Adsorption on a Surface with Varying Properties
We propose a self-consistent model taking into account variations in adsorption properties of the adsorbent surface in the process of adsorption–desorption of molecules of gas on it. We introduce a dimensionless coupling parameter that characterizes the interaction of an adsorbed molecule with polarized medium. It is established that the system can be bistable if the coupling parameter is greater than a critical value and the concentration of gas belongs to a certain interval. We show that the adsorption isotherms obtained within the framework of the proposed model essentially differ from the Langmuir isotherms and establish that the Zeldovich hysteresis is possible. The kinetics of the surface coverage is analyzed in detail. We show that taking account of variations in adsorption properties of the surface in the process of adsorption–desorption leads to new phenomena: a “quasistationary” state in the case of the overdamped approximation and damped self-oscillations of the system in the general case.
pacs:68.43.-h; 68.43.Mn; 68.43.Nr
The study of adsorption of molecules on the surface of different bodies covers an extremely wide class of problems of physics and chemistry and is one of the most important problems both from the theoretical point of view and for practical applications. The results of numerous investigations show that adsorption of molecules on surfaces of bodies leads to changes in various physical and chemical characteristics of these bodies. The detailed analysis of changes in the properties of the surface due to adsorption is given in (1); (2); (3); (4); (5); (6); (7); (8); (9); (10). The results of investigation of changes in properties of the surface due to adsorption–desorption processes are also widely used in the design of various sensors (physical, chemical, and biological) (11); (12) whose action is based on the use of the change in a certain characteristic of a sensitive element of the sensor due to adsorption of molecules on its surface.
The results of the theory of adsorption are extremely important for investigation of heterogenous-catalytic reactions because processes of adsorption and desorption are integral stages of these reactions.
The classical Langmuir theory that describes adsorption of a gas on solid surfaces is based on several assumptions. Numerous theoretical investigations, which, to a large extent, were stimulated by many experimental data that did not agree with conclusions of the Langmuir theory, were aimed at the construction of more general models free of one or several restrictions of the Langmuir theory. An extensive material obtained on the basis of these models and applications to various problems of adsorption and catalysis are widely presented in the literature (see, e.g., (7); (13); (14); (15); (16); (17); (18)). In particular, it is established that taking account of lateral interactions between adsorbed molecules can lead to a qualitative change in adsorption isotherms, namely, to a hysteresis of isotherms and to structural changes in the surface of bodies (surveys of theoretical and experimental results are given, e.g., in (3); (6); (7); (8); (9); (18)).
At the same time, as early as in 1938, in (19), Zeldovich has suggested an idea on a change in the surface in the course of adsorption and desorption due to the presence of adsorbed molecules on it. Using this idea, he has predicted a hysteresis of adsorption isotherms if the typical time of adsorption and desorption is much less than the relaxation time of the surface.
Note that a change in adsorption isotherms due to lateral interactions between adsorbed molecules can also be interpreted as a consequence of a certain change in adsorption properties of the surface caused by adsorption. However, as far as we know, the problem of variation in adsorption properties of the surface itself in the course of adsorption and desorption of molecules of gas on it in the absence of interactions between adsorbed molecules and the possibility of hysteresis of adsorption isotherms in the this case remains open.
The present paper is devoted to investigation of specific features of the behavior of adsorption isotherms and the kinetics of the surface coverage by adsorbate molecules with regard for variations in adsorption properties of the surface in the course of adsorption–desorption.
In Sec. 2, we propose a self-consistent model taking into account variations in adsorption properties of the surface in the process of adsorption–desorption of molecules of gas on it. We introduce a dimensionless coupling parameter that characterizes the interaction of an adsorbed molecule with polarized medium. We obtain adsorption isotherms and establish that their behavior essentially depends on the value of this parameter (Sec. 3). It is shown that, within the framework of the proposed model, the Zeldovich hysteresis is possible. In Sec. 4, we investigate specific features of the kinetics of the surface coverage. It is established that variations in adsorption properties of the surface in the course of adsorption–desorption cardinally change the Langmuir kinetics.
Ii Model of the Surface with Varying Adsorption Properties
We consider a problem of adsorption of molecules of a one-component gas on the surface of a solid adsorbent. According to the classical Langmuir theory, molecules of gas are adsorbed on adsorption centers located on the adsorbent surface and the number of centers does not change with time. Furthermore, all centers have equal adsorption activity (energy-uniform surface), do not interact with each other, and each adsorption center can be bound only with one adsorbate molecule. The Langmuir kinetics of the quantity of adsorbed substance is described by the differential equation (13)
where is the surface coverage by adsorbate, is the total number of adsorption centers, and are, respectively, the numbers of occupied and free () adsorption centers at the time , and are the adsorption and desorption rate constants, respectively, and is the concentration of molecules in the gas phase that is kept constant.
