Physico-Chemistry of Solid-Gas Interfaces: Concepts and Methodology for Gas Sensor Development

By Rene Lalauze

Fundamental elementary facts and theoretical tools for the interpretation and model development of solid-gas interactions are first presented in this work. Chemical, physical and electrochemical aspects are presented from a phenomenological, thermodynamic and kinetic point of view. The theoretical aspects of electrical properties on the surface of a solid are also covered to provide greater accessibility for those with a physico-chemical background. The second part is devoted to the development of devices for gas detection in a system approach. Methods for experimental investigations concerning solid-gas interactions are first described. Results are then presented in order to support the contribution made by large metallic elements to the electronic processes associated with solid-gas interactions.

• Language: English
• Publisher: Wiley
• Release date: Mar 1, 2013
• ISBN: 9781118623862

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Preface

Produced with the collaboration of Christophe Pijolat and Jean Paul Viricelle, this book is the fruit of research carried out over a long period of time by the Microsystems, Instrumentation and Chemical Sensors department at the Ecole des Mines, Saint Etienne, France.

The abilities of this laboratory on the subject of modeling and instrumentation on heterogenous systems have enabled us to develop and study different devices for the detection of gas.

The theoretical models based on kinetic concepts constitute the course of reflection and progress in a scientific area that is still little understood.

A large part of this book refers to PhD and scientific reports. My thanks go out to all the authors.

I would also like to thank the translators of this book from French, Zineb Es-Skali and Matthieu Bourdrel.

Chapter 1

Adsorption Phenomena¹

1.1. The surface of solids: general points

The concept of form, which can be associated with that of surface, is characteristic of a solid.

On a crystallographic level, every solid can be identified by its atomic or molecular arrangement. This arrangement, which is specific to each solid, constitutes a solid phase.

Generally, the identification of such a structure (atomic positions, cohesive energy) is defined in the hypothesis of an infinite crystal, which implies a similar environment for all atoms. Near the surface, this is no longer true and it is important to imagine a new local structure of atoms or electrically charged species.

In the particular case of ionic species, to submit to the local electroneutrality, it will often be necessary to take the solid’s environment into account. The material and the different phases in contact with it will thus reach equilibrium.

Thus appears the concept of interface: a privileged area of the solid, from which all interactions likely to occur between a solid and different surrounding compounds upon its contact will start and develop.

Depending on the nature of these compounds, there will be talk of solid-solid, solid-liquid or gas-solid reactions.

To conceptualize the solid-gas reactions on which we will concentrate, it is essential to start by simply picturing a molecule of gas bonding with a solid. The bonded molecule could remain independent from its support or react with it.

In the first hypothesis, the reversible process at work is one of adsorption, which then constitutes the overall reaction. It is called the adsorption-desorption phenomenon (see Figure 1.1a).

In the second hypothesis, adsorption will be the first step of a more complex process. It has, in this case, a non-reversible character due to which a new compound, GS for instance, will form.

The nature of the observed phenomenon will depend on the thermodynamic conditions (pressure, temperature) as well as on the chemical affinity of the present species.

It is also possible in adsorption phenomena to distinguish between physical and chemical adsorption. Chemical adsorption or chemisorption is characterized by a simple electron transfer between the gas in physisorbed state and the solid. This transfer results in the forming of a reversible chemical bond between the two compounds (see Figure 1.1b). Once again, the appearance of the chemisorption process is directly related to the environment’s thermodynamic conditions.

Figure 1.1. The different interaction modes between a gas and a solid: a) physical adsorption, b) chemisorption, c) non-reversible reaction

ch1-image-1.gif

1.2. Illustration of adsorption

Volumetric and gravimetric methods are the most explicit and common methods used to display and quantify adsorption.

1.2.1. The volumetric method or manometry

In a closed system, the bonding of a gas molecule with a solid contributes to lowering the partial pressure of the gas and measuring the variation of this pressure is enough to access the necessary information.

