Parameter | Meaning | General | Sphere | Rhombic dodecahedron |
---|---|---|---|---|
E | Energy function or Hamiltonian | J s+λ s (s−S)2+λ v (v−V)2 | ||
J | Edhesion energy (per contact length) | J | J | J |
s | Cell surface | k s l 2 | 4π r 2 | \(8\sqrt {2}l^{2}\) |
v | Cell volume | k v l 3 | \(\frac {4}{3}\pi r^{3}\) | \(\frac {16}{3\sqrt {3}}l^{3}\) |
S | Rest surface area | \(k_{s}{L_{s}^{2}}\) | \(4\pi {R_{s}^{2}}\) | \(8\sqrt {2}{L_{s}^{2}}\) |
V | Ttarget cell volume | \(k_{v}{L_{v}^{3}}\) | \(\frac {4}{3}\pi {R_{v}^{3}}\) | \(\frac {16}{3\sqrt {3}}{L_{v}^{3}}\) |
λ s | Surface constraint | λ s | λ s | λ s |
λ v | Volume constraint | λ v | λ v | λ v |
l | Basic length scale | l | r (radius) | l |
k s | Surface scaling factor | \(\frac {s}{l^{2}}\) | 4π | \(8\sqrt {2}\) |
k v | Volume scaling factor | \(\frac {v}{l^{3}}\) | \(\frac {4}{3}\pi \) | \(\frac {16}{3\sqrt {3}}\) |
L s | Rest surface area, using basic length scale | \(\sqrt {\frac {S}{k_{s}}}\) | \(R_{s}=\frac {1}{2}\sqrt {\frac {S}{\pi }}\) | \(\sqrt {\frac {S}{8\sqrt {2}}}\) |
L v | Target cell volume,using basic length scale | \(\sqrt [3]{\frac {V}{k_{v}}}\) | \(R_{v}=\sqrt [3]{\frac {3V}{4\pi }}\) | \(\sqrt [3]{\frac {3\sqrt {3}V}{16}}\) |
E | Energy function or Hamiltonian, using basic length scale | \({Jk}_{s}l^{2}+\lambda _{s}(k_{s}l^{2}-k_{s}{L_{s}^{2}})^{2} +\lambda _{v}(k_{v}l^{3}-k_{v}{L_{v}^{3}})^{2}\) | \(4\pi Jr^{2}+\lambda _{s}(4\pi r^{2}-4\pi {R_{s}^{2}})^{2} +\lambda _{v}(\frac {4}{3}\pi r^{3}-\frac {4}{3}\pi {R_{v}^{3}})^{2}\) | \(8\sqrt {2}Jl^{2}\) \(+\lambda _{s}(8\sqrt {2}l^{2}-8\sqrt {2}{L_{s}^{2}})^{2} +\lambda _{v}(\frac {16}{3\sqrt {3}}l^{3}-\frac {16}{3\sqrt {3}}{L_{v}^{3}})^{2}\) |
\(\frac {\partial E}{\partial l}\) | Energy variation per length change | \(2k_{s}l\left (\gamma -\frac {3k_{v}}{2k_{s}}l\Pi \right)\) | 4π r(2γ−r Π) | \(16\sqrt {2}l\left (\gamma -\frac {l}{\sqrt {6}}\Pi \right)\) |
γ | Interfacial tension | \(J+2k_{s}\lambda _{s}(l^{2}-{L_{s}^{2}})\) | \(J+8\pi \lambda _{s}(r^{2}-{R_{s}^{2}})\) | \(J+16\sqrt {2}\lambda _{s}(l^{2}-{L_{s}^{2}})\) |
Π | Pressure | \(-2k_{v}\lambda _{v}(l^{3}-{L_{v}^{3}})\) | \(-\frac {8}{3}\pi \lambda _{v}(r^{3}-{R_{v}^{3}})\) | \(-\frac {32}{3\sqrt {3}}\lambda _{v}(l^{3}-{L_{v}^{3}})\) |
\(\frac {\partial E}{\partial l}\) | Energy variation per length change, full expansion | 2k s (a l 5+b l 3−c l 2+τ l) | 8π(a r 5+b r 3−c r 2+τ r) | \(16\sqrt {2}\left (al^{5}+bl^{3}-cl^{2}+\tau l\right)\) |
a | Aggregate parameterin \(\frac {\partial E}{\partial l}\) equation | \(\frac {3{k_{v}^{2}}\lambda _{v}}{k_{s}}\) | \(\frac {4}{3}\pi \lambda _{v}\) | \(\frac {16}{9}\sqrt {2}\lambda _{v}\) |