The solution of this equation with zero initial condition has the form (16)
is the stationary surface coverage (Langmuir isotherm), which is defined by the single dimensionless quantity (dimensionless concentration) , is the adsorption–desorption equilibrium constant for the given concentration of gas,
is the time taken for the surface coverage to reach the stationary value , and is the typical lifetime of a complex (adsorption center + adsorbed molecule).
According to (2) and (3), there is a single-valued correspondence between the gas concentration and the surface coverage. At the same time, in (19), Zeldovich has suggested an idea on a change in adsorption properties of the surface in the process of adsorption and desorption and, using this idea, predicted a hysteresis of adsorption isotherms if the typical time of adsorption and desorption is much less than the relaxation time of the surface.
In the present paper, within the framework of the Langmuir model, we take into account variations in properties of the surface of a homogenous adsorbent with plane boundary in the process of adsorption and desorption of molecules of a one-component gas on it. We introduce the Cartesian coordinate system centered at the surface of the adsorbent with -axis directed into the adsorbent perpendicularly to its surface so that the adsorbent and the gas occupy the regions and , respectively. Each adsorption center is simulated by a one-dimensional linear oscillator of mass that oscillates perpendicularly to the surface about its equilibrium position . In the absence of adsorbate, the motion of an absorption center is described by the well-known equation of motion of a free linear oscillator
where is the restoring force constant, is the friction coefficient, and is the coordinate of the oscillator.
In the general case, due to occupation of the adsorption center with a molecule of adsorbate, the electron structure of the center changes, which results in a change in the interaction of the center with neighboring atoms of the adsorbent, i.e., to a local polarization of the adsorbent. As a result, the medium acts on the complex with a certain force , where is the running coordinate of the complex, which is the reaction of the medium on the electron reconstruction of the adsorption center. Under the action of this force, the complex tends to a new equilibrium position different from the equilibrium position of the free center. If the polarization of adsorbent caused by the formation of the complex is axially symmetric about the axis passing through the complex and parallel to the -axis, then this force has only the component normal to the boundary, , where is the unit vector along the -axis; for convenience, the center has the coordinate .
The force either acts on each oscillator of the system if the oscillator is occupied with a molecule of gas or does not act if it is free, i.e., the oscillator interacts with the polarized medium only for discrete time intervals. Instead, we consider an approximation where the oscillator–medium interaction is continuous in time and the oscillator is permanently bound with molecule with the time-dependent probability equal to the surface coverage . In this case, . This approximation is analogous to the mean-field approximation used in problems of adsorption with regard for lateral interactions between adsorbed particles (7).
We represent the force in the form , where is the potential energy of interaction of the polarized medium with the complex. Expanding the quantity in the Taylor series and keeping only the linear term, we obtain
where the parameter of complex–polarized medium interaction is the force acting on the complex by the medium polarized by this complex.
Ignoring the internal motion in the bound molecule–center system, i.e., considering the motion of the complex as a whole, and taking into account a change in the mass of the oscillator in the process of adsorption–desorption within the framework of this approximation, we obtain the following equation of motion for the oscillator:
where is the effective mass of the complex that varies in the process of adsorption–desorption and is the mass of an adsorbate molecule. Since , the effective mass of the complex does not exceed its total mass .
It follows from Eq. (7) that bonding of an adsorbate molecule with center leads to a shift of the equilibrium position of the oscillator by and to a change in the potential energy of the free oscillator equal to by . In the limiting case where all centers are bound, i.e, , the equilibrium position of the bound oscillator is maximally distant from the surface and the potential energy of the oscillator at this equilibrium position is minimal and equal to a half of the energy of interaction of the bound oscillator with the polarized medium, , .
Thus, due to the interaction of adsorbate molecules with adsorption centers, centers shift relative to the nonperturbed surface of the adsorbent, i.e., this interaction leads to the formation (for adsorption) and healing (for desorption) of local defects of the surface. For , these defects are “pits” (for ) or “hills” (for ) whose depth and height depend on the properties of both the adsorbate and the adsorbent. In the special case where all atoms of the surface are adsorption centers, this interaction leads to a shift of the surface of the adsorbent either inwards (for ) or outwards (for ), i.e., to the relaxation of the surface (9). In other words, the processes of adsorption and desorption result in a deformation of the surface of the adsorbent, which leads, in the general case, to changes in the adsorption and desorption rates and, as a consequence, the surface coverage. Within the framework of the Langmuir theory of kinetics on the nondeformable surface (), the adsorption and desorption rate constants and do not depend on the concentration of gas and are defined by the Arrhenius relations
where and are the activation energies for adsorption and desorption, respectively, and are the preexponential factors, is the absolute temperature, and is the Boltzmann constant.
A molecule bound with center, due to its interaction with polarized medium, is in a deeper potential well than in the case of the nondeformable surface. Therefore, for its desorption, the molecule requires an energy greater than by the value , where is an additional energy that the bound molecule must acquire to break the bond with polarized medium.
