Roughness at the Film-Liquid Interface

Roughness, generally speaking, is not easily incorporated into the multilayer formahsm because it violates the assumption of lateral heterogeneity. On the other hand, it is certainly essential. The chapter (M. Urbakh et al. 2006, in this volume) describes various ways to deal with roughness. Roughness may very well occur not only at a quartz-Hquid interface, but also at a film-liquid interface. There is a logical extension of the formahsm treated to the case of multilayers (M. Urbakh et al. 2006, in this volume). One uses an impedance of the hquid of the form  

The term in front of the bracket is the well-known Kanazawa expression (Eq. 61). The correction terms inside the brackets take care of roughness at the interface of the hquid k is the lateral correlation length of roughness (where the spectriun of spatial frequencies is assiuned to be Gaussian), hr is the root-mean-square roughness, and 8 = 2r]l pco)) / is the penetration 

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