Domain growth in lipid membranes and the budding instability
We investigate the growth of a circular domain in a phase-separating lipid membrane and its relation to the budding instability. We describe the in-plane dynamics with a conserved, time-dependent Ginzburg-Landau model under radial symmetry. Numerical solutions show a crossover of the domain size $D$ as a function of time $t$ from $D\sim t^{1/3}$ at early times to $D\sim t^{1/2}$ at later times, followed by saturation due to finite system size. The time-dependent line tension $σ(t)$, evaluated from the evolving concentration profile, rises rapidly before approaching the planar-interface value. This behavior distinguishes an early line-tension-dominated regime from a later domain-size-dominated regime. To estimate the budding instability, we combine $D(t)$ and $σ(t)$ with a quasistatic criterion for the loss of stability of a partially budded domain. The budding-instability time decreases with increasing membrane spontaneous curvature and increases near the critical point. Thus, the evolution of both domain size and line tension is essential for determining the time required for the budding instability.