First-Passage Exponent in Two-Dimensional Soap Froth
We report an experimental measurement of the temporal dependence of the area ${A}_{\mathrm{us}}$ in a two-dimensional soap froth which has not been swept by the passage of soap films up to time $t$, as the froth coarsens from an initial time ${t}_{0}$ within the scaling regime. We find ${A}_{\mathrm{us}}$ scales with the mean cell area $〈A〉$ as ${A}_{\mathrm{us}}\ensuremath{\propto}〈A{〉}^{\ensuremath{-}{\ensuremath{\theta}}^{\ensuremath{'}}}$, with a first-passage exponent ${\ensuremath{\theta}}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.16\ifmmode\pm\else\textpm\fi{}0.02$; and for the average perimeter $〈P〉$: $〈P〉\ensuremath{\propto}〈A{〉}^{{\ensuremath{\psi}}^{\ensuremath{'}}}$ and ${A}_{\mathrm{us}}\ensuremath{\propto}〈P{〉}^{{\ensuremath{\varphi}}^{\ensuremath{'}}}$ with values of the scaling exponents ${\ensuremath{\psi}}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.49\ifmmode\pm\else\textpm\fi{}0.01$ and ${\ensuremath{\varphi}}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.38\ifmmode\pm\else\textpm\fi{}0.02$. We also find a scaling function in ${A}_{\mathrm{us}}(t)/{A}_{\mathrm{us}}({t}_{0})$ versus $〈A(t)〉/〈A({t}_{0})〉$.