To conduct an experiment, one uses two containing vessels A and B (see Figure 1.2) are used. Vessel A is connected to a device that measures pressure in it or in vessel A+B if A and B are joined by a valve V1. Gas is introduced in vessel A using valve V2 under pressure Pa. The solid sample is put in vessel B. A simple gas expansion in vessel A+B is enough to allow us to measure pressure Pa+b.

Figure 1.2. Adsorption-measuring device using the volumetric method

ch1-image-2.gif

Generally, the number of gas molecules introduced, n, is given, either by:

ch1-image-3.gif

when vessel A is isolated from vessel B, or by:

ch1-image-4.gif

after the expansion of the gas in vessel A+B.

Va and Vb are, respectively, the volumes of vessel A and B.

If there is no solid sample in vessel B, we naturally find that:

ch1-image-5.gif

If there is a solid sample, we generally note that:

ch1-image-6.gif

The difference n1-n2 is the amount of gas bonded to the solid.

This experiment, if conducted under different gas pressure conditions, gives us adsorption isotherms, which plot n against P at a given temperature.

1.2.2. The gravimetric method or thermogravimetry

When a molecule of gas bonds with a solid, it changes the mass of the solid and simply weighing the solid gives us information about the bonded amount given the system’s parameters. This method allows us to easily verify that the process is reversible (see Figure 1.3).

Figure 1.3. Evolution of a solid sample’s mass gain Δm under changing pressure: if t < t0: P = P0 if t0 < t < t1: P = P1 > P0 if t > t1: P = P0 a) reversible process; b) non-reversible process

ch1-image-7.gif

1.3. Acting forces between a gas molecule and the surface of a solid

1.3.1. Van der Waals forces

By analogy with molecular interactions, we can use forces known as Van der Waals forces to interpret the source of the physical adsorption processes which are favored by very low temperatures. These forces, denoted for instance, are associated with a scalar potential φ:

ch1-image-8.gif

The scalar potentials are additive and the global scalar potential is the sum of the potential of attraction φa and the potential of repulsion φr :

ch1-image-9.gif

where:

ch1-image-10.gif

and:

ch1-image-11.gif

r represents the intermolecular distance, while the constant C consists of three contributions:

– the Keesom interaction or Keesom force, which only applies to polar molecules and originates from the attraction between several molecules’ permanent dipoles;

– the induction interaction or Debye force, which originates from a molecule’s polarizability. It is caused by the attraction between permanent dipoles and other dipoles that are induced by the permanent dipoles;

– London’s dispersion force, which originates from the attraction between molecules’ instantaneous dipoles. This is generally the most powerful attraction.

As for the expression of φr, this is an empirical expression for which we generally choose n = 12.

The global scalar potential φ between two molecules is thus given by:

ch1-image-12.gif

If we take into account the fact that the potential reaches a minimal value, φo, at equilibrium, meaning for an intermolecular distance r0, we then obtain:

ch1-image-13.gif

1.3.2. Expression of the potential between a molecule and a solid

In the hypothesis that all of the supposed semi-infinite crystal’s n molecules interact with the gas molecule, and that the potentials are additive, the global potential Φ can be expressed as follows:

ch1-image-14.gif for the attraction potential

and:

ch1-image-15.gif for the repulsion potential

These summations can be replaced with integrals:

ch1-image-16.gif

and:

ch1-image-17.gif

where dn = N dv, and N represents the number of molecules per volume unit.

The volume element used in the integrals is the volume between the spherical caps of radius r and r+dr, (see Figure 1.4), so:

ch1-image-18.gif

Ω is the solid angle, and if α is the maximum angle formed by the sphere’s radius and the solid’s surface normal, then:

ch1-image-19.gif

where cos α = Z /r , and Z is the distance between the molecule G and the solid.