Generally speaking, the polarization of the medium can also affect the number of free molecules of gas that can overcome the adsorption barrier , i.e., a peculiar activation of free molecules of gas occurs and varies in the process of adsorption–desorption. Here, we do not take into account a change in the activation energy for adsorption (some results obtained with regard for a decrease in the activation energy for adsorption in the process of adsorption–desorption are presented in Appendix A). Supposing that the preexponential factor is not changed, we obtain the following expression for :
It is worth noting that this quantity already depends on the concentration of gas because it is defined by the current state of the surface (the quantity ) that depends on the concentration of gas. Therefore, adsorption and desorption of molecules proceed on the surface whose adsorption characteristics vary with time.
Introducing the dimensionless coordinate of oscillator , we obtain the following autonomous system of nonlinear differential equations, which describes the kinetics of the quantity of adsorbed substance with regard for variations in adsorption properties of the surface in the process of adsorption–desorption:
where the dimensionless parameter , which characterizes the interaction of an adsorbed molecule with polarized medium, can be called a coupling parameter. In the absence of interaction (the linear case) where (), the parameter .
Note that system (10)–(11), in many respects, is analogous to the system of equations given in (20), which describes a transport of electrons in a system of molecules of biological nature with regard for electron-conformation interaction.
Iii Stationary Case
In the stationary case, it follows from Eq. (11) that . Therefore, the equilibrium state of system (10)–(11) is defined not by the pair of quantities (), as is typical of dynamical systems of two equations (21); (22), but only by one quantity , which is a solution of the equation
In the general case, it is hardly possible to solve the transcendental equation (12) in the explicit form. Nevertheless, based on this equation, we can make a qualitative conclusion on the influence of a change in adsorption properties of the surface on the surface coverage. To this end, note that the ratio is equal to and the quantity is also the ratio but in the linear case. Rewriting relation (12) in the form
we see that a change in properties of the surface caused by adsorption and desorption leads to an increase in the surface coverage for any concentration of gas. The difference between the numbers of bound centers in the nonlinear () and linear cases increases with the coupling parameter .
This conclusion can also be made by taking into account that, in the stationary case, the desorption rate characteristic (9) has the form
where the surface coverage is a solution of Eq. (12). For a system whose adsorption properties vary in the process of adsorption–desorption, the adsorption–desorption equilibrium constant
is greater than the classical adsorption–desorption equilibrium constant . Therefore, the equilibrium of the system shifts towards an increase in the number of adsorbed molecules.
It follows from (14) that the interaction of adsorbed molecules with polarized medium results in an increase in the typical lifetime of complex
and, hence, an increase in the surface coverage.
To analyze solutions of Eq. (12), we use a standard procedure (22), namely: we consider the plane and take into account that and (Fig. 1). The required solutions of Eq. (12) are the abscissas of the points of intersection of a horizontal line corresponding to the given concentration (the left-hand side of Eq. (12)) with the curve (the right-hand side of Eq. (12)), which is shown in the figure for different values of the parameter . Since the behavior of the function is essentially different for and , where , it is convenient to represent the parameter in the form , where . For (Fig. 1a), the function monotonically increases and lies to the right of curve 1 for the linear case (). Thus, as in the linear case, for any given concentration, the surface coverage has the unique value , which agrees with conclusion made above on the basis of relations (13) and (15). With increase in , the curve becomes more deformed and its deviation from curve 1 increases, which leads to an increase in the difference between the values of the surface coverage in nonlinear and linear cases. For , the function (curve 1 in Fig. 1b) has an inflection point for for the concentration .
For , the behavior of the function essentially changes: for the concentrations and () depending on the value of the parameter , the function has a minimum and a maximum at the points and , respectively, which are roots of the quadratic equation
and are equal to
The concentrations and corresponding to these surface coverages are defined as follows:
In Fig. 1b, the concentrations and and the surface coverages and for them are shown with the use of dashed straight lines for the special case . Thus, for concentrations , Eq. (12) has three solutions , furthermore, only the first solution lies near the linear . With increase in , the concentrations and decrease and the difference between maximum and minimum solutions increases.
Analysis of the system of equations (10)–(11) shows that its stationary solutions and are asymptotically stable and the solution is unstable.
If the concentration tends to the end point of the interval (to the value or ), then the stable (or ) and unstable solutions approach each other and, in the limit (or ), coalesce into one solution (or ) (in Fig. 1b, for , these cases are shown for curve 2.) Therefore, and are the bifurcation concentrations for which the dynamical system (10)–(11) is structurally unstable (21); (22) and has the compound (double) equilibrium states and . These special cases should be investigated in their own rights.
Using relations (12) and (17), we plot a bifurcation curve in the plane of parameters (). This curve defined in the parametric form
is shown in Fig. 2. For any point of this plane lying between the branches of the bifurcation curve, the system of equations (10)–(11) has three structurally stable equilibrium states: two states are stable and one is unstable. If a point lies outside these branches, then the system has one structurally stable equilibrium state. At any point of the bifurcation curve, except for the cusp (, ), the system has two equilibrium states: one is structurally stable and another is double (22). At the cusp, the system of equations (10)–(11) has one triple equilibrium state (22).