b | Aggregate parameterin \(\frac {\partial E}{\partial l}\) equation | 2k s λ s | 8π λ s | \(16\sqrt {2}\lambda _{s}\) |
c | Aggregate parameterin \(\frac {\partial E}{\partial l}\) equation | \(\frac {3{k_{v}^{2}}\lambda _{v}{L_{v}^{3}}}{k_{s}}\) | \(\frac {4}{3}\pi \lambda _{v}{R_{v}^{3}}\) | \(\frac {16}{9}\sqrt {2}\lambda _{v}{L_{v}^{3}}\) |
τ | Length-independent component of interfacial tension | \(J-2k_{s}\lambda _{s}{L_{s}^{2}}\) | \(J-8\pi \lambda _{s}{R_{s}^{2}}\) | \(J-16\sqrt {2}\lambda _{s}{L_{s}^{2}}\) |
ϕ | \(\frac {b}{6a}\) | \(\frac {{k_{s}^{2}}\lambda _{s}}{9{k_{v}^{2}}\lambda _{v}}\) | \(\frac {\lambda _{s}}{\lambda _{v}}\) | \(\frac {3\lambda _{s}}{2\lambda _{v}}\) |
ψ | \(\frac {c}{8a}\) | \(\frac {{L_{v}^{3}}}{8}\) | \(\frac {{R_{v}^{3}}}{8}\) | \(\frac {{L_{v}^{3}}}{8}\) |
ν (when ϕ>0) | \(\frac {\tau }{12a\phi ^{2}}\) | \(\frac {\left (J-2k_{s}\lambda _{s}{L_{s}^{2}}\right)9{k_{v}^{2}}\lambda _{v}}{4{k_{s}^{3}}{\lambda _{s}^{2}}}\) | \(\frac {\left (J-8\pi \lambda _{s}{R_{s}^{2}}\right)\lambda _{v}}{16{\pi \lambda _{s}^{2}}}\) | \(\frac {\left (J-16\sqrt {2}\lambda _{s}{L_{s}^{2}}\right)\lambda _{v}}{48\sqrt {2}{\lambda _{s}^{2}}}\) |
μ(when ϕ>0) | \(\frac {\psi }{\phi ^{\frac {3}{2}}}\) | \(\frac {27{k_{v}^{3}}\lambda _{v}^{\frac {3}{2}}{L_{v}^{3}}}{8{k_{s}^{3}}\lambda _{s}^{\frac {3}{2}}}\) | \(\frac {{R_{v}^{3}}\lambda _{v}^{\frac {3}{2}}}{8\lambda _{s}^{\frac {3}{2}}}\) | \(\frac {{L_{v}^{3}}\lambda _{v}^{\frac {3}{2}}}{6\sqrt {6}\lambda _{s}^{\frac {3}{2}}}\) |
ν ′(when ϕ=0) | \(\frac {\tau }{12a}\) | \(\frac {{Jk}_{s}}{36{k_{v}^{2}}\lambda _{v}{\vphantom {\frac {1}{2}}}}\) | \(\frac {J}{16\pi \lambda _{v}}\) | \(\frac {3J}{64\sqrt {2}\lambda _{v}}\) |
μ ′(when ϕ=0) | ψ | \(\frac {{L_{v}^{3}}}{8}\) | \(\frac {{R_{v}^{3}}}{8}\) | \(\frac {{L_{v}^{3}}}{8}\) |
Bifurcation 1 (γ(l ∗)=0) | Transition from negative to positive interfacial tension at equilibrium | \(\nu =-\frac {9{k_{v}^{2}}\lambda _{v}{L_{v}^{2}}}{2{k_{s}^{2}}\lambda _{s}}\) | \(\nu =-\frac {\lambda _{v}{R_{v}^{2}}}{2\lambda _{s}}\) | \(\nu =-\frac {\lambda _{v}{L_{v}^{2}}}{3\lambda _{s}}\) |
ν ′=0 | ν ′=0 | ν ′=0 | ||
Bifurcation 2 (pseudo-transcritical) | Transition of l ∗=0 from unstable to stable | ν=0 | ν=0 | ν=0 |
ν ′=0 | ν ′=0 | ν ′=0 | ||
Bifurcation 3 (fold) | Transition from 2 to 0 non-trivial equilibria | ν=f(μ)(μ−f(μ)), where \(f(\mu)=\sinh \left (\frac {1}{3}\text {arcsinh} \left (\mu \right)\right)\) | ν=f(μ)(μ−f(μ)) | ν=f(μ)(μ−f(μ)) |
\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\) | \(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\) | \(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\) |