Chapter 10.1: Langmuir Adsorption to the Fundamental Diagram

Why Surface Chemistry Explains Traffic

In 1918, Irving Langmuir derived the equilibrium coverage of molecules adsorbing onto a surface with a finite number of identical, independent sites. A century later, the same mathematics describes the occupancy of a single-lane road by vehicles. The key insight is that the exclusion principle—no two molecules can occupy the same site, no two cars can occupy the same stretch of road—produces identical nonlinear dynamics.

This chapter develops the Langmuir–traffic isomorphism in full, derives the Greenshields fundamental diagram as a direct consequence, and extends the framework to mixed traffic classes via competitive Langmuir adsorption.

10.1.1 Langmuir Adsorption Kinetics

Consider a surface with \(N_s\) adsorption sites. Let \(\theta\) denote the fraction of occupied sites (the coverage). Molecules from a gas phase at concentration\(c\) adsorb onto empty sites with rate constant \(k_a\) and desorb with rate constant \(k_d\).

The Rate Equation

The adsorption rate is proportional to the gas-phase concentration\(c\) and the fraction ofempty sites \((1-\theta)\). Desorption depends only on the occupied fraction\(\theta\):

$$\frac{d\theta}{dt} = k_a \, c \, (1-\theta) - k_d \, \theta$$

The first term represents adsorption: molecules arrive at rate proportional to\(c\), but can only stick to empty sites. The factor \((1-\theta)\) is the exclusion factor—it enforces the constraint that each site holds at most one molecule. The second term represents desorption: occupied sites release their molecules at a rate independent of the gas-phase concentration.

Equilibrium Solution

Setting \(d\theta/dt = 0\) and defining the equilibrium constant \(K = k_a / k_d\):

$$k_a \, c \, (1-\theta^*) = k_d \, \theta^* \quad \Longrightarrow \quad \theta^* = \frac{K c}{1 + K c}$$

This is the Langmuir isotherm. At low concentration (\(Kc \ll 1\)), coverage grows linearly:\(\theta^* \approx Kc\). At high concentration (\(Kc \gg 1\)), the surface saturates:\(\theta^* \to 1\).

Time-Dependent Solution

The ODE is linear in \(\theta\) and can be solved exactly. Defining \(\lambda = k_a c + k_d\):

$$\theta(t) = \theta^* \bigl(1 - e^{-\lambda t}\bigr) + \theta_0 \, e^{-\lambda t}$$

The relaxation time \(\tau = 1/\lambda\) decreases with increasing concentration—busier roads equilibrate faster, just as busier surfaces reach steady-state coverage more quickly.

10.1.2 The Exclusion Factor IS the Car-Following Constraint

In traffic flow, a single-lane road is discretised into cells of length\(\ell\) (roughly one car length plus a safety buffer). Each cell is either occupied (a vehicle present) or empty. The density\(\rho\) (vehicles per unit length) maps directly to coverage:

$$\theta = \frac{\rho}{\rho_{\max}} \quad \text{where} \quad \rho_{\max} = \frac{1}{\ell}$$

A vehicle can only advance into the next cell if that cell is empty. The probability that the next cell is empty is\((1 - \theta) = (1 - \rho/\rho_{\max})\). This is precisely the Langmuir exclusion factor. The mean-field velocity becomes:

$$v(\rho) = v_{\max}\!\left(1 - \frac{\rho}{\rho_{\max}}\right)$$

This is the Greenshields linear speed-density relation (1935)—the oldest and simplest traffic model. It arises purely from site exclusion, without any explicit car-following psychology.

Physical Correspondence Table

Langmuir

Traffic

Coverage θ

Normalised density ρ/ρmax

Concentration c

Inflow rate / demand

Adsorption rate ka

On-ramp injection rate

Desorption rate kd

Off-ramp extraction rate

Exclusion (1-θ)

Headway constraint

Saturation θ→1

Gridlock ρ→ρmax

10.1.3 Deriving the Fundamental Diagram

The fundamental diagram of traffic is the flow-density relation\(q(\rho) = \rho \cdot v(\rho)\). Substituting the Greenshields velocity:

$$q(\rho) = v_{\max} \, \rho \left(1 - \frac{\rho}{\rho_{\max}}\right)$$

This is an inverted parabola with maximum flow (capacity) at\(\rho_c = \rho_{\max}/2\):

$$q_{\max} = \frac{v_{\max} \, \rho_{\max}}{4}$$

Now compare with the Langmuir adsorption flux. In the Langmuir model, the net flux of molecules onto the surface is:

$$J_{\text{ads}} = k_a \, c \, \theta(1-\theta) \propto \theta(1-\theta)$$

In the lattice gas picture, the particle current on a lattice with exclusion is exactly\(J = p \, \theta(1-\theta)\) where\(p\) is the hopping rate. This is the same parabolic form as the Greenshields fundamental diagram, confirming the isomorphism.