The -shaped adsorption isotherm depicted in Fig. 3 for (curve 1) essentially differs from the Langmuir isotherm (curve 2) and, on the qualitative level, reproduces the Zeldovich hysteresis predicted in (19).
With quasistatic increase in the concentration from zero, the surface coverage, at the initial section of the lower stable branch of the isotherm, coincides with Langmuir one. For these concentrations, a released adsorption center manages to relax to the nonperturbed state before it binds with other molecule, furthermore, . With increase in the concentration up to the bifurcation value , the difference between the typical lifetimes of complex and increases. In this case, a free adsorption center can bind with a subsequent molecule before it relaxes to the nonperturbed state. In this section of the lower stable branch of the isotherm, occupation of the surface by adsorbate molecules is determined by two factors: an increase in the concentration of gas and a change in adsorption properties of the surface. Due to the last factor, the isotherm deviates from the Langmuir isotherm, and this deviation increases with concentration. The pattern cardinally changes as soon as the concentration negligibly exceeds . In this case, the lower stable branch of the isotherm disappears and a new (unique) equilibrium state of the complex is considerably more distant from the surface than the previous one for . Furthermore, the passage to this state is performed for a constant concentration, i.e., solely due to a change in adsorption properties of the surface of adsorbent (according to the terminology used in (19), a slow adsorption occurs). This passage can require many molecules that successively take part in the process of adsorption–desorption on the same adsorption center. Thus, in this stage, a certain interaction between the molecule leaving the adsorption center and the molecule binding with it occurs. In Fig. 3, this stage of a sharp increase in the surface coverage for a constant concentration is shown by the dashed straight line .
In passing to a stable equilibrium state lying on the upper stable branch of the isotherm (the point in Fig. 3), the majority of adsorption centers is bound. As a result, a subsequent increase in the concentration of gas slightly affects an increase in the surface coverage. Such a “saturation” of the surface with adsorbate, which rapidly increases with parameter (Fig. 3b), occurs for concentrations considerably less than those in the linear case.
In passing through the bifurcation concentration , the conditions of desorption for adsorbed molecules become essentially worse due to a considerable increase in the depth of the potential well and the displacement of the equilibrium state of bound adsorption centers from the surface. As a result, for returning the system to the lower branch of the isotherm, the concentration should be considerably less than . With quasistatic decrease in the concentration, the surface coverage decreases slightly and only, in approaching the bifurcation value , a variation in becomes noticeable. In passing through the bifurcation concentration , the upper stable branch of the isotherm disappears and an equilibrium state of the complex lies considerably closer to the surface than the previous state for . As a result, the surface coverage sharply decreases for the fixed concentration due to a change (restoration) in properties of the adsorbent surface. The transition of the system from the upper branch of the isotherm to its lower branch is shown by the dashed straight line in Fig. 3. Note that this stage of drop of the quantity is absent in (19). A subsequent decrease in the concentration is accompanied by a decrease in the surface coverage along the lower branch of the isotherm, which, in fact, coincides with Langmuir isotherm.
This behavior of the adsorption isotherm corresponds to the principle of perfect delay (23) according to which a system, which is in a stable state at the initial time, with variation in a parameter (concentration in the case at hand), remains in this state until the state exists.
As the parameter increases, the bifurcation concentration rapidly vanishes (see Fig. 2). Using the results of calculation, we can say that, for , a change in adsorption properties of the adsorbent in the process of adsorption–desorption leads to a peculiar adaptation of the system to a state in which the majority of adsorption centers are bound up to very low concentrations.
Note that the isotherms obtained above for the surface whose adsorption properties vary in the course of adsorption–desorption (Fig. 3) are similar to the isotherms obtained with regard for lateral interactions between adsorbed molecules on a nondeformable surface (3); (7); (18) and to the Hill–de Boer isotherms derived on the basis of the Hill–de Boer equation of state for adsorbed molecules (a two-dimensional analog of the Van der Waals equation) (24).
In analysis of adsorption isotherms with regard for lateral interactions between adsorbed molecules (see, e.g., (25)), for investigation of possible surface phase transitions, a critical temperature is introduced (7); (8); (9); (24). For the model considered in the present paper, using the expression for the coupling parameter , the critical value , and the analysis of adsorption isotherms performed above, the critical temperature is defined as follows: . For a system of adsorbed molecules, one stable state occurs for , whereas, for , two stable states are possible. The corresponding phase diagram for the adsorbed layer in the “surface coverage–critical temperature” coordinates is determined by the relation
and, as in the case taking into account lateral interactions between adsorbed molecules on a nondeformable surface within the framework of the mean-field approximation (7), is symmetric about .