Figure 1.4. Domain of integration in solid

ch1-image-20.gif

The potential of attraction then becomes:

ch1-image-21.gif

and the potential of repulsion now is:

ch1-image-22.gif

Thus, in the case of a gas molecule interacting with a solid, the 1/r³ Van der Waals potential becomes a 1/r⁶ potential of attraction, and a 1/r¹² potential becomes a 1/r⁹ potential of repulsion. The solid seems to be a thousand times more attractive or repulsive than a simple molecule.

1.3.3. Chemical forces between a gas species and the surface of a solid

In an upcoming chapter, we will go into more detail about this physico-chemical aspect, which is crucial in explaining the workings of chemical sensors.

For now, we will merely point out that if a gas atom has free electrons, a chemical bond between the gas and the solid becomes a possibility, and there are two extreme polarization possibilities that can be observed, either:

ch1-image-23.gif

or:

ch1-image-24.gif

1.3.4. Distinction between physical and chemical adsorption

The difference between physical and chemical adsorption is due to the difference between the natures of forces that keep the gas molecules on the solid’s surface.

Let us analyze the Φ = f ® curve: it goes through a minimum defined by Φ0 and r0. In physical adsorption (see Figure 1.5), the value of Φ0 is so much smaller than that observed for chemical adsorption (1 instead of 5 or 6 Joules per mole); r0, on the contrary, is lower for chemical adsorption.

At last, physical adsorption can be represented as a non-activated and therefore spontaneous process that is likely to take place at very low temperatures.

On the contrary, chemisorption is an activated process and the Φ = f ® curve goes first through a maximum marked by the activation energy value EA.

The necessity of activation is related to the fact that electron transfer, from the gas or the solid, requires an energy input; this implies the existence of a kinetic process.

Figure 1.5. Plot aspect of Φ = f ® in case of a): physisorption; b): chemisorption

ch1-image-25.gif

1.4. Thermodynamic study of physical adsorption

1.4.1. The different models of adsorption

In order to build a thermodynamic model of physical adsorption, it is important to take note of a few experimental results. The adsorption isotherms acquired through the volumetric or the gravimetric method allow us to make sure that the quantity n of bonded molecules is a function of gas pressure and temperature. These results involve a divariant system. Thereby, at the very least, all proposed models will have to meet this condition.

It is important to note that the adsorption process can in no case whatsoever be identified with a simple condensation process. Indeed, condensation is a monovariant process that owes its origin to gas saturation. Saturation is achieved at a pressure P greater than or equal to the saturation vapor pressure P0, which varies with temperature only. On the other hand, adsorption is observed at gas pressure values that are lower than P0.

The various thermodynamic models that have been proposed are grounded on such considerations.

1.4.2. The Hill model

To take into account a system’s divariance, Hill deems it necessary to take surface effects into consideration. With this aim in mind, he supposes that the adsorption film, that is to say the adsorbent + adsorbed block, is easily assimilated to a solution where the adsorbent is formed by the free sites on the solid’s surface, and the adsorbed species are the gas molecules that have settled on those sites.

In this case, the possible variables are pressure, temperature and the quantities of matter for the adsorbent (ns) and the adsorbed (na) species.

There are 2 independent components (adsorbent + adsorbed + gas – an equilibrium relation between these three components).

There are 2 exterior parameters (P and T) and there are 2 phases (solid and gaseous).

Thus, the variance v is:

ch1-image-26.gif

We can therefore plot:

– isotherms na = f (P) where T = constant;

– isobars na = f (P) where P = constant;

– isosters P = f (T) where na = constant.

1.4.3. The Hill-Everett model

Unlike the previous model, where the surface acts through the surface ns of its species, Hill and Everett describe surface effects using a physical parameter, namely the force field of the solid.

