Tobias
Dickbreder
*,
Ralf
Bechstein
and
Angelika
Kühnle
Physical Chemistry I, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany. E-mail: dickbreder@uni-bielefeld.de
First published on 6th August 2021
Desorption of molecules from surfaces constitutes an elementary process that is fundamental in both natural and application-oriented fields, including dewetting, weathering and catalysis. A powerful method to investigate desorption processes is temperature-programmed desorption (TPD), which offers the potential to provide mechanistic insights into the desorption kinetics. Using TPD, the desorption order, the energy barrier as well as the entropy change upon desorption can be accessed. In the past, several analysis methods have been developed for TPD data. These methods have in common that they rely on the Polanyi–Wigner equation, which requires proposing a desorption mechanism with a single (or at least dominating) desorption path. For real systems, however, several coupled desorption paths can be easily envisioned, which cannot be disentangled. Here, we analyse the influence of exchange between the first and the second adsorbate layer on the desorption process. We present a kinetic model, in which molecules can desorb directly from the first layer or change into the second layer and desorb from there. Interestingly, considering this additional desorption pathway alters the desorption spectrum considerably, even if the transient second-layer occupation remains as small as 4 × 10−6 monolayers. We show that the impact of this layer exchange can be described by a modified Polanyi–Wigner equation. Our study demonstrates that layer exchange can crucially impact the TPD data.
(1) |
(2) |
Based on the Polanyi–Wigner equation, the kinetic parameters of the desorption process, namely ΔEd, ΔSd and n, can be obtained. At this point, it is important to note that the Polanyi–Wigner equation only considers a single desorption path. Applying the above-mentioned analysis methods to a TPD measurement, thus, assumes that the investigated desorption process is a single process or – in case of several steps – that one step is dominating over the others. Whether this assumption is justified or not will surely require to consider the properties of the specific system of interest. However, in real systems desorption often involves coupled desorption pathways. For example, even if single-crystal surfaces are considered, an exchange between various adsorption positions like step edges, kink sites, and defects is possible. Moreover, molecules can jump from the first to the second layer and adopt adsorption positions in the second layer. Since adsorbates can typically form more than one layer, layer exchange is a rather common phenomenon. This leads to coupled desorption pathways if the barrier for molecules to jump into the second layer is comparable to (or smaller than) the desorption barrier. Such coupled desorption pathways can be addressed by a kinetic model as has been shown in various other cases in literature.12–15,22–31 Here, we evaluate the impact of this layer exchange on TPD data analysis using a generic kinetic model. To this end, we model the desorbing molecules (or atoms) on a support surface. The elementary processes such as desorption from the first (second) layer as well as layer exchange from the first (second) to the second (first) layer are described by their respective rate constants. Depending on the balance between the desorption barriers and the barriers for layer exchange, three different cases can be identified. If the barriers for layer exchange are (a) significantly higher than the barriers for desorption, the layer exchange is kinetically hindered. In this case, two independent desorption processes occur, manifesting in two desorption peaks that can be treated separately with the Polanyi–Wigner equation. Note, however, that in this case the common method to determine the coverage by observing the appearance of the second-layer peak fails. If the barriers for layer exchange are (b) similar to the barriers for desorption, the analysis of the kinetics becomes difficult as layer exchange sets in at the same temperature as desorption occurs. If the barriers for layer exchange are (c) small compared to the barriers for desorption, layer occupation is in quasi-equilibrium throughout the desorption process. In this case, second-layer occupation becomes important even for coverages as small as 4 × 10−6 monolayers (ML). The resulting TPD spectra differ significantly from a TPD curve as expected based on the Polanyi–Wigner equation. We show that this situation cannot be described correctly based on the Polanyi–Wigner equation. Instead, a correction is required that can be expressed by a simple modification of the Polanyi–Wigner equation.