Iv Nonstationary Case
iv.1 Overdamped Approximation
First, we investigate the kinetics of system (10)–(11) within the framework of overdamped approximation where the masses of adsorption center and molecule are low and the friction coefficient is so large that the first term on the left-hand side of Eq. (11) can be neglected as compared with the second term. Using the well-known results for a linear free oscillator of constant mass (21), this approximation is correct if
where , is the oscillation frequency of an oscillator of mass , and is the typical relaxation time of a massless oscillator. Since is the maximally possible effective mass, condition (19) is even somewhat high. In this approximation, the system of equations (10)–(11) is simplified to the form
where is the dimensionless time and .
Analysis of system (20)–(21) performed on the basis of the qualitative theory of differential equations (22); (26) shows that the stable equilibrium states of the system and are stable nodes and its unstable equilibrium state is a saddle. For the bifurcation concentration (or ), the system is structurally unstable and has a compound equilibrium state (or ), namely, a saddle-node with two saddle sectors and one stable nodal sector. The system is also structurally unstable for the critical concentration and . In this case, the system has one equilibrium state , which is stable triple node.
The numerical analysis of the system of equations with initial conditions for
shows that, for any values of the parameters , , and , the system monotonically evolutes to the nearest stable equilibrium state. Therefore, for the bistable system ( and ), the stable equilibrium state is inaccessible. The time taken for attaining the equilibrium state considerably depends on parameters, first of all, on the concentration.
Let us investigate the kinetics of the surface coverage for a system that can be bistable for . In this case, and ; and .
In Fig. 4, the kinetics of the surface coverage is shown for concentrations less (, Fig. 4a) and greater (, Fig. 4b) than the bifurcation concentration . For comparison, the Langmuir kinetics is shown in this figure by curve 2. For , the behavior of is analogous to that in the Langmuir case: the quantity monotonically increases from zero to the nearest stationary value that lies near the linear value (Fig. 4a). With increase in the concentration, this behavior remains true up to the bifurcation value (moreover, both the stationary value and the time taken for its attaining increase).
For , the system has only one stable equilibrium state, furthermore, in this state, the surface coverage, which is close to the maximum possible value, is essentially greater than that in the linear case. Moreover, both the shape of the kinetic curve and the time taken for attaining the stationary value considerably differ from the Langmuir ones (Fig. 4b). With increase in the concentration, the time taken for attaining the stationary value decreases.
The value of the parameter affects only the time taken for the system to attain the stationary value but does not qualitatively change the kinetics of both for and for . This time decreases if decreases and increases if increases, which is quite natural because a variation in is equivalent to a variation in the friction coefficient .
For concentrations near the bifurcation concentration , the behavior of qualitatively changes. The behavior of the quantity for , for low values of the relative concentration , is shown in Fig. 5. If the concentration slightly exceeds the bifurcation concentration (curves 2 and 3), then the evolution of can be conditionally divided into three stages: (i) from the initial zero value to a value of corresponding to the bifurcation concentration ; (ii) a very slow (in the limit , infinitely slow) variation in the neighborhood of ; (iii) from to the stationary value . For low values of , the time taken for attaining the stationary level is determined mainly by the second (“quasistationary”) stage in which the system, in fact, does not change (curve 2 in Fig. 5b). This behavior of the surface coverage is caused by the well-known effect of slowing down of a system near a singular point for the bifurcation value of a parameter (23); (27); (28); (29) in the case where a phase trajectory of the system moves near this point.
This behavior of system (20)–(21) can be clearly explained on the basis of analysis of its phase trajectories in the phase plane . The phase trajectories of the system with zero initial condition (22) are shown for concentrations less (Fig. 6a), equal (Fig. 6b), and slightly greater (Fig. 6c) than the bifurcation concentration . The dashed lines in these figures stand for the main isoclines of the system: the isocline of horizontal slopes and the isocline of vertical slopes . The points of intersection of these isoclines are singular points of the system.
For (Fig. 6a), the singular points and are stable (stable nodes) and the singular point is unstable (saddle). The phase trajectory in Fig. 6a starting from the origin of coordinates and going to the nearest singular point completely lies between the main isoclines. Moreover, the immediate analysis of system (20)–(21) shows that all phase trajectories of system (20)–(21) with initial values belonging to the domain bounded by the sections of the main isoclines before their intersection at the point also completely lie between the main isoclines. A change in the parameter does not qualitatively change the behavior of the phase trajectories and only shifts them to one of the main isoclines: for and , the phase trajectories are closely pressed to the isoclines of horizontal and vertical slopes, respectively.
For , the singular points and coalesce into one (compound) singular point , which is a point of tangency of the main isoclines (Fig. 6b). In this case, the phase trajectory is analogous to that in the previous case. For , the system has only one (stable) singular point (Fig. 6c). If the relative concentration is low, then a gap between the main isoclines in the neighborhood of their point of tangency for is also small. Since a phase trajectory does not leave the domain bounded by the main isoclines, it goes through the gap and, in a neighborhood of the point , its motion becomes slower. Furthermore, the less the relative concentration, the narrower the gap between the main isoclines and the closer the phase trajectory approaches the point and, hence, the more its slowing down near the point. This behavior of the system corresponds to the effect of critical slowing down near a degenerate critical point (23); (27); (28).
