Table 1.

Estimated attachment energies E_A (i) for spherical particles of radius r_p attached to an oil–water interface with a typical interface tension $\gamma =28\mathrm{m}\mathrm{N}{\mathrm{m}}^{-1}\approx 6.75{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\mathrm{n}{\mathrm{m}}^{-2}$ [35], assuming that the interface tensions of the particle with both phases are identical and (ii) for spherical particles that are half-wrapped by a membrane with bending rigidity $\kappa \approx 2{10}^{-19}\mathrm{J}\approx 50k_{\mathrm{B}}T$ [36, 37]. We use the particle-membrane adhesion strength $w\approx 0.2k_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ between DMPC and silica ($w=0.5-1\mathrm{m}\mathrm{J}{\mathrm{m}}^{-2}=0.12\mathrm{\text{–}}0.24k_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ is reported in [38]), the adhesion strength $w\approx 7\cdot {10}^{-3}{k}_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ for receptor-ligand bond-mediated interaction ($w=0.03\mathrm{m}\mathrm{J}{\mathrm{m}}^{-2}=7\cdot {10}^{-3}$$k_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ is reported in [39]), and the adhesion strength $w\approx 2.5\cdot {10}^{-6}{k}_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ between DOPC/DOPG and glass ($w={10}^{-5}\mathrm{m}\mathrm{J}{\mathrm{m}}^{-2}=2.4\cdot {10}^{-6}{k}_{\mathrm{B}}T\mathrm{n}{\mathrm{m}}^{-2}$ is reported in [40]). The particle-membrane interaction is only attractive beyond a threshold particle radius.

System	$r_{\mathrm{p}}$	$E_{\mathrm{A}}$
Oil–water interface	$10\mathrm{n}\mathrm{m}$	$-2\cdot {10}^3{k}_{\mathrm{B}}T$
Oil–water interface	$1\mu \mathrm{m}$	$-2\cdot {10}^7{k}_{\mathrm{B}}T$
DMPC-silica	$10\mathrm{n}\mathrm{m}$	$375k_{\mathrm{B}}T$
DMPC-silica	$20\mathrm{n}\mathrm{m}$	$-375k_{\mathrm{B}}T$
DMPC-silica	$50\mathrm{n}\mathrm{m}$	$-5650k_{\mathrm{B}}T$
Receptor-ligand	$50\mathrm{n}\mathrm{m}$	$410k_{\mathrm{B}}T$
Receptor-ligand	$100\mathrm{n}\mathrm{m}$	$-250k_{\mathrm{B}}T$
DOPC/DOPG-glass	$1\mu \mathrm{m}$	$600k_{\mathrm{B}}T$
DOPC/DOPG-glass	$5\mu \mathrm{m}$	$-160k_{\mathrm{B}}T$