The paper is organized as follows. In the next section, we describe the concept of the kinetic model. In Section 3 we present the results of the model and discuss consequences of the layer exchange and second-layer occupation on the TPD data analysis. We show that a modified Polanyi–Wigner equation can be derived that accounts for the effect of the layer exchange. We will conclude with the finding that exchange between layers can crucially impact TPD data.
(3) |
(4) |
(5) |
The total desorption rate is the sum of the individual layers' desorption rates and reads as eqn (6).
(6) |
Now, we can combine the expressions for the net layer exchange rate (eqn (3)) and the desorption rates (eqn (4) and (5)) to obtain differential equations for the first- and second-layer coverage, respectively (eqn (7) and (8)).‡
(7) |
(8) |
This set of two coupled differential equations describes how the model system evolves with time in terms of layer exchange and desorption in case of diffusion equilibrium. Hence, we can use these equations to calculate the coverages of the first and second layer during a simulated TPD experiment numerically. The desorption rate can be calculated with eqn (6).
Additionally, our model requires the knowledge of the elementary steps' kinetic rate constants as well as their temperature dependence to calculate desorption spectra (see eqn (7) and (8)). Here, we assume that the rate constants kx (x denotes the process, e.g., x = d for desorption) are given by transition state theory according to eqn (9). The quantities ΔEx and ΔSx are the corresponding potential energy barriers and entropy changes between the initial and transition states.
(9) |
The Gibbs free energy barrier ΔGx of a process in relation to the kinetic parameters ΔEx and ΔSx is given by the Gibbs–Helmholtz-eqn (10). Here, it is important to note that eqn (10) assumes that energy and enthalpy are identical for condensed phases.33
ΔGx = ΔEx − TΔSx | (10) |
Our model includes one kinetic rate constant for every elementary step considered explicitly, i.e., for every process discussed here except for diffusion, because diffusion is considered to be in equilibrium. As we describe each of these rate constants with two parameters ΔEx and ΔSx in terms of transition state theory, this makes a total of eight independent model parameters. For applications of our model, however, it can be desirable to minimize the number of model parameters, because it simplifies both the fit of our model to experimental data and the interpretation of our model. Hence, we consider the following strictly optional assumptions, which can be used in order to reduce the number of independent model parameters. The layer exchange can be interpreted as one process, but in two directions, so we assume that the layer exchange takes place over the same transition state independent from the initial state of the molecule. For desorption, it is reported in literature that the desorption process is reversible with a negligible energy barrier for many cases of molecular adsorption.34 In this case the interaction between adsorbate and surface is negligible in the transition state,34 so we assume that desorption from the first and second layer share one transition state. Considering these assumptions and the energy balance between the involved processes, the kinetic rate constants of layer exchange and desorption are related by eqn (11). This reduces the number of independent model parameters from eight to six.
(11) |
In terms of Gibbs free energy, the case of quasi-equilibrium layer exchange corresponds to a significantly smaller barrier of layer exchange from the first to the second layer ΔGle,1→2 than desorption from the first layer ΔGd,1. Moreover, the barrier of layer exchange from the second to the first layer ΔGle,2→1 needs to be significantly smaller than the barrier of desorption from the second layer ΔGd,2. This situation is shown schematically in Fig. 2. However, under the already-made assumptions it is sufficient to consider either ΔGle,2→1 ≪ ΔGd,2 or ΔGle,1→2 ≪ ΔGd,1.§
Here, we focus on systems, where ΔGle,2→1 ≪ ΔGd,2, because we expect this to be relevant for a wide variety of real adsorbate–substrate systems. For desorption it is necessary to break all existing bonds of a particle as the particle is removed from the surface. In contrast, for layer exchange the moving particle can still interact with adjacent particles and the surface, e.g., via hydrogen bonds or dispersion interactions. These interactions can stabilize the particle in the transition state of layer exchange compared to desorption, which is why we expect that the activation barrier for layer exchange is often significantly smaller than for desorption. Consequently, the case of quasi-equilibrium layer exchange is especially relevant for the analysis of TPD experiments. Additionally, this is the case where layer exchange is the most relevant for the kinetics of desorption, because the rate of layer exchange is high compared to the rate of desorption. Thus, the case of quasi-equilibrium layer exchange can reveal the maximum impact of layer exchange on the kinetics of desorption.
