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Modeling the Kinetics of Gas Adsorption in Multilayer Porphyrin Films Alex J. McNaughton,* Alan Dunbar, William Barford,† and Tim H. Richardson Department of Physics & Astronomy, UniVersity of Sheffield, Sheffield S3 7RH, U.K. ReceiVed July 28, 2006. In Final Form: October 5, 2006 A kinetic model is proposed to describe the diffusion and adsorption behavior of gas in multilayer films. Numerical solutions are attained on time scales of seconds using a finite differencing approximation to the kinetic equations. Predictions of this model are compared to experimental data for the case of NO2 diffusing through a porphyrin film. The model predicts a binding energy for the NO2 porphyrin interaction of 0.72 eV. It also predicts that for this system diffusion is the limiting factor for the adsorption response time of the film, although the recovery time is determined by both the diffusion coefficient and NO2 binding energy. Comparison with experiment gives a predicted diffusion coefficient of ∼10-14 m2‚s-1.
1. Introduction Environmental concerns are leading to stricter regulations on toxic gas output from industry and automobiles. If these regulations are to be enforced, then there is a need for more sensitive gas-sensing devices that are able to detect toxic gases in the sub-parts per million range. Recent research has investigated the gas-sensing properties of thin films of various materials.1-3 Gas-sensing devices typically measure the change in some physical property (e.g., electrical conductivity or optical absorbance) of a thin film upon exposure to an analyte gas. One method of fabricating these thin films is the Langmuir-Blodgett technique.4 This method allows careful control of properties such as film thickness and leads to a highly organized layer structure. Understanding the response of a multilayer film to an applied gas is of vital importance when considering what factors will influence the effectiveness of such a gas-sensing device. The effect that a gas has on the properties of a film depends on the ease with which gas molecules can adsorb to active sites in the film. Factors such as the diffusivity of the gas through the film, the binding energy of the gas to an active site, and the collision rate of gas molecules with active sites can affect the speed of the film response and the equilibrium properties of the exposed film. In this article, a model is proposed that describes the behavior of gas in a multilayer film. Methods of determining the input parameters for the model are described, and predictions of the model are compared with experimental results for a real gas-sensing system.
2. Theoretical Model In this section, we introduce a mean-field theory for the kinetics of gas adsorption/desorption through a thin film. A thin film can be modeled as a series of discrete layers through which gas can diffuse. Each layer will contain a number density of active sites, N, onto and from which gas molecules are allowed to adsorb and * Corresponding author. E-mail:
[email protected]. † Present address: Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, OX1 3QZ, U.K. (1) Richardson, T. H.; Dooling, C. M.; Jones, L. T.; Brook, R. A. AdV. Colloid Interface Sci. 2005, 116, 81-96. (2) Valli, L. AdV. Colloid Interface Sci. 2005, 116, 13-44. (3) Hanwell, M. D.; Heriot, S. Y.; Richardson, T. H.; Cowlam, N.; Ross, I. M. Colloids Surf., A 2006, 284-285, 379-383. (4) Roberts, G., Ed. Langmuir-Blodgett Films; Plenum Press: New York, 1990.
