tutorial about isosteric heat

docs/make.jl

@@ -16,7 +16,7 @@ makedocs(;
     "Home" => "index.md",
     "Background" => "tutorials/background.md",
     "Getting Started" => "tutorials/getting_started.md",
-    "Tutorials" => Any["tutorials/tutorial.md"],
+    "Tutorials" => Any["tutorials/tutorial.md", "tutorials/isosteric_heat.md"],
     "Supported models" => "models/models.md",
     "Reference" => "reference.md",
   ],

docs/src/models/models.md

@@ -25,28 +25,26 @@ Langmuir.ZeroIsotherm
 Langmuir.Henry
 ```
 
-## Two-parameter Isotherm Models
+## Three-parameter Isotherm Models
 
 ```@docs
 Langmuir.LangmuirS1
-Langmuir.Freundlich
 ```
 
-## Three-parameter Isotherm Models
+## Four or more parameter Isotherm Models
 
 ```@docs
+Langmuir.Freundlich
 Langmuir.RedlichPeterson
 Langmuir.Toth
-Langmuir.Sips
-Langmuir.LangmuirFreundlich
 Langmuir.Unilan
 Langmuir.Quadratic
 Langmuir.BrunauerEmmettTeller
+Langmuir.Sips
+Langmuir.LangmuirFreundlich
+Langmuir.MultiSite
+Langmuir.ThermodynamicLangmuir
 ```
 
-## Four and more parameters Isotherm Models
 
-```@docs
-Langmuir.MultiSite
-```
 

docs/src/tutorials/isosteric_heat.md

@@ -0,0 +1,111 @@
+# [Comparing the estimation of isosteric heat from isotherms with Langmuir.jl](@id isosteric)
+
+During adsorption, heat is released as adsorbate molecules transition to a lower energy state on the adsorbent surface compared to the bulk gas phase. This released heat partially accumulates in the adsorbent, causing a temperature rise on its surface, which in turn can decelerate the adsorption process - as adsorption is an exothermic process.
+
+Predicting equilibrium loading at a given temperature and pressure often receives greater focus. However, accurately predicting the energy release as a function of these same variables is equally critical, as it is as impactful as loading in the adsorption process.
+
+On an energetically heterogeneous surface, the isosteric heat decreases as surface loading increases. Isotherms like the single-site Langmuir model, which assume a constant heat of adsorption regardless of surface loading, are therefore often inadequate for accurately representing experimental data in many cases.
+
+Other isotherms can account for surface heterogeneity such as Toth:
+
+$n = \frac{MKP}{(1 + M(K P)^f)^{1/f}}$ 
+
+However, exhibits an unrealistic infinitely large negative value at high surface loading. below you can see the behavior of the isosteric heat for the toth isotherm as a function of the loading. You can also see the comparison of the analytical expression (which is very tedious to derive) and the one given by `Langmuir.jl` with automatic differentiation.
+
+```@example Toth
+using Langmuir, Plots
+toth = Toth(7.464, 3.6e-7, 8.3144*-5649.81, 0.5, 50.22)
+p_range = 1e-5:10.0:10*101325.0 |> collect
+loading_350_t = loading_at_T(toth, p_range, 350.0)
+loading_300_t = loading_at_T(toth, p_range, 300.0)
+ΔH_350_t = map(p -> isosteric_heat(toth, p, 350.0), p_range)
+ΔH_300_t = map(p -> isosteric_heat(toth, p, 300.0), p_range)
+plot(loading_300_t/toth.M, ΔH_300_t, framestyle=:box, title = "Toth", xlabel = "covered fraction", ylabel = "ΔH (J/mol)", label = "Automatic Differentiation - 300 K")