In general, however, there are three cases for the relations between the rates of layer exchange and desorption: (a) desorption is significantly faster than layer exchange, (b) the rates of layer exchange and desorption are of the same order as well as (c) layer exchange is significantly faster than desorption. Thus, our model also yields the two other cases of (a) kinetically-hindered layer exchange and (b) balance between layer exchange and desorption in addition to the case of (c) quasi-equilibrium layer exchange. In the following, we first describe the characteristics of desorption spectra in cases (a) and (b). After that, we discuss case (c) and provide a strategy for the analysis of TPD data in case of quasi-equilibrium layer exchange. Guided by the situation found in many real systems,4,11,35–40 we consider a model system in which the first layer binds more strongly to the substrate than the second layer to the first for all three cases. Moreover, we consider well separated peaks for desorption from the first layer and second layer as this enables us to differentiate between the effects of layer exchange on desorption from the first layer and desorption from the second layer, respectively. An interactive visualisation of simulated desorption spectra can be found online (https://doi.org/10.4119/unibi/2955951). Additional simulations for the two cases of kinetically-hindered layer exchange and balance between layer exchange can be found in the ESI.†
Only in this case, the integral over the first (second) layer desorption signal corresponds to the respective initial coverage in the first (second) layer. The individual desorption signals can be analysed with standard Polanyi–Wigner based analysis methods each.16,20,21
Especially, desorption from the first layer can take place via two different desorption paths: particles can either desorb directly from the first layer or hop on top of other particles prior to desorption. The presence of these two desorption paths causes a larger peak width and flatter leading edge than observed for a comparable first-order desorption peak. Consequently, a Polanyi–Wigner approach according to eqn (1) cannot describe the kinetics of desorption in this case. Instead, we need to consider the full kinetics of layer exchange and desorption as given by eqn (6)–(8). In the analysis of TPD data, this could, e.g., be realised by optimisation of the model parameters and initial layer coverages to match the experimental data.
Moreover, the similar barriers for layer exchange and desorption cause that layer exchange from the second to the first layer and desorption from the second layer to be activated in a similar temperature range. Hence, the appearance of the desorption spectrum depends on the initial coverages in the first and second layer like in case (a). However, in this case the initial coverage distribution also influences the observed kinetics of the second-layer desorption peak. For a partial first-layer and second-layer occupation as obtained by dosing the shape of the second-layer desorption peak is determined by layer exchange from the second to the first layer and desorption from the second layer. This causes a shift in the desorption maxima to higher temperatures with increasing coverage. In contrast, the second-layer desorption signal is solely determined by desorption for a layer-by-layer initial coverage distribution as obtained by annealing. This simpler desorption kinetics is caused by the fact that layer exchange from the second to the first layer cannot take place if the first layer is completely occupied. Additionally, this simpler desorption kinetics is advantageous for the analysis of experimental TPD data as it enables the use of standard Polanyi–Wigner based analysis methods for second-layer desorption.
Next, we discuss the case of quasi-equilibrium layer exchange on the example of simulated TPD data shown in Fig. 3. To ensure that the simulated TPD data presented here are relevant for real systems as well, we based our simulation on kinetic parameters of desorption similar as those reported for TPD experiments in literature.3–9,38–49 Specifically, we chose kinetic parameters of ΔEd,1 = 1.0 eV and ΔSd,1 = 10kB for desorption from the first layer, which are in the typical range for the desorption of molecularly adsorbed molecules from metals or metal oxides.3,4,8,40 Concerning desorption from the second layer, we consider a system where the particles in the second layer bind much weaker to the first layer than the first layer to the substrate. Hence, we chose an energy barrier of ΔEd,2 = 0.5 eV for desorption from the second layer, which corresponds to a significant binding energy difference of ΔE1,2 = 0.5 eV between the first and second layer within the already-made assumptions. As the particles in the second layer are bound significantly weaker than in the first layer, we assume that they are more mobile as well. Thus, we consider a significantly smaller entropy difference between the adsorbed and transition state than for desorption from the first layer. We chose an entropy change of ΔSd,2 = 2kB.