desorb. Gas molecules are introduced to the film by applying a fixed concentration of gas to the surface layer and then allowing it to diffuse between adjacent layers. In this model, the state of the film at any given time is defined by the concentration of gas in each layer and the number of sites onto which a gas molecule has adsorbed in each layer. The reaction of the film to a gas is characterized by the fraction of active sites in the film that have adsorbed a gas molecule. If the density of adsorbed sites in the ith layer at time t is n(i, t), then the rate of adsorption per unit volume for that layer will be given by
kaF(i, t)
(N - n(i, t)) N
where ka is the adsorption rate for a single gas molecule in a layer of vacant sites and F(i, t) is the concentration of gas molecules in the ith layer at time t. If there is no energy barrier to overcome for adsorption to occur, then every collision between a gas molecule and an active site will result in the adsorption of the gas molecule. ka is then simply the collision rate between a single gas molecule and the active sites in a layer. The rate of desorption of gas molecules from the active sites in a layer is given by
kdn(i, t) where kd is the rate of desorption for a single active site. kd will depend on factors such as temperature and the binding energy of the gas-film interaction. Combining the previous two expressions gives eq 1, which describes the rate of change of adsorbed sites per unit volume in a given layer:
∂n(i, t) (N - n(i, t)) ) kaF(i, t) - kdn(i, t) ∂t N
(1)
The change in the gas concentration in the ith layer arises from two processes: diffusion of gas to neighboring layers and the flux of gas molecules adsorbing/desorbing from active sites in that layer. The diffusion of gas between layers is described by a simple diffusion equation, and the adsorption/desorption of gas molecules is described by eq 1. Combining these two terms
10.1021/la062231a CCC: $37.00 © 2007 American Chemical Society Published on Web 01/11/2007
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gives the time dependence of the concentration of gas in the ith layer
mean-field theory model parameters
∂F(i, t) ∂2F(i, t) ∂n(i, t) )D ∂t ∂t ∂x2
ka kd D
or
∂2F(i, t) (N - n(i, t)) ∂F(i, t) )D + kdn(i, t) - kaF(i, t) 2 ∂t N ∂x
(2)
(N - n(i, t)) ) kdn(i, t) N
which can be rearranged into the form
n(i, t) ) N
1+
1 kdN
( ) kaF(i, t)
A comparison of this expression with the Gibbs distribution for a two-state system in equilibrium with a particle reservoir,
n 1 ) N 1 + e( - µ)β gives
kd )
kaF(i, t) e( - µ)β N
where is the energy of a state, µ is the chemical potential of the gas reservoir, and β ) (kT)-1. Finally the chemical potential is related to the concentration of gas particles by
eµβ )
F F0
where
F0 )
( ) mkT 2πp2
3/2
is the quantum concentration of the gas. The desorption rate can therefore be expressed as
F0eβ kd ) k N a
6.06 × 1011 2.37 × 109 2.71 × 10-9
atomistic model parameters p(A) p(D) p(E)
0.606 0.00237 0.00828
Comparative Model
using eq 1. Equations 1 and 2 describe the diffusion and adsorption of gas molecules through a series of layers. To predict the correct equilibrium conditions, these equations must satisfy detailed balance. In the steady state, the number of adsorbed gas molecules and the concentration of gas in a layer must be constant, implying that the left-hand side of both equations along with the diffusion term in eq 2 must be zero for all layers. This gives the condition
kaF(i, t)
Table 1. Input Parameters for the Comparison of the Two Kinetic Models
(3)
For the case of a gas molecule binding to an active site in a film, is the binding energy of the gas molecule to the active site. Solving eqs 1 and 2 subject to the condition imposed by eq 3 will show how the state of the film evolves with time. Appendix A describes a numerical approach to solving these equations.
The model outlined in the previous section describes a meanfield theory for finding the state of a film. A more atomistic approach is to treat each gas molecule in the film separately. The behavior of each gas molecule is determined by the probability of adsorption, desorption, or diffusion in a given time step. This alternative model is described in more detail in Appendix B. The atomistic model is limited to small time steps (on the order of picoseconds); therefore, it is not a practical method for systems that take seconds to reach equilibrium. However, a comparison of the two models has been made for a model system with parameters chosen so as to make it reach equilibrium on a much shorter time scale. The values of parameters used to make this comparison are given in Table 1. The probabilities of adsorption, desorption, and escape from a layer were calculated as described in Appendix B. Figure 1 shows exposure and recovery cycles predicted by the two models. During the recovery stage, the incoming gas concentration is set to zero. The agreement between these two models shows that the mean-field theory approach to modeling this film, which can be used for time scales of any length, gives predictions consistent with the behavior of discrete gas molecules.