In effect, this model considers that adsorption can be described as a localized condensation process that tends to progressively cover the entire surface when the gas pressure increases. A fluctuation in the fraction of covered surface necessarily induces a fluctuation in the energy of the contact between the two phases. This energy term is the product of an extensive quantity As, which is the contact surface between the two phases, with an intensive quantity PS, which is comparable with a surface pressure. PS is identified with the variation of the surface tension coefficient γ during the covering, that is to say:

ch1-image-27.gif

In this case, there are 3 exterior parameters: pressure, temperature and surface pressure. Assuming that there is only one independent component, the variance is given by:

ch1-image-28.gif

There is in effect only one component: a solid + a gas – an equilibrium relation.

At P = P0 (saturation vapor pressure), we will assume that the entire surface is covered and that there is no longer any change in the solid’s force field; the conditions we now have are those of a simple condensation, for which v = 1 + 2 – 2 = 1.

1.4.4. Thermodynamics of the adsorption equilibrium in Hill’s model

1.4.4.1. Formulating the equilibrium

In the adsorbed phase, the change dG in enthalpy G is given by Gibbs’ equation:

ch1-image-29.gif

Where: ch1-image-30.gif and: ch1-image-x001.gif

For component 1 (the adsorbed species) in the solution, we have:

ch1-image-31.gif

being the differential molar entropy of component 1, and the differential molar volume of component 1, we know that:

ch1-image-32.gif

Let us suppose that dn2 = 0, which would imply that the quantity n2 of adsorbent is a constant. This assumption works all the better for the fact that the value of n1 is very much lower than that of n2 .

As to the gas phase, it is pure, so we have:

ch1-image-33.gif

SG and VG representing the molar entropy and the molar volume of the gas.

At equilibrium, necessarily, d μ1 = d μG, hence the following equilibrium equation:

ch1-image-34.gif

1.4.4.2. Isotherm equation

To make things simpler, we will make a few hypotheses:

– the volume of the gaseous phase is far greater than that of the adsorbed phase:

ch1-image-35.gif

– the gas is ideal:

ch1-image-36.gif

The process being isothermal dT = 0 leads to:

ch1-image-37.gif

if: μ1 = μ0 + RT Ln nS, where nS is the number of fixed moles per surface unit.

Then:

ch1-image-38.gif

thus:

ch1-image-39.gif

which brings us to:

ch1-image-40.gif

in which we recognize Henry’s law.

1.4.5. Thermodynamics of adsorption equilibrium in the Hill-Everett model

In this case, the effects due to the surface tension between the condensed phase and the solid need to be taken into account when using Gibbs’ equation.

To this end, we can start by expressing the internal energy changes dUS of the condensed phase:

ch1-image-41.gif

A new additional energy term appears when we distance ourselves from the classic model, which is a function of P and T only, that is, PS dAS.

SS, VS, AS and dnS stand for entropy, volume, contact surface and the condensed phase’s number of moles.

We can finally give the expressions for US and GS using the fact that, for a compound in a pure phase, GS = μS nS:

ch1-image-42.gif

and:

ch1-image-43.gif

If we consider G to be an exact differential, then we arrive at:

ch1-image-44.gif

Since:

ch1-image-45.gif

then:

ch1-image-46.gif

where ch1-image-47.gif and

If Γ denotes the quantity of matter per surface unit, as in 1/as, the equilibrium condition:

ch1-image-48.gif

becomes:

ch1-image-49.gif

This equation expresses the equilibrium condition in a Hill-Everett system.

1.5. Physical adsorption isotherms

1.5.1. General points

Physical adsorption isotherms are generally obtained experimentally using a volumetric or gravimetric method. Before we try to obtain meaningful results from the theoretical expressions we have arrived at using Hill’s or Hill and Everett’s hypotheses, we can already point out a few things concerning the mobility of the layers that become adsorbed on the surface of a solid.

There are two borderline cases to be considered:

– molecules that are adsorbed in the form of mobile layers;

– molecules that are perfectly located on some of the solid’s sites.

The existence of these borderline cases can be explained by the non-uniformity of the solid’s surface potential and the fact that this is typically related partly to the periodicity of a crystal lattice and the nature of the solid’s constituents.