Fig. 3 Comparison of simulated desorption spectra (a) and total coverage as a function temperature (b) for a two-layer system with quasi-equilibrium layer exchange according to our model (violet) and a first order Polanyi–Wigner process with the same kinetic parameters as used for desorption from the first layer (grey). For the two-layer model the contributions of desorption from the first (red) and second (blue) layer are shown in (a) and the corresponding coverages in (b). An approximate desorption spectrum according to eqn (13) is displayed in black. It coincides with the model desorption spectrum in violet within the accuracy of this figure. The TPD data was calculated with the kinetic parameters for desorption given in the text. |
Concerning the numerical simulation, we also need to consider the different timescales of layer exchange and desorption in case of quasi-equilibrium layer exchange. Due to the high difference in rates of layer exchange and desorption, the accurate numerical treatment of layer exchange requires a very small time step on the timescale of desorption. Therefore, the simulation of desorption spectra based on the model differential equations (eqn (7) and (8)) is inefficient. Instead, we integrated eqn (6) as a differential equation of the total coverage with a fourth order Runge–Kutta algorithm. For this treatment it is necessary to know the first-layer and second-layer coverages as functions of the total coverage. We calculated these layer coverages from the total coverage and the equilibrium condition for layer exchange (rle ≈ 0) numerically with a Newton–Raphson algorithm. This strategy yields excellent results for sufficiently small energy barriers of layer exchange, while still being numerically efficient. Details of the numerical simulation procedure are described in the ESI.†
In Fig. 3 we present the desorption spectrum and corresponding coverages gained from our kinetic two-layer model for an initial coverage of 1.0 ML. The modelled desorption spectrum (violet curve) in Fig. 3(a) shows a single asymmetric peak with a larger peak width, lower maximum desorption rate and more pronounced low-temperature rise than the first-order process displayed for comparison (grey curve). It is evident that a first-order Polanyi–Wigner approach according to eqn (1) cannot describe the modelled desorption spectrum and, thus the modelled kinetics of desorption.
To understand the origin of the larger peak width and flatter low-temperature rise of the modelled desorption spectrum, we analyse the contributions of first and second layer desorption to the total desorption signal (red and blue curves in Fig. 3(a)). This analysis reveals a significant contribution of second-layer desorption to the spectrum, even though only a very small fraction of particles (≈4 × 10−6 ML) resides in the second layer over the full temperature range (see Fig. 3(b)). We conclude that this contribution originates from a two-step desorption path where particles hop on top of other particles prior to desorption. As this desorption path depends on both, desorption and layer exchange, layer exchange is pivotal for understanding the kinetics of desorption. Consequently, it is necessary to identify whether or not layer exchange needs to be considered in the analysis of TPD data. Otherwise it is impossible to ensure correct results.
Next, we present a simple strategy whereby desorption via hop on top can be included in the Polanyi–Wigner based analysis of TPD data. Fig. 3 reveals, that only a very small fraction of particles resides in the second layer (θ2 ≪ θ1) when desorption from the first layer takes place. The reason for this is that we investigate a system with a high free energy difference between the first and second layer ΔG12 (see Fig. 2), so the second layer is thermodynamically highly unfavourable. Indeed, the second-layer coverage is generally much smaller than the coverage in the first layer during desorption from the first layer if the free energy difference between the layers is sufficiently high. Hence, we can approximate θ1 − θ2 ≈ θ1 and θ ≈ θ1. Using these assumption in eqn (3), we derive for the quasi-equilibrium case () eqn (12) for the approximate coverage in the second layer.
