Statistical analysis of the drying pattern of coffee

TL;DR

This paper investigates the statistical properties of coffee stain patterns, analyzing how sugar content affects the surface roughness, crack statistics, and fractal dimensions, revealing transitions towards Gaussian free field behavior at higher sugar levels.

Contribution

It provides a detailed statistical and multifractal analysis of coffee stain patterns with varying sugar content, linking physical properties to pattern complexity and crack behavior.

Findings

Fractal dimensions approach Gaussian free field exponents at high sugar levels.

Mass fractal dimension decreases with increasing sugar, indicating more sparse patterns.

Crack statistics and surface roughness are quantitatively characterized.

Abstract

In this study, we experimentally study the dried pattern droplets of coffee with and without sugar. We statistically analyze the rough surface formed after the stain becomes dried. The amount of sugar is controlled by the mass $m$. Along with the formation of the coffee ring, we discuss the Marangoni effect, in the system, and also analyzed the statistics of the cracks. For large enough $m$ values, the exponents approach to the ones for the Gaussian free field (GFF) (the loop fractal dimension $\frac{3}{2}$, loop and gyration radius distribution exponents $\tau_l=\frac{7}{3}$ and $\tau_r=3$ respectively). Using the multifractal analysis (MA) for the mass configuration of the dried pattern, we numerically show that, the mass-fractal dimension is $1.76\pm 0.04$ for the case without sugar, which decreases increasing the sugar. This is explained by the fact that the droplet becomes more hydrophilic, resulting in more sparse spatial patterns, in agreement compatible with the contact angle analysis.