Context. We present new insight into the long-standing problem of plasma heating in the lower solar atmosphere in terms of collisional dissipation caused by two-fluid Alfvén waves. Aims. Using numerical simulations, we study Alfvén wave propagation and dissipation in a magnetic flux tube and their heating effect. Methods. We set up 2.5-dimensional numerical simulations with a semi-empirical model of a stratified solar atmosphere and a force-free magnetic field mimicking a magnetic flux tube. We consider a partially ionized plasma consisting of ion + electron and neutral fluids, which are coupled by ion-neutral collisions. Results. We find that Alfvén waves, which are directly generated by a monochromatic driver at the bottom of the photosphere, experience strong damping. Low-amplitude waves do not thermalize sufficient wave energy to heat the solar atmospheric plasma. However, Alfvén waves with amplitudes greater than 0.1 km s −1 drive through ponderomotive force magneto-acoustic waves in higher atmospheric layers. These waves are damped by ion-neutral collisions, and the thermal energy released in this process leads to heating of the upper photosphere and the chromosphere. Conclusions. We infer that, as a result of ion-neutral collisions, the energy carried initially by Alfvén waves is thermalized in the upper photosphere and the chromosphere, and the corresponding heating rate is large enough to compensate radiative and thermal-conduction energy losses therein.