Reference: "Principles of Adsorption and Reaction on Solid Surfaces" by Richard I. Masel [Wiley, 1996]
The elementary Redhead model introduces the fundamental features that apply in extensions to more realistic desorption models. These include interaction between gas molecules and adsorption sites on the solid surface, simple reaction mechanisms for desorption (e.g., first or second order), and the Arrhenius type temperature dependence of rate constants. These ideas can be readily extended to model more complex TPD spectra.
A typical TPD spectrum shows multiple peaks. One possible explanation is that the adsorbed molecules are not all equivalent. Rather, they may exist as subpopulations with different adsorption rates. Here we illustrate an extention of Redhead's model and the multiple peaks arising in TPD spectra.
Consider the single adsorbate species, A, that adsorbs at two different surface sites
or environments. In the first site we denote the adsorbed species as A and at the second
site as A' while the desorbed species is denoted A(g). Then we assume an desorption
mechanism given by four reaction steps:
Step 1, desorption from first site: A --> A(g) with rate constant k0(T)
Step 2, desorption from second site: A' --> A(g) with rate constant k1(T)
Steps 3 and 4, interconversion between adsorbates A and A':
.....A --> A', rate constant k2(T), and
.....A' --> A, rate constant k3(T).
Each step in the mechanism has its own reaction order, pre-exponential, and activation
energy: ki(T) = ki exp(-Ei/RT) Xni (X is
either A or A'). For instance, suppose their values are as follows.
.....n0 = 1, k0 = 1013 s-1, E0 = 26 kcal mol-1;
.....n1 = 1, k1 = 0.5 k0, E1 = 23 kcal mol-1;
.....n2 = 1, k2 = 0.2 k0, E2 = 26 kcal mol-1;
.....n3 = 1, k3 = 0.2 k0, E3 = E2 - (E0 - E1) = 23 kcal mol-1.
The activation energy for desorption from the second site is smaller than from the first site: A is more strongly bound at the first site. The frequency to approach the activation barrier is larger at the first site than the second by a factor of 2. Interconversion between A and A' is also governed by Arrhenius factors and the activation energies are comparable to those for desorption as if all steps must pass through the same activated state.
Again we use a linear temperature ramp: T = T0 + bt.
Finally, the problem is to solve for the rates of desorption from A and A'. From the preceding assumptions, we have the following rate equations:
r0 = -dA/dt = k0 exp(-E0/RT) An0 - k2 exp(-E2/RT) An2 + k3 exp(-E3/RT) A'n3,
for the rate of desorption from first type of adsorption site (A), and
r1 = -dA'/dt = -k3 exp(-E3/RT) A'n3 - k1 exp(-E1/RT) A'n2 + k2 exp(-E2/RT) A'n2
where also T' = dT/dt = b.
Let the initial surface coverages be A0 = 0.5 and A'0 = 0.3, and choose T0 = 250K and b = 10 K s-1.
Then a plot of the combined rates versus T is the model TPD spectrum: -d(A+A')/dt vs. T.
Several computer software products are available to solve the given set of coupled differential equations. You can use your favorite method and then compare with the following.
This solution uses the dynasys software. Parameterization was modified slightly as follows. Pre-exponential factors were absorbed into the exponential: k exp(-E/RT) = exp(-E/RT + ln(k)). Thus, pre-exponential parameters are replaced by lnk0 = 29.9336, lnk1 = 29.2405, and lnk2 = lnk3 = 28.3242. Also, the value of the quotient E/R (-e/k) is used instead of E and R separately: e0ok = e2ok = 13090, e1ok = e3ok = 11580. Also, for dynasys six functions were used; three defined via differentials: T' = dT/dt = b, A' = -w, B' = -x, and three functions defined explicitly: w, x and z as shown in the dynasys solution file. Initial values used in the solution were A(0) = 0.5, B(0) = 0.3, w(0) = 4.436 10-9, x(0) = -1.588 10-8, and z(0) = -1.144 10-8.
A mathcad template for solving this extended Redhead model is available. Click on the graph: . Note the TPD model spectrum (in green) shows two peaks; the lower temperature peak corresponds to the A' adsorption site with the smaller activation energy and the higher temperature peak corresponds to the A site with larger activation energy. Also shown are the coverages at the two sites. While A' is being desorbed it is also being converted into A as shown by the peak in the A coverage curve (in red).