The Characteristic Wave Speed

The characteristic (kinematic wave) speed is:

$$c_k = \frac{dq}{d\rho} = v_{\max}\!\left(1 - \frac{2\rho}{\rho_{\max}}\right)$$

For \(\rho < \rho_c\), perturbations travelforward (free flow). For \(\rho > \rho_c\), they travel backward (congested regime). The sign change at\(\rho = \rho_c\) is the hallmark of the free-flow to congested phase transition.

10.1.4 Competitive Langmuir: Mixed Traffic

Real roads carry a mix of vehicle types—cars, trucks, buses—each with different effective lengths and speeds. In surface chemistry, the analog is competitive Langmuir adsorption where multiple species compete for the same sites.

Two-Species Competitive Adsorption

For two species A (cars) and B (trucks) with equilibrium constants\(K_A, K_B\) and gas-phase concentrations\(c_A, c_B\):

$$\theta_A = \frac{K_A c_A}{1 + K_A c_A + K_B c_B}, \qquad \theta_B = \frac{K_B c_B}{1 + K_A c_A + K_B c_B}$$

The total coverage is:

$$\theta_{\text{tot}} = \frac{K_A c_A + K_B c_B}{1 + K_A c_A + K_B c_B}$$

In traffic terms, trucks have a larger \(K_B\)because they occupy more road space per vehicle (the passenger car equivalent, PCE, is typically 1.5–3.0 for trucks). The presence of trucks reduces the capacity available for cars:\(\theta_A\) decreases as\(c_B\) increases.

Mixed-Traffic Fundamental Diagram

The total flow in the competitive case becomes:

$$q_{\text{tot}} = v_A \rho_A\!\left(1 - \frac{\rho_A + \eta \rho_B}{\rho_{\max}}\right) + v_B \rho_B\!\left(1 - \frac{\rho_A + \eta \rho_B}{\rho_{\max}}\right)$$

where \(\eta = K_B/K_A\) is the PCE factor. Increasing the truck fraction \(\rho_B / (\rho_A + \rho_B)\) shifts the peak of the fundamental diagram downward and to the left—capacity degrades.

10.1.5 Simulation: Fundamental Diagram & Langmuir Overlay

The following code plots the Greenshields fundamental diagram alongside the Langmuir isotherm, demonstrating their mathematical equivalence. It also shows how mixed traffic degrades capacity.

Fundamental Diagram & Langmuir Isotherm Overlay

Python
script.py81 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

10.1.6 Mixed-Traffic Degradation Analysis

The competitive Langmuir framework provides a quantitative prediction for how road capacity degrades with truck fraction. Let \(f\) be the fraction of trucks. The effective jam density seen by the traffic mix is:

$$\rho_{\max}^{\text{eff}} = \frac{\rho_{\max}}{1 - f + \eta f}$$

And the reduced capacity:

$$q_{\max}^{\text{eff}} = \frac{v_{\max} \, \rho_{\max}}{4(1 - f + \eta f)}$$

For \(\eta = 2\) and\(f = 0.2\), the capacity drops by a factor of \(1/(1 + 0.2) = 0.833\)—a 17% reduction. This matches the empirical Highway Capacity Manual correction factors remarkably well.

Mixed-Traffic Capacity Degradation

Python
script.py52 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key Takeaways

  • The Langmuir exclusion factor \((1-\theta)\) is mathematically identical to the car-following headway constraint in traffic flow.
  • The Greenshields fundamental diagram \(q = v_{\max}\rho(1 - \rho/\rho_{\max})\) is the lattice gas current with exclusion.
  • Competitive Langmuir adsorption models mixed traffic (cars + trucks) and predicts capacity degradation consistent with engineering handbooks.
  • The mapping \(\theta \leftrightarrow \rho/\rho_{\max}\) is exact in mean-field and serves as the foundation for the TASEP models in the next chapter.
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