This periodicity (see Figure 1.6) can be described as there being E1 sites of lower energy, separated by higher E2 energy levels.

The probability of a molecule moving from one stable site to another is thus proportional to E2 – E1. The E2 – E1 difference represents the energy barrier related to this species’ surface movement.

If kT >> E2 – E1, the probability of a hopping step is high, and it is then said that the layer is mobile.

If kT << E2 – E1, the probability of a hopping step is low, and it is then said that the layer is localized.

Figure 1.6. Periodic change in surface energy of crystal of parameter a

ch1-image-50.gif

Beyond this classification, it is also relevant to consider the case of monolayer adsorption as well as that of multilayer adsorption.

1.5.2. Adsorption isotherms of mobile monolayers

In such a case, the solution analogy is perfect. Hill’s model can therefore be applied here. The equation for the isotherm is as previously stated, that is to say:

ch1-image-51.gif

In fact, this model does not take into consideration the interaction between adsorbed molecules. However, the truth is that, generally, the hypothesis of a dilute solution cannot be used when studying adsorption phenomena; therefore, the thermodynamic model for solutions will rarely be of any help to us.

1.5.3. Adsorption isotherms of localized monolayers

The adsorbed molecule is bound in a low energy position; this position constitutes an active site.

The solid’s surface is now made up of identical active sites that we will assimilate to active species, which we will denote by s.

Adsorption can then be expressed as an actual chemical reaction for which we will use the following notations:

– [A] stands for a compound A in gaseous state;

– << A >> is for a constituent A in a solid or liquid solution;

– < A > is for a pure solid or liquid phase of a compound A.

In the situation we are considering, the adsorption reaction is expressed by:

ch1-image-52.gif

<< G – s >> appears here as the new species, and we therefore have a G – s solution in s. This model is fully compatible with Hill’s model.

To express the equation of the isotherm, there are two complementary and equally effective methods available.

1.5.3.1. Thermodynamic method

If we apply the mass action law to the previous equilibrium, we have:

ch1-image-53.gif

where K is the equilibrium constant.

It is given by:

ch1-image-54.gif

Adsorption is an exothermic process, which leads to ΔH° < 0. A negative value for ΔH° means that it is the reverse reaction that takes place if the temperature increases; therefore, adsorption is more likely to take place at low temperatures.

In an ideal solution, if S denotes the number of free sites, S0 denotes the number of sites, and θ represents the fraction of sites that are in use, which is expressed as:

ch1-image-55.gif

then:

ch1-image-56.gif

which brings us to:

ch1-image-57.gif

Thus:

ch1-image-58.gif

This relation, which is called the Langmuir isotherm, demonstrates that the percentage coverage of the surface is a homographic function of pressure. This function is an accurate way to represent most adsorption-related experimental results.

Note that, at low surface coverage fractions ( θ << 1), a proportionality law not unlike Henry’s law is obtained:

ch1-image-59.gif

1.5.3.2. The kinetic model

An equilibrium is reached if the variable resulting from two dynamic processes is brought to zero:

– adsorption, which has a rate RF;

– desorption, which has a rate RD.

These two processes can be seen as two elementary steps that do not involve any intermediate reactions.

In such a case, the RF rate is proportional to the number of shocks υ, which is the number of molecule and surface impacts per time unit, as well as to the number of free sites on the surface.

Thus:

ch1-image-60.gif

where:

ch1-image-61.gif

Therefore:

ch1-image-62.gif

Rd is proportional to the number of already adsorbed molecules, so:

ch1-image-63.gif

At equilibrium, VF = VD, which leads to:

ch1-image-64.gif

where:

ch1-image-65.gif

1.5.4. Multilayer adsorption isotherms

During physical adsorption, each adsorbed molecule forms a new active site for the remaining gas molecules, but there is no reason why this process should limit itself to only one layer. The Brunauer-Emmet-Teller theory, which originates from the Langmuir theory, allows us to obtain a relation (BET equation) involving a parameter that expresses the influence of the solid’s global surface area, that is to say, the area exposed to gas action.