As a result, for low values of , the function in Fig. 5b has the form of a double step (curves 2 and 3). The first plateau of the step corresponds to the quasistationary state and the second corresponds to the stable state .
For , the kinetics of the surface coverage, which is shown in Figs. 4 and 5 for the intermediate case , can be analyzed in a standard way with the use of a potential (23); (27). It follows from Eq. (21) that . Substituting this relation into (20), we obtain the following equation for :
where the potential can be represented as the sum
is the parabolic potential for the Langmuir kinetics (1) and
is the potential caused by the action of the polarized medium on the complex.
Thus, the behavior of the quantity is completely defined by the form of the potential as a function of . Analysis of the potential shows that its form essentially depends on values of the parameters and . For , the function , like in the linear case, has one minimum for a certain . With increase in , the value increases and the minimum of decreases. The behavior of the potential essentially changes for , . In this case, the function has the form of a double well with local minima at and separated by a maximum at . As the concentration varies from to , the positions of the extrema, the depths of the wells, and the barrier between them vary. For concentrations near , the second well with minimum at is rather flat, essentially shallower as compared with the first well with minimum at , , and corresponds to a possible metastable state of the system. As the concentration increases, the second well becomes deeper and the barrier between the wells decreases. At a certain concentration, the depth of the second well becomes equal to the depth of the first one. For higher concentrations, the second well is deeper than the first, , i.e., the system is in a metastable state in the first well and in a stable state in the second well. With a further increase in the concentration, the modulus of the difference between the minima of the wells increases and the barrier decreases (the slope of the first well between and is close to zero) and disappears for . As soon as the concentration becomes greater than the bifurcation value , the first minimum disappears, furthermore, for concentrations near (), the potential , in a neighborhood of , has almost a zero slope, which leads to the well-known critical slowing down of the system (23); (27); (28). The behavior of this gradient dynamical system corresponds to the well-known principle of perfect delay (23) according to which a transition of the bistable system between two stable states of equilibrium is absent.
iv.2 Influence of the Masses of Center and Molecule on the Kinetics of the Surface Coverage
Now we investigate the kinetics of the surface coverage taking into account the masses of adsorption center and molecule. The estimating condition (19) for the overdamped approximation can be represented in the form , where and are, respectively, the -factor and the damping constant of a free oscillator of mass . We also introduce the dimensionless quantities and , where and is the oscillation frequency of a free oscillator of mass . The quantity can be represented in the form , where and are, respectively, the -factor and the damping constant of a free oscillator of mass .
Analysis of the singular points , , and of the system of equations (10)–(11) carried out in the phase space gives the following: For very low values of the -factor , , and are, as in the overdamped case, stable nodes and a saddle, respectively (30). With increase in the -factor , starting from certain values that depend on values of the parameters , and , and become stable node-focuses. In the interval , near , there appears a domain such that if falls within this domain, then the singular point is a saddle-focus, otherwise, it is a saddle. With increase in the -factor, this domain rapidly grows and covers the entire interval so that the unstable singular point is always a saddle-focus.
Below, we give the results of numerical analysis obtained for , i.e., for a system with possible bistability, and .
The curves shown in Fig. 7 illustrate the influence of the masses of adsorption center and molecule on the kinetics of the surface coverage for the concentration lying in the middle of the bistability interval , where and . Here, for comparison, we present the quantity without regard for the masses of adsorption center and molecule (curve 1) and the stationary level for this concentration (dashed line). For low values of masses (curve 2 in Fig. 7a), as in the overdamped case, the number of molecules adsorbed on the surface monotonically increases with time and reaches its maximum value . With increase in the coefficient , which is proportional to the mass of adsorption center, the behavior of the quantity changes. The surface coverage reaches its stationary value only after several oscillations about it (Fig. 7b). The amplitude and the number of oscillations as well as the time taken for attaining the stationary value increase with (curves 2 and 3 in Fig. 7b). This behavior is caused by the inertia of the oscillator that overshoots its equilibrium position and the deviation of the oscillator from the equilibrium position increases with its mass. Therefore, taking account of the masses of adsorption center and molecule changes only the character of attainment of the nearest equilibrium state by the system.
The behavior of the system can essentially change if the concentration is near the end point of the bistability interval. This case is shown in Fig. 8 where the stationary states of the system (the stable states and and the unstable state ) are shown by dashed lines. The curves in Fig. 8a show that, for low values of , the behavior of the system, to a large extent, is analogous to its behavior for low masses considered above (Fig. 7a). Due to inertia, the system penetrates into the domain , which is a domain of attraction of the attractor (domain I) (22). As in the overdamped case, with time, the system attains its stationary level . However, as soon as the mass of the complex reaches a value for which the kinetic energy of the complex is sufficient to overcome the “barrier” , the behavior of the system qualitatively changes (cf. curves 2 and 3 in Fig. 8b). Having fallen into the domain , which is a domain of attraction of the attractor (domain II), the system moves toward its second stable stationary state . Having reached this state, the system oscillates about it with decreasing amplitude. As increases, the time of attainment of the stationary state decreases because the system, in fact, is not delayed in the neighborhood of the unstable state (curve 4 in Fig. 8b). Thus, unlike the overdamped case for which the stable equilibrium state is not attainable, due to the masses of adsorption center and molecule, the system can be in this state rather than in the state .