+plot!(loading_350_t/toth.M, ΔH_350_t, framestyle=:box, label = "Automatic Differentiation - 350 K")
+
+function Q_st1(model::Toth, n1, T)
+    E1 = model.E
+    β = model.β
+    R = 8.3144  # Assuming Rgas gives the gas constant for the type of T
+    f = model.f₀ - model.β / T
+    n1_0 = model.M  # Assuming saturation loading as reference loading
+
+    # Calculate the two terms inside the brackets in the equation
+    term1 = log.(n1 ./ n1_0) ./ (1 .- (n1 ./ n1_0).^f)
+    term2 = log.((n1 ./ n1_0) ./ (1 .- (n1 ./ n1_0).^f).^(1/f))
+
+    # Calculate Q_st1 using the main formula
+    Q_st1_value = E1 .+ (β * R / f) * (term2 .- term1)
+    return Q_st1_value
+end
+
+ΔH_analytical = Q_st1(toth, loading_350_t, 350.0)
+
+plot!(loading_350_t/toth.M, ΔH_analytical, label = "Analytical - 350 K", color = :slateblue2)
+```
+
+It can be noticed that the isosteric heat for the Toth isotherm blows for high surface coverages. 
+
+
+Multi-site Langmuir can also account for surface heterogeneity. Below you can see the behavior of the isosteric heat as a function of the surface coverage.
+
+```@example Multisite
+using Langmuir, Plots #hide
+dualsite = MultiSite(LangmuirS1(2.337, 6.6e-11, 8.3144*-5340.87),
+ LangmuirS1(3.490, 3.4e-11, 8.3144*-4273.13))
+p_range = 1e-5:10.0:10*101325.0 |> collect
+loading_270 = loading_at_T(dualsite, p_range, 270.0)
+loading_350 = loading_at_T(dualsite, p_range, 350.0)
+ΔH_270 = map(p -> isosteric_heat(dualsite, p, 270.0), p_range)
+ΔH_350 = map(p -> isosteric_heat(dualsite, p, 350.0), p_range)
+plot(loading_270/(2.337 + 3.490), ΔH_270, framestyle=:box, title = "Dualsite Langmuir", xlabel = "covered fraction", ylabel = "ΔH (J/mol)", label = "270 K")
+plot!(loading_350/(2.337 + 3.490), ΔH_350, framestyle=:box, label = "350 K")
+```
+It can bee seen that the isosteric heat presents an s-shape varying from more energetic to less energetic sites.
+
+[Some literature](https://doi.org/10.1007/s10450-020-00296-3) points that these behaviors are non-physical and potentially problematic when trying to model thermal effects in adsorption.
+
+To overcome it, the adsorption Nonrandom Two-Liquid (aNRTL) activity coefficient model into an activity-based formulation for Langmuir isotherm, [Chang et al. 2020](https://doi.org/10.1007/s10450-019-00185-4) proposed a thermodynamic Langmuir (tL) which seems to have superior properties for predicting the isosteric heat compared to Toth and Multisite Langmuir.
+
+The equation for the loading $n_i$ in terms of $\gamma_i$ and $\gamma_\phi$ is:
+
+$n_i = \frac{M K_i P}{\frac{\gamma_i}{\gamma_\phi} + K_i P}$
+
+where $K_i$ is also a function of temperature $K_i = K_i^\circ\exp{\frac{-E}{RT}}$, and the activity coefficients $\gamma_i$ and $\gamma_\phi$ are given by:
+
+$\gamma_i = \exp\left(\frac{d \frac{g^E}{RT}}{d \theta_i}\right)$
+
+$\gamma_\phi = \exp\left(\frac{d \frac{g^E}{RT}}{d \theta_{\phi}}\right)$
+
+The Gibbs excess free energy term, $\frac{g^E}{RT}$, is expressed as:
+
+$\frac{g^E}{RT} = \frac{\theta_i \theta_\phi \tau_{i\phi} (G_{i\phi} - 1)}{\theta_i G_{i\phi} + \theta_\phi}$
+
+where:
+
+$B_{i\phi}$ is a model parameter, $\tau_{i\phi} = B_{i\phi} / T$, $G_{i\phi} = \exp(-0.3 \cdot \tau_{i\phi})$.
+