1.5.4.1. Isotherm equation

This derives from the application of the Langmuir theory to the piled up layers.

A given site can be covered by 0, 1, 2,… i,… layers of localized gas molecules that do not interact.

We will denote by S0, S1, S2,… Si the respective fractions of surface covered by 0, 1, 2,… i layers (see Figure 1.7). We will also hypothesize that the number of layers can be infinite. We will discuss the latter point further in a following section.

At equilibrium, each fraction is constant.

Now we must consider the different layers:

– Layer 0

The rate of the disappearance of S0 (condensation over S0) is equal to the rate of adsorption over S0, which is a1PS0.

The rate of the appearance of S0 is equal to the rate of desorption of S1, which is b1S1.

Figure 1.7. Parceling of the solid’s surface depending on the number of layers

ch1-image-66.gif

At equilibrium, the rates of appearance and disappearance of S0 are equal, so:

ch1-image-67.gif

a1 and b1 are kinetic constants relative to adsorption and desorption.

This equation is similar to that obtained using the monolayer model and the Langmuir hypothesis.

– Layer 1

The rate of the disappearance of S1 is equal to the rate of adsorption over S1, which is a2PS1, to which the rate of desorption of S1 is added, that is: b1S1.

The rate of the appearance of S1 is equal to the rate of adsorption over S0, which is a1PS0, to which the rate of desorption of S2 is added, that is: b2S2.

At equilibrium:

ch1-image-68.gif

which, given the relation obtained for layer 0, becomes:

ch1-image-69.gif

– Layer i-1

The same reasoning as before leads to:

ch1-image-70.gif

We therefore have a system of i equations and i + 1 variables S0, S1, S2,… Si, where:

ch1-image-71.gif

In addition:

ch1-image-72.gif the global surface area of the solid.

Moreover, if V0 is the volume per cm² of fixed gas for an assumed complete theoretical monolayer, then the total volume is:

ch1-image-73.gif

If Vm refers to the SV0 product, which is the theoretical volume obtained in the case of a monolayer that is assumed complete, then:

ch1-image-74.gif

To find the solutions, we will assume that ai and bi are respectively equal to a and b for every i but 1.

Thus:

ch1-image-75.gif

for i ≠ 1 and by analogy:

ch1-image-76.gif

for i = 1 where:

ch1-image-77.gif

if we consider that when i = 1, a = a1 and b = b1.

The first layer is different from the rest by the fact that the solid is in direct contact with the gas.

The system of equations then becomes:

ch1-image-78.gif

Thus:

ch1-image-79.gif

Assume that 0 < x < 1 (we will soon explain why). We therefore have:

ch1-image-80.gif

which leads to:

ch1-image-81.gif

In a similar way, we arrive at:

ch1-image-82.gif

since, if 0 < x < 1:

ch1-image-83.gif

which gives us:

ch1-image-84.gif

thus:

ch1-image-85.gif

We must now analyze the role of x. If: x → 1, we note that in effect V → ∞, which means that the gas liquefies and that P = P0 at infinite V. This brings us to:

ch1-image-86.gif

thus:

ch1-image-87.gif

which leads to:

ch1-image-88.gif

Since P is necessarily less than P0 during adsorption, then we meet the requirement of x being less than 1.

The isotherm equation is therefore expressed as follows:

ch1-image-89.gif

It is possible, by applying the appropriate mathematical transform to this equation and making comparisons with the experimental results, to prove the validity of such an expression and also calculate the values of the related parameters, namely Vm and c.

It is important to note that knowing Vm gives us access to the value of the solid’s specific surface area. Moreover, the BET equation allows access to the porosity of the solid. This book will not go into details on these two points.

Overall, the previous expression’s domain