3. Predictions of the Mean-Field Model To predict the response of a film to gas, some appropriate values for the parameters must be input into the simulation depending on the system being modeled. The temperature, gas concentration, and binding energy of the gas-film interaction will all affect the equilibrium state of the film. If the binding energy of the interaction is unknown, then an estimated binding energy can be found by seeking the best agreement with experiment for the temperature dependence or concentration dependence of the equilibrium state of the film. The kinetic response of the film will also depend on the values of the adsorption rate and diffusion coefficient used in the model. Typically, one of these two parameters will limit the speed of response of the film. The recovery rate of a film (i.e., the rate at which it returns to its initial state once the gas supply is halted) depends on the rate of diffusion of gas out of the film and on the desorption rate, which is linked to the adsorption rate and the binding energy through eq 3. Stronger binding energies therefore lead to longer recovery times, unless recovery of the film is carried out at elevated temperatures. 3.1. NO2 Gas Sensing with Porphyrin Films: An Example System. A system in which we have had recent interest is the interaction of NO2 with porphyrin.5 Porphyrin is known to interact with NO2, and films of porphyrins have been seen to undergo fast reversible reactions to low concentrations of NO2 gas.6 The structure of the porphyrin molecule used in this work is shown in Figure 2. The absorption spectrum of porphyrin switches notably upon exposure to NO2. This change is shown in Figure 3. The experimental effect of NO2 on porphyrin films is well (5) McNaughton, A. J.; Richardson, T. H.; Barford, W.; Dunbar, A.; Hutchinson, J.; Hunter, C. A. Colloids Surf., A 2006, 284-285, 345-349. (6) Richardson, T. H.; Dooling, C. M.; Worsfold, O.; Jones, L. T.; Kato, K.; Shinbo, K.; Kaneko, F.; Tregonning, R.; Vysotsky, M. O.; Hunter, C. A. Colloids Surf., A 2002, 198-200, 843-857.
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Figure 1. Comparison of the mean-field theory with the atomistic model for a 10-layer film. The top graph shows the predicted response to an applied gas, and the bottom graph shows the predicted recovery of the film after the gas supply has been stopped. Dashed lines show the fraction of switched sites for the mean-field theory, and solid lines show the fraction of switched sites for the atomistic model. For both theories, the black lines represent the overall state of the film, and the gray lines show the state of each layer.
the variation in the mean square displacement of the NO2 molecules as
D)
Figure 2. Molecular structure of the porphyrin used in this work.
characterized, so this provides a good system with which to test the model described above. The films investigated experimentally were created using the Langmuir-Blodgett technique, so the idea of a layered film structure should be valid for this system. 3.2. Determination of ka and D. To simulate this system, some estimates of the values of the diffusion coefficient D and the adsorption coefficient ka are required. Molecular dynamics simulations have been performed on a system of porphyrin and NO2 molecules using Accelrys modeling software. In these simulations, a periodic cell was used containing 10 porphyrin molecules and 10 NO2 molecules. The system was allowed to equilibrate for 100 ps before performing the production run for another 100 ps. The diffusion coefficient of NO2 was found from (7) Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; Oxford University Press: Oxford, U.K., 2002.
1 d
N
∑[ri(t) - ri(0)]2 6N dt i)1
(4)
The value of D was found to be 2.71 × 10-9 m2‚s-1. These molecular dynamics simulations also gave the density of porphyrin molecules to be 1.89 × 1027 m-3. This was used as a value for N in this model, making the assumption that each porphyrin molecule contains one active binding site. The value of ka, which is simply the collision rate between an NO2 molecule and active sites in the film, can be approximated as described in Atkins.7 If the porphyrin molecule presents a cross-sectional area A to the gas molecule and the gas molecule moves with average speed cj through a film with porphyrin density N, then the collision rate can be expressed as
ka ) AcjN ka ) πr2cjN
(5)
where in this article r was taken to be the radius of a porphyrin ring. This approximation may introduce inaccuracies into the model because the NO2 might have to collide with a particular active site on the porphyrin molecule in order to adsorb, reducing the effective cross section presented to the gas molecule. This approximation has also assumed that the gas molecule is traveling perpendicular to the plane of the porphyrin ring, whereas any other orientation of the porphyrin molecules will reduce the
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Figure 3. Effect of NO2 on the observed absorption spectrum of a porphyrin film.