A further increase in is accompanied by an increase in the amplitude of oscillations of the system about the stationary level and the number of oscillations (high- system). However, the system does not leave domain II (curve 2 in Fig. 8c). If the amplitude of oscillations exceeds , then the system falls into domain I and its subsequent behavior can be different. The system can remain in this domain and, after a time, it attains the stable stationary level (curve 3 in Fig. 8c) as in the overdamped case (curve 1) and in the case of low values of masses (curves 2–4 in Fig. 8a and curve 2 in Fig. 8b).
For a somewhat greater value of , the kinetic energy of the complex can be sufficient for the system to overcome the “barrier” for the second time. As a result, the system again penetrates into domain II and attains the stationary level (curve 4 in Fig. 8c). It is clear that, with increase in mass (the value of ), the system can again return into domain I, etc. Therefore, the finial stable state of the bistable system ( or ) depends on the number of intersections of the unstable state by the system in the process of its evolution with time. The value of the parameter , which is proportional to the friction coefficient, also affects overcoming of the “barrier” by the system. Decreasing this parameter, i.e., increasing the -factor of the system, it is possible, in principle, to realize a mode in which the system visits each of two stable stationary states many times. This behavior of the system under study qualitatively agrees with the well-known behavior of a Newton gradient system whose potential energy has two minima separated by a barrier in the case of a low value of the damping parameter (23).
Since the -factor of the system is determined both by the mass of adsorption center and by the mass of adsorbed molecule, these two characteristics affect (but different in rights) the possibility of the system to overcome the “barrier” . The curves depicted in Fig. 9, which describe the kinetics of the surface coverage for systems with equal masses of adsorption centers but different masses of adsorbed molecules, visually illustrate this conclusion. The behaviors of systems that returned from domain II into domain I are different. The kinetic energy of the complex with more light-weight molecule (curve 2) is insufficient for the complex to overcome the “barrier” for the second time, and the system is stabilized at the stable level . At the same time, the complex with heavier molecule is able to overcome this “barrier” and the system returns into domain II and attains the stable level (curve 3).
Note that the behavior of the kinetic curves shown in Figs. 7–9, on the qualitative level, agrees with conclusion on the kinetics of the surface coverage made in Appendix B with the use of the effective potential in the special case where the relaxation time of the quantity and the characteristic times of the quantity are essentially different.
For concentrations that slightly exceed the bifurcation value , taking account of the masses of adsorption center and molecule can also qualitatively change the kinetics of the surface coverage (Fig. 10). For equal masses of adsorption center and molecule () and for values (i.e., for ), the influence of masses on the behavior of is insignificant (curves 1 and 2, in fact, coincide). With increase in the mass, the plot of the function in the form of a double step typical of the overdamped case remains true (curve 3 in Fig. 10b). However, the residence time of the system in the “quasistationary” state somewhat decreases, i.e., the delay of the system in a neighborhood of the singular point corresponding to the bifurcation concentration is less than in the overdamped case. As the mass increases, the residence time of the system in the “quasistationary” state rapidly decreases (curve 4 in Fig. 10b) and, starting from certain values of , the system moves to the stable stationary state without delay near the “quasistationary ” state (curve 5). Thus, the possibility of an intermediate “quasistationary” state for the system essentially depends on relations between the inertial and dissipative characteristics of the system.
In the present paper, we have proposed the self-consistent model taking into account variations in adsorption properties of the adsorbent surface in the process of adsorption–desorption of molecules of gas on it. Within the framework of this model, we have introduced the dimensionless coupling parameter that characterizes the interaction of an adsorbed molecule with polarized medium. We have established that the system can be bistable if this parameter is greater than critical and the concentration of molecules in the gas phase belongs to a certain interval. We have investigated bifurcation concentrations for which stable states of the system appear and disappear. We have obtained adsorption isotherms that essentially differ from the classical Langmuir isotherms. It is established the possibility of the Zeldovich hysteresis within the framework of the proposed model. It is shown that variations in adsorption properties of the surface in the course of adsorption–desorption can lead to a peculiar adaptation of the system to the state in which the majority of adsorption centers is bound up to very low concentrations.
The detailed analysis of the kinetics of the surface coverage established that taking account of variations in adsorption properties of the surface in the course of adsorption–desorption leads to new phenomena: a “quasistationary” state for the overdamped approximation and damped self-oscillations of the system in the general case.
