

also increases by one. For example, at some point in the middle of the loop, x1 might equal

342/1000. On the next iteration, xb values will be calculated for x1 = 343/1000 and also for x1 =

343/1001. The scenario that produces the smaller xb, meaning the chemical potential is lower

and the system is more favorable, will be selected. The capacity (or denominator) of the surface

will then be adjusted accordingly, and the loop will continue.

Analysis of Model Simulations

This model finds the degree of adsorption compression, the final lattice dimensions, and

the final number of vacancies for a system given the Єs, Єa, Єb, geometric coefficients, and ratio

of surface area to number of sites. This last item can be used to simulate what happens when the

adsorbed molecules are larger than the sites created by the surface geometry, or a change from a

cubic to hexagonal lattice.

This first simulation ran was for a square lattice with an Єb of -.5, an Єa of -1, a 1:1 ratio

of sites to surface area, and varying Єs.

39

Here is the simulation output for final lattice dimensions:

Figure 33: Final dimensions with different Єs values. With a larger surface energy, there is a higher degree of adsorption compression.

Figure 32: Isotherm from variable capacity model with different Єs values. As expected, the isotherms with a higher Єs show a higher degree of adsorption compression.

x1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

Isotherm: Varying Es

xb

x1

Es = -1

Es = -2

Es = -5

Es = -10

xb

x1

Єs

Єs = -1

Єs Єs = -2

Єs = -5

Єs = -10

40

Depending on the strength of the surface energy, an extra row is added to several of the systems.

Figure 34: Change in vacancies over adsorption from variable capacity model. Initially one vacancy is lost for each

additional molecule adsorbed. Eventually, it is more favorable to add a site when a molecule is adsorbed. This

manifests itself in the graph with a horizontal jump.

It was also found, as Abaza’s simulation indicated, that at a certain point it is more

favorable for the surface to increase its number of sites when a molecule is added. These site

additions can be seen in the plot of vacancies against molecules adsorbed by the horizontal

jumps, meaning a molecule was added but an empty site was not lost. Eventually the number of

vacancies becomes constant, which continues until xb reaches unity. The number of sites versus

the number of molecules adsorbed is actually the same with different Єs values, but when the Єs is

higher, the curve extends further or the final capacity is higher.

Another parameter that can vary is the Єa, or the magnitude of the Lennard-Jones

function.

0 50 100 1500

20

40

60

80

100Conservation of Vacancies

Particles Adsorbed

Vacancie

s

41

Total Sites at Different Єa Values

With different Єa values, the extra sites are added in different places

0 50 100 1500

20

40

60

80

100

Vacancies at Different Ea values

Particles

Vacancie

s

Ea = -0.5

Ea = -1

Ea = -2

Ea = -3

Figure 35: Adsorption isotherms from variable capacity model with different Єa values. At low adsorbed density, adsorption is higher when Єa is large because of higher attractive forces. At higher densities, the repulsive force is higher for a large Єa, and adsorption is lower.

Figure 36: Change in vacancies over adsorption from variable capacity model. Depending on the value of Єa, new sites are added at different adsorbed densities.

Figure 37: Change in sites over adsorption from variable capacity model. Depending on the value of Єa, sites are added at different points of the adsorption. Ultimately, more sites are added when Єa is smaller because of the lower inter-molecule repulsions at high adsorbed density.

0 50 100 150100

110

120

130

140

150

Vacancies at Different Ea values

Particles

Adsorp

tion S

ites

Ea = -0.5

Ea = -1

Ea = -2

Ea = -3

Єa

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

Isotherm: Varying Ea, zoom view

xb

x1

Ea = -0.5

Ea = -1

Ea = -2

Ea = -3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

Isotherm: Varying Ea

xb

x1

Ea = -0.5

Ea = -1

Ea = -2

Ea = -3

Єa Єa

x1 x1

xb xb

42

An Alternate Definition of Capacity and Adsorbed Fraction

Adsorbed fraction or x1 is defined as the volume fraction of the adsorbent surface which

is filled with adsorbate molecules. Since capacity has always been considered a constant, x1 is

often alternatively defined as the fraction of surface sites filled. However, this poses a problem

in a variable capacity model. Two systems can have the same fraction of sites filled with one of

the systems having more sites, and therefore higher inter-molecule repulsions. In order to

distinguish between these two systems, this paper has defined x1 in all graphs as

This way, graphs can be used to see the extent of adsorption compression. However, this is not

the definition used for Ono-Kondo calculations. The Ono-Kondo entropy term is derived based

Figure 38: Final dimensions of adsorbed layer and degree of adsorption compression with different Єa values. The degree of adsorption compression is greater for a system with a lower Єa value due to lower inter-molecule repulsions at high adsorbed density.

43

on the number of potential states that a system can be in based on the number of sites in the bulk

and on the surface. Since the number of sites on the surface increases in the variable capacity

model, the number of configurations is affected by this. As such, the definition of x1 used for

Ono-Kondo calculations is

Both of these variables are crucial in describing adsorption compression systems. Since the

definition in equation 22 a. is closer to the classical definition, this paper will define the variable

capacity adsorbed fraction as

This is the variable used for all equations involving Ono-Kondo theory.

New Ono-Kondo Coordinates and Back-Calculating Єs and Initial Sites

Another useful way to plot xb and x1 is by using new Ono-Kondo Coordinates. This is done by

plotting the entropy term (log[x1*(1-xb)/(xb*(1-x1))]) versus x1, which gives a plot whose y

intercept is Єs and whose initial slope is Єa, a relation you can see from the Ono-Kondo equation

by rearranging so the entropy term is on one side and the enthalpy term is on the other side (and

ignoring Єb*xb which is often negligible).

