Screen-wick heat pipes are efficient passive heat transfer devices. The heat transfer capacity is determined by the capillary pressure, which is related to the film curvature inside the screen wick. A capillary evaporating film model was developed for the screen wick that divided the film into a capillary pressure microlayer region and a macroscopic film. The film surface shape in the capillary pressure microlayer region is mainly influenced by the long-range molecular forces (disjoining pressure) and the surface tension, while the film surface shape in the macroscopic film is affected by the wick geometry. The numerical results show that the present model more accurately predicts the three-dimensional characteristics of the liquid film inside the screen wick than the classical model. The wire opening distance and the wire diameter affect the capillary pressure and the flow path in the wick. As the liquid film recedes into the wick, the capillary force first increases and then decreases. The film dynamics in the wick are related to the liquid height, which automatically adapts to the required evaporation and pressure balance for a given heat load. High operating temperatures can lead to a mismatch between the capillary head and the pressure loss, leading to dryout in the wick. Thus, this work provides a reference for operating alkali-metal wick heat pipes.