Laws of 4D Printing

Three-dimensional (3D) printing is an additive manufacturing process. Accordingly, four-dimensional (4D) printing is a manufacturing process that involves multiple research fields. 4D printing conserves the general attributes of 3D printing (such as material waste reduction, and elimination of molds, dies, and machining) and further enables the fourth dimension of products to provide intelligent behavior over time. This intelligent behavior is encoded (usually by an inverse mathematical problem) into stimuli-responsive multi-materials during printing, and is enabled by stimuli after printing. The main difference between 3D- and 4D-printed structures is the presence of one additional dimension, which provides for smart evolution over time. However, currently there is no general formula for modeling and predicting this additional dimension. Herein, by starting from fundamentals, we derive and validate a general bi-exponential formula with a particular format that can model the time-dependent behavior of nearly all 4D (hydro-, photochemical-, photothermal-, solvent-, pH-, moisture-, electrochemical-, electrothermal-, ultrasound-, etc. responsive) structures. We show that two types of time constants are needed to capture the correct time-dependent behavior of 4D multi-materials. We introduce the concept of mismatch-driven stress at the interface of active and passive materials in 4D multi-material structures, leading to one of the two time constants. We develop and extract the other time constant from our unified model of time-dependent behavior of nearly all stimuli-responsive materials. Our results starting from the most fundamental concepts and ending with governing equations can serve as general design principles for future research in the field of 4D printing, where time-dependent behaviors should be properly understood, modeled, and predicted.