( )

( )

The following graph has isotherms from the variable capacity model with different values

for in New Ono-Kondo coordinates.

44

Figure 39: Isotherms in New Ono-Kondo coordinates for varying Єs. The initial slopes are all equal because inter-molecule forces are equal. The different y-intercepts correspond precisely with the different surface energies. The shift from positive to negative slope denotes the change from inter-molecule attractions to repulsions.

The y-intercept is the value of that isotherm. The initial slopes are all equal, with m = 4,

because the initial Єa is -1, and therefore = 4. An important note to point out is that x1

values used here are equal to the number of molecules adsorbed divided by the initial number of

sites, 100. When using a number other than 100, the slopes and y intercepts no longer directly

correspond with Єa and Єs. Since the initial number of sites is generally unknown, a method is

needed for determining initial sites from the New Ono-Kondo coordinates.

This next plot shows the effect of varying Єa on isotherms in New Ono-Kondo

Coordinates.

0 0.5 1 1.5-5

0

5

10

15

20Isotherm in New Ono-Kondo Coordinates: Varying Es

x1

log[x

1*(

1-x

b)/

(xb*(

1-x

1))

]

Es = -1

Es = -2

Es = -5

Es = -10

Єs

s

x1

Єs = -1

Єs = -2

Єs = -5

Єs = -10

Єa =

45

In this plot, all the isotherms have the same y-intercept of one because they were obtained using

the same Єs value. The initial slopes correspond to the Єa values from the LJ equation used.

The y-intercept and slopes will change if a different value for initial sites is used.

However, we found that using the incorrect number of sites gave predictable values which follow

a pattern. The following table shows the y-intercept and initial slope found when different

values for initial sites are used; 100 is the correct value because that is the value used in the code

for the Lennard-Jones capacity.

Figure 40: Isotherms in New Ono-Kondo coordinates for varying Єa. The y-intercepts are all equal because surface energies are equal in each system. The different slopes correspond precisely with the different Єa values. The shift from positive to negative slope denotes the change from inter-molecule attractions to repulsions.

Єa

x1 0 0.2 0.4 0.6 0.8 1 1.2-10

-5

0

5

10

15Isotherm in New Ono-Kondo Coordinates: Varying Ea

x1

log[x

1*(

1-x

b)/

(xb*(

1-x

1))

]

Єa = -0.5

Єa = -1.0

Єa = -2.0

Єa = -3.0

Єa =

Єa

x1

46

What you can see from the table is that when the number of sites used for the denominator of x1

is increased by a regular interval, it increases the slope and decreases the y intercept by a regular

interval as well. The opposite occurs when decreasing the number of guessed initial sites: the

slope decreases and the y-intercept increases.

The problem with this is that there are three unknowns: Єs, Єa, and sites, but only two

known values: slope and y-intercept. However, Єs is related to Henry’s constant, which is a

value that is often known. So if Єs is known, we can simply guess the initial number of sites, and

then look at the y-intercept. If the y-intercept does not correspond with the Єs suggested by the

known Henry’s constant, simply increase or decrease the guess for initial number of sites until

the two values converge. At this point, the true number of sites and the initial Єa value are

known.

47

Conclusions

The soft molecule correction for the Ono-Kondo model utilized by Donohue and

Aranovich is able to accommodate adsorption compression by defining surface capacity as the

number of sites at xb equals unity, but it assumes that this capacity existed from the beginning.

From an entropic perspective, this is not accurate since the number of vacancies did not continue

to decrease as xb increased. To account for this, we developed a model that can determine when

the system will choose to compress a row of molecules, thereby producing an extra site and

increasing the capacity. There is also a distinction made between the entropic capacity, or

temporary number of sites, and the enthalpic capacity, which is a constant from the Lennard-

Jones potential.

Furthermore, the model is used to show that given raw adsorption data, New Ono-Kondo

coordinates can be used to calculate the surface energy, inter-molecule energy, and initial

number of molecules, given that one of the three parameters is known. It can also be seen from

the isotherm exactly where the system underwent rearrangements based on the change in sign of

the slope.

This research provides two key contributions for future adsorption research. New Ono-

Kondo coordinates could help researchers to better understand inter-molecule interactions in the

adsorption systems they study. Furthermore, the expandable capacity algorithm provides a more

accurate mechanism for adsorption compression. If future adsorption models use this

mechanism, they will be better able to utilize adsorption for commercial processes.

48

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Appendix

2 Site Model (Mathematica)

51

Varying Parameters for soft molecule model (Mathematica)

52

Variable Capacity Code (Matlab)

53

54

55

56

57

Curriculum Vitae

Daniel Bier was born on April 25th

, 1993 in New Jersey. He attended Johns Hopkins

University where he received his Bachelor’s degree in Chemical and Biomolecular Engineering

in 2015 and his Master’s in 2016.

Daniel started research his sophomore year in the Institute for NanoBioTechnology with

Professor Peter Searson. His project involved synthesizing and testing quantum dots for

pancreatic cancer imaging. He also worked with Professor Donohue his junior and senior year

on a project to model the effects of combining chemotherapy and cannabis to treat brain tumors.

While in graduate school, he served as the teaching assistant for Projects in the Design of a

Chemical Car.

Daniel ran for four years on the Johns Hopkins Varsity Cross Country and Track and

Field teams, specializing in the 800m and 1500m. He was also a member of the Johns Hopkins

Engineers Without Borders Guatemala chapter, and traveled to Chicorral, Guatemala his

freshman year to help construct a solar-powered water delivery system.