COMMENT ON KINETIC ROUGHENING IN SLOW COMBUSTION OF PAPER

In a recent Letter, Maunuksela et al. present experiments on the combustion of paper [1]. They observe the roughening of the burning front and study the scaling properties of the correlation function of the height of the burning front. Maunuksela et al. find good agreement between the measured exponents characterizing the front roughening and the predictions of the Kardar-Parisi-Zhang (KPZ) equation [2], χKPZ = 1/2 (χ is the roughness exponent which characterizes the scaling of the saturated height-height correlation function G(r) with distance G(r) ∼ r). Reference [1] also comments on the results of earlier experiments on paper burning by Zhang et al. [3] which measured χ ≈ 0.71, but offers no explanation for the difference in the measured value of the exponent. Here, we show that the results of Maunuksela et al. [1] and of Zhang et al. [3] may be both understood under the framework of interface motion in disordered media [4–7]. For many experimental cases, the dominant source of noise in the dynamics is the disorder in the medium which is not time dependent. The universality classes for interface motion in disordered media have been identified and the values of the exponents are known, especially for (1 + 1) dimensions [4–7]. One of the universality classes for interface motion in disordered media [7] can be described by the directed percolation depinning (DPD) model [4–6]. For a driving force F smaller than a critical value Fc, which defines the depinning transition, the interface eventually stops, and the scaling ofG(r) is characterized by an exponent χDPD ≈ 0.63 [4]. If, on the other hand, F > Fc, the interface moves with an average velocity v ∼ (F −Fc) θ (where θ is a critical exponent), and G(r) presents two scaling regimes in the steady-state. For length-scales smaller than the correlation length given by the disorder of the medium ξ ∼ (F − Fc) −ν‖ (where ν‖ is a critical exponent), G(r) scales with an exponent χDPD ≈ 0.75, while for length-scales larger than ξ, G(r) scales with the KPZ exponent, χKPZ = 1/2 [6]. To check that the results reported by Maunuksela et al. are consistent with the above theory, we digitize the data reported in [1] and plot it in Fig. 1 along with the predictions of the DPD model (see Refs. [4–6]). It is visually apparent that the experimental and theoretical data sets have the same two scaling regimes as discussed in Ref. [6]. Furthermore, the same theory may explain why Zhang et al. found no crossover to a KPZ dominated regime. Zhang’s experiments were performed near the depinning transition, which implies that ξ was nearly as large as the system size. Thus, the quenched disorder dominates the scaling of G(r) leading to the observation of only one scaling regime with a roughness exponent χ ≈ χDPD. On the other hand, in the experiments reported by Maunuksela et al., a shorter correlation length ξ is found, so that quenched disorder dominates only for small length scales where an exponent with values close to χDPD was measured. For length scales larger than ξ, time-dependent disorder dominates, and the KPZ exponent χKPZ was measured, in agreement with the theory [6]. These results suggest that the DPD universality class may provide a compelling explanation of the experimental findings of Refs. [1,3].