Figure 4. Comparison of experimental data with the predicted temperature dependence of the equilibrium state of films with different binding energies for 100 ppm gas concentration at 293 K.
effective area that the gas molecule encounters. Both of these effects will lower the value of ka, but it will be seen later that this inaccuracy in ka is unlikely to have a significant effect on the predictions of the model for this system. ka was calculated in this way to have a value of 6.06 × 1011 s-1. 3.3. Predictions and Experimental Results. Comparison of the equilibrium state of the modeled film with that of a real film can give an estimated binding energy for the NO2-porphyrin system. The binding energy affects the temperature dependence of the equilibrium state of the film. The fraction of interacting sites in a film can be found from the experimental absorption spectrum from the equation
abs0 - absF n ) N abs0 - abssat where abs0 is the absorbance at a particular wavelength when no gas is present, absF is the absorbance for gas concentration F, and abssat is the absorbance if the gas concentration is high enough to saturate the film so that all active sites have adsorbed a gas molecule. This relationship is true provided the system is a two-state system. An estimated binding energy of the interaction is the one that gives the best agreement between the predicted temperature dependence of the fraction of interacting sites in the film and the experimental observations. Figure 4 shows the temperature profile of the equilibrium state of a modeled film for a range of binding energies, along with experimental data. This Figure shows that the best agreement occurs for a binding energy of 0.72 eV.
Using this value for the binding energy, the other parameters, ka and D, can be varied to determine which of these is the limiting factor in the response time of the NO2-porphyrin system. This is important to know in order to understand how film response times might be improved. If diffusion is the limiting factor, then response times could be improved by creating a more open structured film that will allow gas to diffuse through more readily. This could be achieved by making mixed films of porphyrin contained within some matrix material that itself allows faster diffusion. If, however, the collision rate is the limiting factor, then creating mixed films would have an adverse effect on the response time because a mixed film would have a reduced density of active sites, giving a lower collision rate and hence slower response times. Figures 5 and 6 show the effect of varying ka for constant D and varying D for constant ka, respectively. It can be seen from Figure 5 that for the NO2-porphyrin system variations in ka of a few orders of magnitude have no effect on the response time of the film. The gas in a given layer of the film is reaching equilibrium on a much faster time scale than the length of the simulation. However, altering D has a significant effect on the response time of the film. For this system, the model predicts that the diffusion of the gas through the film is the limiting factor in the time in which a film can react to gas. To improve the response of such a gas sensor, the diffusion of gas through the film must be improved. It is therefore expected that response times could be improved by creating mixed films of porphyrin with a more open structured material.
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Figure 5. Effect of varying ka on the response time of a simulated film.
Figure 6. Effect of varying the diffusion coefficient on the response time of a simulated film.
Figure 7. Predicted and experimental kinetic response of a 10-layer film to 100 ppm NO2. Smooth lines depict the predicted response, and noisy lines show the experimental response.
3.4. Kinetic Response. The kinetic response of the film has been modeled using the input parameters calculated in subsection 3.2 for both exposure to gas and recovery of the film after the gas supply has been turned off. For a binding energy of 0.72 eV, the modeled film reaches saturation upon exposure to 100 ppm gas after 0.01 s at 293 K and completely recovers in just over 1 s at the same temperature. In practice, films take about 10 min to react completely to an applied gas at room temperature and take days to recover completely at room temperature. This discrepancy suggests that the diffusion coefficient used in this model is too large. Better agreement with experiment is achieved if a diffusion coefficient of 5 × 10-14 m2‚s-1 is used. Figures 7 and 8 show the predicted and experimental response and
recovery of a film to 100 ppm gas using this empirical diffusion coefficient in the model. There is good agreement between predictions and experiment for the kinetic response to gas at room temperature, with worse agreement at higher temperatures. The higher-temperature films had longer response times, indicating a reduction in the diffusion coefficient of the gas under these conditions. This is contrary to the expected temperature dependence of the diffusion coefficient according to the StokesEinstein relation, in which the diffusion coefficient increases linearly with temperature. It was observed experimentally that the response time to gas depended on the thermal history of the film, with longer response times measured for films that had previously been heated to temperatures above room temperature
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Figure 8. Predicted and experimental recovery of a 10-layer film. Smooth lines depict the predicted response, and noisy lines show the experimental response.
Figure 9. Illustration of a 10-layer system.