Acknowledgements.The author expresses the deep gratitude to Prof. Yu. B. Gaididei for the statement of the problem, valuable remarks, and useful discussions of results.
Appendix A A
A change in the activation energy for adsorption caused by polarization of the medium in the process of adsorption–desorption depends on various factors connected both with adsorbent and adsorbate. As an example, we consider the case where, due to polarization of the medium, the activation energy for adsorption decreases by the quantity equal to the increment of the activation energy for desorption caused by polarization. Supposing that the preexponential factor is not changed, we obtain the following expression for the adsorption rate characteristic :
which, like the desorption rate characteristic defined by relation (9), depends on the concentration of gas. As a result, we obtain a system of equations that describes the kinetics of the quantity of adsorbed substance and differs from system (10)–(11) derived above under the assumption that the activation energy for adsorption does not vary in the process of adsorption–desorption only by the replacement of Eq. (10) by the equation
The factor in the first term on the right-hand side of this equation takes into account a change in the activation energy for adsorption in the process of adsorption–desorption of molecules of gas.
In the stationary case, we obtain the same equation (12) for determination of the quantity but with function with changed parameter
In addition, we obtain
Comparing (A4) and (15), we obtain a natural result that a decrease in the activation barrier in the process of adsorption–desorption leads to a shift of the equilibrium of the system towards an increase in the number of adsorbed molecules.
Since the behavior of function (A.3) is identical to the behavior of function (12) with replacement of the critical value of the coupling parameter by , the results of analysis of adsorption isotherms carried out in the the third section of the present paper remains also true in the case at hand with replacement . By analogy, the critical temperature below which the system can be bistable is changed, .
The kinetics of the surface coverage is analogous to the kinetics of established above without regard for a change in the activation energy for adsorption (Figs. 4, 5, and 7–10). However, in the considered case, it is somewhat faster, which is quite natural because a decrease in the barrier favors a faster filling of the surface with molecules of gas. The specific features of the kinetics of depicted in Figs. 8 and 9 for a bistable system also occur, furthermore, they are realized for lesser values of masses of adsorption centers and molecules.
Appendix B B
Here, we investigate the behavior of the dynamical system (10)–(11) that describes the kinetics of adsorption of molecules on the surface whose adsorption properties vary in the process of adsorption–desorption in the special case where the relaxation time of the quantity is much less than the characteristic times of the quantity , i.e., the variables and are fast and slow, respectively. Performing the adiabatic elimination of the fast variable (27), namely, setting in Eqs. (10) and (11), we obtain the following representation for the surface coverage versus the slow variable :
The dimensionless coordinate of oscillator is determined as a solution of the nonlinear differential equation
that describes the motion of the oscillator with effective variable mass
in the effective potential
Note that the second term in relation (B.3) for the effective mass disappears in the absence of polarization of the adsorbate in the process of adsorption–desorption, i.e., .
Therefore, we reduced the problem of investigation of the kinetics of the surface coverage to the problem of study of the motion of an oscillator of variable mass in potential (B.5). Since the quantity is the dimensional coordinate of a bound adsorption center, in terms of the coordinate of this center, the equation of motion for it has the form
Note that the effective potential (B.8) is analogous to the potential derived in the adiabatic approximation in (31) where the structural regulation of functioning of a macromolecular in repeating cycles of reactions is investigated.
Analysis of the potential shows that, for , , it has the form of a double well with local minima at and separated by a maximum at , where , , and are the stationary surface coverages investigated in Sec. 3 that satisfy Eq. (12). For and any as well as for and or , the potential has one minimum.
The curves presented in Fig. 11 for clearly illustrate the essential influence of the concentration on the form of the potential. For concentrations lying outside the interval , the potential has a single minimum (curves 1 (for ) and 5 (for )), furthermore, the equilibrium position of the oscillator for is considerably more distant from the nonperturbed position than that for . Curves 2–4 illustrate the double-well character of the potential for the concentrations and deepening of its wells (especially, the second well) with increase in the concentration.
In the case of the double-well potential , the motion of the oscillator described by Eq. (B.2), which was initially at rest at the point , can be different depending on the contributions of the inertial and dissipative terms. For small masses and large values of the friction coefficient, the oscillator rolls down into the nearest well of the potential and, finally, is stabilized in the steady state at the point corresponding to a minimum of the potential. With increase in mass and/or a decrease in the friction coefficient, the kinetic energy of the oscillator may be sufficient to overcome the potential barrier between the wells and the oscillator falls in the second well. Depending on the values of masses and the friction coefficient, the oscillator can both remain in the second well with subsequent stabilization at its minimum and return to the first well. For very small values of , the oscillator can many times visit each well before stabilization in one of them.
The surface coverage has a similar behavior. Therefore, the kinetics of the surface coverage shown in Figs. 7–9 and obtained without additional assumptions on fast and slow variables, on the qualitative level, agrees with conclusions made above on the basis of the motion of an oscillator in a double-well potential.
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