(12) |
Insertion of the approximate second-layer coverage (eqn (12)) and the relation between the rate constants (eqn (11)) in the total desorption rate (eqn (6)) yields eqn (13) as an the approximate total desorption rate. The given approximation reveals that in the limit of small second layer coverage (and quasi-equilibrium layer exchange) the desorption rate depends on two contributions. The first term (kd,1θ) corresponds to the expected first-order Polanyi–Wigner description of desorption from the first layer. The second term (kd,1θ2/(1 − θ)) describes second-layer desorption via the hop on top desorption mechanism.
(13) |
Interestingly, the ratio between these two contributions to the submonolayer desorption rate does not depend on the free energy difference between the layers, as long as this energy difference is sufficiently high (i.e., a small fraction of particles resides in the second layer). However, the relative contribution of desorption via hop on top increases with coverage. For small coverages nearly all particles desorb directly from the first layer (θ > θ2/(1 −θ)) as there are only very little adsorption sites in the second layer. With increasing coverage the number of adsorption sites in the second layer increases and more particles desorb via hop on top. For coverages greater than 0.5 ML more particles desorb via hop on top than directly from the first layer (θ < θ2/(1 −θ)). Consequently, the total contribution of desorption via hop on top increases with increasing initial coverage.¶
A further comparison of eqn (13) with the Polanyi–Wigner eqn (1) shows that both equations only differ in the coverage function, which is θn for the Polanyi–Wigner-equation and for our model. Therefore we expect that most standard analysis procedures for TPD data like leading edge20 or complete analysis16,21 work with eqn (13) as well.
For other geometries we find similar expressions for the desorption rate, where the second contribution in eqn (13) varies with the expected number of adsorption positions in the second layer. We describe this in detail in the ESI.†
Concerning the accuracy of eqn (13), Fig. 3(a) shows that the desorption spectrum calculated with eqn (13) (black) is identical with the not-approximated model desorption spectrum (violet) within the figure's accuracy. We conclude that eqn (13) is an excellent approximation for the rather complex model equations (eqn (7) and (8)) within the temperature range, where desorption from the first layer takes place. It requires, however, that the system is in a quasi-equilibrium state for layer exchange and the energy difference between the layers is sufficiently high (i.e. only a small fraction of molecules resides in the second layer).
So far, we discussed the three different cases for our model and outlined possible analysis strategies for each case. For real systems, however, the applicable case is generally not known before the analysis. Within the framework of our model, one strategy to overcome this problem is to classify the investigated system based on characteristic features of the desorption spectrum. We discussed before that the appearance of the second-layer desorption signal before the first-layer desorption signal is saturated is an indicator for a kinetic-hindrance of layer exchange. In contrast, a significantly broader first-layer desorption peak with a flatter low temperature rise indicates that layer exchange influences the desorption spectrum. Based on these information we conclude that a systems which shows the former (latter) but not the latter (former) most likely belongs to case (a) (case (c)) and can be analysed as such. Experimental desorption spectra showing indications for both, a (partly) hindered layer exchange and the influence of layer exchange on the desorption signals, can be classified and analysed as case (b). Here, it is important to note that systems with, e.g., lateral interactions or additional adsorption sites might show desorption spectra with similar features. These alternative origins need to be considered as well.
Footnotes |
† Electronic supplementary information (ESI) available: Quantitative discussion on case stability, additional desorption spectra for cases (a) and (b), additional information on simple chain geometry, two-layer model for a shifted-chain geometry, coverage and geometry dependence of desorption via hop on top, details of the numerical calculations. See DOI: 10.1039/d1cp01924d |
‡ From these two equations it can be seen that the Curtin–Hammett principle does not apply here as these equations deviate from first-order kinetics required in the Curtin–Hammett principle.32 |
§ We already assume that eqn (11) holds true and, thus, ΔGle,1→2 − ΔGd,1 = ΔGle,2→1 − ΔGd,2. As both differences between the barriers of layer exchange and desorption are equivalent either both or neither are sufficiently high. |
¶ We provide additional information on the coverage and geometry dependence of desorption via hop on top in the ESI.† |
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