(i.e., in a previous exposure/recovery of the film) and then allowed to cool back down to room temperature. It is therefore thought that higher temperatures may cause some permanent restructuring of the film, creating a more closely packed structure leading to a decrease in gas mobility. The agreement between prediction and experiment of the recovery stage is less convincing at lower temperatures. Experimentally, an initial slow recovery was observed, followed by an increase in the recovery rate. This was not seen in the predictions of the model. At higher temperatures, there is better agreement between the predicted recovery and the experimentally observed recovery. It was thought that the large difference in the diffusion coefficient calculated from molecular dynamics simulations and the value that gives the best fit with experiment could be due to the way in which porphyrin molecules order themselves in Langmuir-Blodgett films. The molecular dynamics simulations were performed on a system containing 10 porphyrin molecules that were randomly aligned. However, because of the fabrication process of LB films the porphyrin molecules in a real film should take on a much more ordered structure, organizing themselves into layers in which all porphyrin molecules lie in the same plane. This could make a much more closely packed film, with a reduced mobility of gas between layers. To test this idea, molecular dynamics simulations have been performed on a more ordered structure of porphyrin molecules arranged in layers all with the same orientation. It was hoped that this would reveal
a much slower diffusion rate perpendicular to the plane of the porphyrin rings. In fact, these simulations have shown that the majority of the gas can still diffuse through the film at the same rate as predicted before but some gas molecules can become trapped at certain sites in the film. NO2 molecules that were situated between the centers of two porphyrin rings were seen to remain in this position for the duration of the molecular dynamics simulation (∼1 ns), whereas gas molecules situated further from the centers of the porphyrin rings were still free to diffuse as before. This could offer a possible explanation for the kinetics of the recovery stage. If the binding site for the NO2porphyrin interaction is in the center of the ring, then gas molecules that desorb could remain trapped close to the active site for long periods of time in some metastable state. Gas molecules trapped in this position would be much more likely to readsorb than if they were free to escape from the active site, giving a slower observed recovery rate. At higher temperatures, thermal fluctuations of the porphyrin molecules could allow gas molecules to escape from the active site much more easily, leading to better agreement with the modeled film in which gas molecules do not get trapped at the active site.
4. Conclusions This article has outlined a model that can be used to describe the kinetics of gas molecules in a multilayer film on a time scale of seconds (and longer if necessary). Results of this model can
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Figure 10. Calculating the likelihood of an analyte molecule either desorbing, adsorbing, diffusing up or down, or remaining in the layer.
be used to predict the binding energy of the interaction between a gas molecule and an active site in the film by comparing the equilibrium state of the film with experimental data. The input parameters to the model can be adjusted to predict which factor will limit the response time of a film to gas. The model has been tested for an NO2-porphyrin system and was found to predict that the diffusion of gas through the film is the limiting factor in the response time of the film to the gas. Mixing porphyrin with some material that allows faster diffusion of gas should therefore improve film response times. One such material is calixarene, a large ring-shaped molecule that should produce a more open structured film. Work with porphyrin-calixarene mixed films has shown that the response times of these films is indeed faster than that of pure porphyrin films.8 The binding energy of the NO2-porphyrin interaction was predicted to be 0.72 eV. Values for the adsorption rate and diffusion coefficient of NO2 were calculated by molecular dynamics simulations and used to simulate the kinetic response of a film to gas. These calculated values for the film parameters were found to predict response and recovery times that were much faster than the experimentally observed response and recovery of films. Better agreement with experiment was found by greatly reducing the diffusion coefficient used in the model. Further molecular dynamics simulations on more ordered porphyrin systems revealed that the slow response times may be due to the inaccessibility of active sites in the film. In these ordered film structures, NO2 molecules were becoming trapped for long periods of time between the centers of two porphyrin molecules. It is thought that the center of the porphyrin ring could be the active site for the NO2-porphyrin interaction and the slow response could be due to the long time taken for gas to diffuse between the rings. This trapping of gas molecules near (8) Dunbar, A.; Richardson, T. H.; McNaughton, A. J.; Barford, W.; Hutchinson, J.; Hunter, C. A. Colloids Surf., A 2006, 284-285, 339-344. (9) Press: W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C++: The Art of Scientific Computing; Cambridge University Press: New York, 2002.
the active sites could also explain the difference in the predicted and experimental recovery kinetics at lower temperatures. The slower experimental recovery at lower temperatures is thought to be due to the existence of these trapped states. Gas molecules that desorb from the center of the ring are not free to move away from the active site at lower temperatures. Any gas molecules that remain trapped near the active site will have an increased chance of readsorbing, giving an overall slower recovery of the film. This trapping of gas molecules is not accounted for in the model, so the predicted recovery is faster than the experimentally observed recovery. At higher temperatures, the thermal motion of the porphyrin molecules makes it easier for gas molecules to escape from these trapped sites. This explains the better agreement between the model and experiment for recovery at higher temperatures. The response times of experimental films were seen to increase for films that had been heated prior to exposure to gas. It is thought that there is some permanent structural reorganization of the films upon heating, leading to slower diffusion of gas through the film or a decrease in the accessibility of the active site on the porphyrin molecule.
