We report a relationship between the order of stress singularity λ and surface free energy γ in the adhesion between epoxy resins and polymers. In previous works, considering the existence of a stress singular field at the bonding edge, the evaluation of adhesion strength has been proposed using both the order of stress singularity λ and the intensity of stress singularity K. The characteristic equation can be solved to determine λ. Studies on the dependence of λ on γ, however, have not yet come to our knowledge, so far. This is probably due to difficulty in establishing a physical model that connectsλ and γ. Here we successfully elucidated the dependence of the order of stress singularity λ on the surface free energy γ of adherend by substituting E₁/E₂ (the ratio of Young’s modulus of adhesive to Young’s modulus of the adherend) with the surface free energy γ in the previously reported relation between λ and E₁/E₂. Consequently, a discernible pattern emerged: λ increased as γ decreased, reaching a saturation value of 0.5 at 20.4 mN/m. Furthermore, the value of λ remained consistently at 0.5 even as γ decreased to 15.3 mN/m, approaching the lower limit of γ for practical materials. The robust outcome unequivocally established the critical value of λ in the low surface free energy region as 0.5.