Abstract The incorporation of natural fibers into polymer matrices poses challenges due to physicochemical incompatibility, which is typically addressed through precursor modification or the use of compatibilizers. Here, we introduce a novel type of composite that overcomes this challenge by utilizing a network of fine, porous cellulosic sheets inter-diffused with a commercial paraffin films. This approach physically adheres the fiber network to the matrix, preserving its structure. Microscopy images confirm the formation of the proposed microstructure, and mechanical testing reveals a gradual increase in modulus and strength with the incorporation of cellulose. The maximum incorporation achieved was 7.6% (w/w) of cellulosic fibers, resulting in a 167% increase (1.67 times improved) in composite stiffness. Moreover, these composites exhibit ductility, with an average deformation of 410 ± 38%, corresponding to 20% reduction in relation to pure matrix. Our findings demonstrate the potential of this approach for developing sustainable materials with improved mechanical properties. incompatibility compatibilizers Here fine interdiffused inter diffused films matrix structure microstructure cellulose 76 7 6 7.6 w/w ww w (w/w 167 1.67 1 67 (1.6 stiffness Moreover ductility 41 38 38% 20 properties 7. 16 1.6 (1. 4 3 2 1. (1 (

In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented. The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering. The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient. Numerical results will be compared with the exact values for several energies. The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition. The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics. A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.