6. Appendix A: Numerical Solutions The partial differential equations (eqs 1 and 2) can be solved numerically by using a finite differencing approximation.9 This method is outlined here. The partial differentials are approximated as follows
∂y(x, t) y(x, t + ∆t) - y(x, t) ) ∂t ∆t ∂2y(x, t) 2
∂x
)
y(x - ∆x, t) - 2y(x, t) + y(x + ∆x, t) ∆x2
where ∆t is some time step and ∆x is a lattice spacing between points at which the differential is to be evaluated. In this model,
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Figure 11. Flow diagram of the calculations performed by the model for each time step.
the equations are evaluated for each layer, so ∆x represents the spacing between layers. ∆t must be chosen so as to give physically realistic results depending on the method used. A simple method to implement is the forward time centered space method. In this method, the change in a variable during a time step is said to depend on the state of the system at the beginning of that time step. Equations 1 and 2 now become
n(i, t′) - n(i, t) N - n(i, t) ) kaF(i, t) - kdn(i, t) ∆t N
F(i, t′) - F(i, t) F(i - 1, t′) - 2F(i, t′) + F (i + 1, t′) )D ∆t ∆x2 N - n(i, t′) - kaF(i, t′) + kdn(i, t′) N which can be rearranged into the following forms
F(i, t′) - F(i, t) F(i - 1, t) - 2F(i, t) + F(i + 1, t) )D ∆t ∆x2 N - n(i, t) + kdn(i, t) - kaF(i, t) N where t′ ) t + ∆t. These can be rearranged to give
n(i, t′) ) n(i, t) + βF(i, t) (N - n(i, t)) - γn(i, t)
N - n(i, t′) n(i, t′) - n(i, t) ) kaF(i, t′) - kdn(i, t′) ∆t N
(6)
F(i, t′) ) F(i, t) + R(F(i - 1, t) - 2F(i, t) + F(i + 1, t)) - β(N - n(i, t)) + γn(i, t) (7) where R ) ∆tD/∆x2, β ) ∆tka/N, and γ ) ∆tkd. These two equations are trivial to solve at each time step but will give unphysical results if the time step is chosen to be too large. This can be seen by examining the term describing the number of gas particles that will adsorb in a layer in a given time step. If βF(i, t)(N - n(i, t)) > N - n(i, t), the number of free active sites in a layer, then the results of the calculation may be nonsensical and cannot be trusted. This will be the case if ∆t >N/(kaF(i, t)). To make it possible to do large time scale simulations in a reasonable amount of computational time, a method is required that will not give unrealistic results for larger time steps. One such method is the fully implicit method in which the change in a variable during a time step is dependent on the state of the system at the end of that time step. In this method, eqs 1 and 2 are written as
n(i, t′) )
n(i, t) + βF(i, t′)N 1 + βF(i, t′) + γ
(8)
F(i, t) + γn(i, t′) ) -RF(i - 1, t′) + (1 + 2R + β(N - n(i, t′)))F(i, t′) - RF(i + 1, t′) (9) where R, β, and γ have the same definitions as before. Equation 9 has been written in this way so that it takes the form of a tridiagonal matrix problem. These equations could be solved if the values of n(i, t′) were known. However, this variable itself depends on the solutions to these equations. One way to find a solution is to solve eqs 8 and 9 self-consistently using an initial guess for n(i, t′) (e.g., assume n(i, t′) ) n(i, t) for the first iteration) to calculate an estimated gas concentration for each layer and then using this estimated concentration to calculate n(i, t′). This can be repeated until the values of n(i, t′) and F(i, t′) have converged.
