Table 1.

Common adsorption isotherms, and corresponding expressions for surface pressure and the Marangoni modulus.

Isotherm	Description	Γ(C)	Π(Γ)	E0(Γ)
Henry	2D ideal gas dilute	KC	kBTΓ	kBTΓ
Freundlich	empirical	(KC)m	$k_{B}T\frac{\Gamma }{m}$	$k_{B}T\frac{\Gamma }{m}$
Langmuir	adsorbs to empty sites no interactions	$\frac{{\Gamma }_{\infty }KC}{1+KC}$	$k_{B}T{\Gamma }_{\infty }\text{ln}\left[\frac{1}{1-\Gamma /{\Gamma }_{\infty }}\right]$	$\frac{k_{B}T\Gamma }{1-\Gamma /{\Gamma }_{\infty }}$
Volmer	finite molecule size	$\frac{{\Gamma }_{\infty }KC}{\text{exp}\left[\frac{\Gamma }{{\Gamma }_{\infty }-\Gamma }\right]+KC}$	$\frac{k_{B}T\Gamma }{1-\Gamma /{\Gamma }_{\infty }}$	$\frac{k_{B}T\Gamma }{{\left(1-\Gamma /{\Gamma }_{\infty }\right)}^2}$
Frumkin	adsorbs to empty sites lateral interactions	$\frac{{\Gamma }_{\infty }KC}{\text{exp}\left[-\frac{\beta \Gamma }{k_{B}T}\right]+KC}$	$k_{B}T{\Gamma }_{\infty }\text{ln}\left[\frac{1}{1-\Gamma /{\Gamma }_{\infty }}\right]-\frac{\beta {\Gamma }^2}{2}$	$\frac{k_{B}T\Gamma }{1-\Gamma /{\Gamma }_{\infty }}-\beta {\Gamma }^2$
van der Waals	finite molecule size lateral interactions	$\frac{{\Gamma }_{\infty }KC}{\text{exp}\left[\frac{\Gamma }{{\Gamma }_{\infty }-\Gamma }-\frac{\beta \Gamma }{k_{B}T}\right]+KC}$	$\frac{k_{B}T\Gamma }{1-\Gamma /{\Gamma }_{\infty }}-\frac{\beta {\Gamma }^2}{2}$	$\frac{k_{B}T\Gamma }{{\left(1-\Gamma /{\Gamma }_{\infty }\right)}^2}-\beta {\Gamma }^2$