Based on the findings of molecular dynamics simulations, we propose a novel diffusion model for the self-assembly in dip-pen nanolithography. A central question in such modeling is how a nascent droplet created below an AFM tip spreads out to form a self-assembled monolayer (SAM) on a substrate later. In the present model, a molecule dropping from the tip pushes a molecule on the substrate out of its original position, and the molecule pushed out in turn pushes its own neighbor. A SAM grows through such a series of push-induced movements. The initial pushing propagates all the way to the periphery where there is no molecule to push out. By contrast, according to the previous hopping-down model, a molecule moves by stepping over molecules trapped on the substrate and occasionally hops down to the substrate. By implementing our model in random walk simulations, we study the structure and growth dynamics of the SAM generated by a fixed tip and the lines and characters created by a moving tip. We investigate how the SAM is influenced by the molecular dripping rate and tip scan speed. Compared to the hopping model, the present model gives a SAM 4 growing faster and more fluctuating in its periphery. A salient feature of our model is its ability to generate various SAMs by changing the directional coherence length of the push-induced displacement. If we choose the coherence length to be zero, each push-induced displacement is random in direction to give a compact circular SAM. As the directional coherence length increases, the SAM becomes a noncircular pattern with distinct branches. In the limit of an infinite coherence length, the SAM becomes a long narrow cross due to the substrate anisotropy.