+Here, $\theta_i$ and $\theta_\phi$ are the coverage terms for the adsorbed species and adsorption sites, 
+respectively ($\theta_i  + \theta_{\phi} = 1$), and $T$ is the temperature.
+
+Below you can see how to iniatilize the thermodynamic langmuir model in `Langmuir.jl` and use it to predict the loading and isosteric heat.
+
+```@example tLangmuir
+using Langmuir, Plots #hide
+tlang = ThermodynamicLangmuir(5.890, 6.1e-11, -4599.86*8.3144, -762.51)
+p_range = 1e-5:10.0:10*101325.0 |> collect
+loading_270 = loading_at_T(tlang, p_range, 270.0)
+loading_350 = loading_at_T(tlang, p_range, 350.0)
+ΔH_270 = map(p -> isosteric_heat(tlang, p, 270.0), p_range)
+ΔH_350 = map(p -> isosteric_heat(tlang, p, 350.0), p_range)
+plot(loading_270/tlang.M, ΔH_270, framestyle=:box, xlabel = "covered fraction", ylabel = "ΔH (J/mol)", title = "tLangmuir", label = "270 K")
+plot!(loading_350/tlang.M, ΔH_350, framestyle=:box, label = "350 K")
+plot!([minimum(loading_270), maximum(loading_270)]./tlang.M,  [-4599.86*8.3144, -4599.86*8.3144], label = "Langmuir")
+```
+
+It is evident that the thermodynamic Langmuir model exhibits behavior that is notably different from both the Toth and multisite Langmuir models. [As highlighted in existing literature](https://link.springer.com/article/10.1007/s10450-019-00185-4), the thermodynamic Langmuir model offers more accurate estimates of isosteric heat while maintaining high predictive accuracy for equilibrium loading. It can also be use to predict binary adsorption. 
+
+Analytical expressions for the isosteric heat can become quite extensive. In fact, there is an [entire manuscript](https://aiche.onlinelibrary.wiley.com/doi/10.1002/aic.17186) dedicated to deriving such expressions for a number of isotherms. 
+
+In `Langmuir.jl`, we leverage automatic differentiation to get accurate estimates of the required derivatives for the isosteric heat without requiring such extensive derivations.
+

src/models/thermodynamic_langmuir.jl

@@ -1,3 +1,31 @@
+"""
+    ThermodynamicLangmuir(M, K₀, E, Bᵢᵩ)
+
+The `ThermodynamicLangmuir` struct represents a thermodynamic Langmuir model with activity coefficients calculated using an aNRTL-like approach.
+
+## Parameters
+
+- `M`: Saturation loading `[mol/kg]`
+- `K₀`: Affinity parameter at high temperature `[1/Pa]`
+- `E`: Adsorption energy `[J/mol]`
+- `Bᵢᵩ`: Interaction energy parameter between adsorbate species `i` and vacant sites `[J/mol]`
+
+## Description
+
+The Langmuir equation is given by:
+
+    nᵢ = (M * K * P) / (γᵢ/γᵩ + K * P)
+
+where `nᵢ` is the adsorbed amount of species `i`, `K` is calculated as:
+
+    K = K₀ * exp(-E / (R * T))
+
+where:
+- `R` is the gas constant
+- `T` is the temperature
+
+The activity coefficients `γᵢ` and `γᵩ` are determined using the Gibbs excess free energy, `gᴱ/RT`, which is calculated based on the surface fractions (`θᵢ`, `θᵩ`) and interaction parameters derived from `Bᵢᵩ`. This free energy value is used in the `activity_coefficient` function to compute the activity coefficients of the adsorbate and phantom molecules.
+"""
 @with_metadata struct ThermodynamicLangmuir{T} <: IsothermModel{T}
     (M::T, (0.0, Inf), "saturation loading")
     (K₀::T, (0.0, Inf), "affinity parameter")