7. Appendix B An alternative model has been developed upon the basis of the probability of the adsorption, desorption, or diffusion of the analyte gas molecules upon interaction with a 10-layer sensor system. A 10-layer sensor system was modeled because the eventual aim was to compare the results with experimental devices fabricated by Langmuir-Blodgett deposition in which 10 layers of analyte-specific sensing molecules have been deposited. In
Kinetics of Gas Adsorption in Porphyrin Films
the model, the number of sensor molecule sites per layer is fixed at a predefined number that is the same for all layers. The model considers the actions of every analyte molecule within each of the 10 layers during a defined time step on the basis of the probabilities of different outcomes occurring; see Figure 9. Initially, there are no analyte molecules in the system anywhere. Free analyte molecules arrive at a fixed rate into the top layer of the sensor system, defined by the concentration and temperature of the analyte gas above the sensor. The model considers what happens to every analyte molecule in each layer every time step, starting at the top layer and working down to the bottom layer. During every time step, if there are any analyte molecules attached to sensor sites in the layer being considered then the model calculates the number likely to desorb and this is added to the number of free analyte molecules in that layer. This is achieved in the model by generating a random number between 0 and 1 for every adsorbed analyte molecule. If the number is less than the probability of desorption p(D), then the analyte desorbs, but if the random number is greater than the probability of desorption, then the analyte remains adsorbed. Likewise, the behavior of the free analyte molecules in each layer is considered. They can then either adsorb onto an empty sensor molecule site, diffuse up or down, or remain in the layer. This is calculated by generating a random number for each analyte molecule and comparing to the probabilities of adsorption p(A) and diffusion p(E). Of course adsorption can occur only if there are free sensor molecule sites to adsorb onto, so p(A) is multiplied by the fraction of free adsorption sites to find the fraction adsorbed f(A). Those analyte molecules that do not adsorb (1 - f (A)) may diffuse. (It is assumed that 50% of diffusing molecules diffuse up and 50% diffuse down.) Finally, those free analyte molecules that neither adsorb nor diffuse remain in the layer. This calculation is represented diagrammatically by Figure 10. Obviously, analyte molecules diffusing down become free molecules in the layer below, and those diffusing up become free analyte molecules in the layer above; however, analyte molecules diffusing up from the top layer escape and are lost from the system, and those diffusing down from the bottom layer “bounce” off of the substrate and remain as free analyte molecules in the bottom layer for the next round of calculation. This process is repeated for each time step with the starting values for the number of free and bound analyte molecules in each layer being renewed depending upon the results of the calculation for the previous time step; see Figure 11.
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When realistic values for the number of sensor molecules per layer, the layer thickness, the number of analyte molecules arriving per time step, and p(D), p(A), and p(E) were put into the model, the weakness of this approach to modeling the interaction of an analyte gas with a 10-layer sensor system was revealed. The probabilities p(D), p(A), and p(E) were calculated using the same diffusion constant D, desorption rate kd, and adsorption rate ka as described earlier in this article according to the following equations
p(D) ) kd∆t p(A) ) ka∆t p(E) )
2D∆t l2
where ∆t is the time step and l is the layer thickness. The arrival rate of analyte molecules was calculated for NO2 at 5 ppm using the impingement rate equation
impingement rate )
P (2mkBT)1/2
where P is the analyte partial pressure, m is the analyte molecular mass, kB is the Boltzmann constant, and T is the temperature of the gas in Kelvin. Bearing in mind that this model will give meaningful results only if the probabilities are all between 0 and 1 and the number of analyte molecules arriving per time step is a whole number, the model was restricted to using very short time steps (on the order of picoseconds) and very large numbers of sensor molecule sites per layer (∼107). Therefore, to model a meaningful period of time (on the order of seconds), a huge number of calculations were required. This meant that calculating the first microsecond of interaction took about 72 h on a desktop PC. Unfortunately, this time is too short to allow meaningful comparison to experiment. It is, however, reassuring that for short time scales this atomistic approach agrees with the meanfield theory described in section 2. Acknowledgment. We thank Professor C. A. Hunter and J. Hutchinson from the Department of Chemistry, University of Sheffield, for synthesizing the porphyrin used in this work. We also acknowledge the EPSRC, ref GR/S96845/01, for providing funding for this work. LA062231A
