Anisotropic Superattenuation of Capillary Waves on Driven Glass Interfaces

Metrological atomic force microscopy measurements are performed on the silica glass interfaces of photonic band-gap fibers and hollow capillaries. The freezing of attenuated out-of-equilibrium capillary waves during the drawing process is shown to result in a reduced surface roughness. The roughness attenuation with respect to the expected thermodynamical limit is determined to vary with the drawing stress following a power law. A striking anisotropic character of the height correlation is observed: glass surfaces thus retain a structural record of the direction of the flow to which the liquid was submitted.

Figure 1

Sketch of the drawing of hollow core photonic band-gap fibers. The two AFM images represent the surface topography (with the same height range of 1.6 nm) of the inner glass surfaces before (preform) and after (fiber) the hot drawing region (in orange). While isotropic before drawing, the subnanometer roughness exhibits clear elongated patterns along the fiber axis after drawing.

Figure 2

(a) 1D PSD of the surface roughness along the drawing direction corresponding to the two AFM images of Fig. 1: the inner core surface of the PBGF is much smoother than the reference preform surface. The slope of the triangle illustrates the $1/f$ evolution from Eq. (2) and the vertical position provides the roughness parameter $T_G/\gamma$. (b) The 2D height autocorrelation of the PGBF surface allows an appreciation of the roughness anisotropy (the AFM scan direction is at 45° from the fiber axis).

Figure 3

rms roughness $w$ of inner fiber surfaces obtained at increasing drawing stress ${\sigma }_M$ measured on $10\times 10\mu {\mathrm{m}}^2$ AFM images. Inset: the roughness attenuation $ϵ=1-w/w_0$ scales as the cubic root of the dimensionless stress ${\alpha }_D$.

Figure 4

Normalized 2D height autocorrelation functions of the inner surface roughness of the silica preform (a) and of three hollow capillary fibers obtained by drawing at increasing levels of stress: ${\sigma }_M=27$ (b), ${\sigma }_M=47$ (c), and ${\sigma }_M=114\mathrm{MPa}$ (d). The degree of anisotropy increases with ${\sigma }_M$ (and the adimensioned control parameter ${\alpha }_D$). The AFM scanning direction is at 45° to the drawing direction to rule out the residual intrinsic anisotropy of the instrument, which is appreciable on the isotropic preform surface shown in (a).

Figure 5

Angular dependence of the prefactor $A_F(\theta )$, i.e., the effective ratio $T_G/\gamma$ obtained from the 2D autocorrelation functions of Fig. 4. While the preform signal presents maxima along the AFM scan axes, the drawn fibers present maxima along the drawing direction (here at 45°) and reflect the physical anisotropy of the surface roughness. Dashed lines represent the fits obtained with the analytical expression (5).