Shape‐Controlled, Self‐Wrapped Carbon Nanotube 3D Electronics

The mechanical flexibility and structural softness of ultrathin devices based on organic thin films and low‐dimensional nanomaterials have enabled a wide range of applications including flexible display, artificial skin, and health monitoring devices. However, both living systems and inanimate systems that are encountered in daily lives are all 3D. It is therefore desirable to either create freestanding electronics in a 3D form or to incorporate electronics onto 3D objects. Here, a technique is reported to utilize shape‐memory polymers together with carbon nanotube flexible electronics to achieve this goal. Temperature‐assisted shape control of these freestanding electronics in a programmable manner is demonstrated, with theoretical analysis for understanding the shape evolution. The shape control process can be executed with prepatterned heaters, desirable for 3D shape formation in an enclosed environment. The incorporation of carbon nanotube transistors, gas sensors, temperature sensors, and memory devices that are capable of self‐wrapping onto any irregular shaped‐objects without degradations in device performance is demonstrated.

This model treats the assembly of the polyimide film and the top pre-stretched polystyrene layer as a composite beam. The problem can be simplified to a onedimensional bending problem assuming the length of the beam is much larger than its thickness and width. For simplicity, additionally we assume the system is purely in the linear elastic regime and any deformation due to shear across the section is neglected, so that the Euler-Bernoulli beam theory can be applied to obtain this analytical solution.
As set up in Figure S1, the coordinate system consists of a vertical y-axis and an x-axis perpendicular to the cross section plane. With the geometric parameters illustrated in Figure S1, A 1 = h 1 w 1 and A 2 = h 2 w 2 give the cross section areas of the polyimide and the polystyrene layers respectively. The centroid for the bottom polyimide layer h 1 is h 1 2 and h 2 for the top polystyrene layer is h 1 + h 2 2 , setting the reference point at the bottom.
Hence, the elastic centroid of the entire cross section of the composite beam is given by, The beam's effective moment of inertia (EI Z ) eff is given by where The bending of the bilayer system induced by the thermal strain can be understood by the superposition of two states. In state I, we can define a term of eigen-strain e * that is the relative reduction in length at a given temperature T when the polystyrene layer is heated alone. For simplicity, we assume the thermal expansion of the polyimide layer is zero.
When we heat the bilayer to temperature T, the pre-stretched polystyrene layer will shrink. Imagine at this time, an external tensile force F can be applied at the centroid of the polystyrene layer to stretch the polystyrene layer towards its original length before heating. A zero-strain state for the whole system can be reached at F = E 2 e * A 2 . The introduction of F will result in a non-zero stress in the polystyrene layer s xx Meanwhile, the stress for the polyimide layer s xx (1) should still be 0.
In the actual system, the above-discussed external force F is not really applied, so this requires the superposition of state II on state I. In state II, we introduce a compression force -F acts on the centroid of polystyrene layer in order to reach a force balance.
Obviously, because this force does not act on the centroid of the composite beam, it generates a bending moment M = F × d 2 . If we set 0 of the y-axis at the centroid as shown in Figure S1, the normal strain perpendicular to the cross section plane is simply given by, The stresses in the polyimide layer and polystyrene layers can be calculated using Hook's law neglecting other strain components: Applying the superposition principle, the total stress in the polyimide layer is the sum of the stresses in state I and II, while the stress in the polystyrene layer is Finally, the curvature of the beam is given by

Derivation of the plastic solution
Definition of the symbols: This model recognizes the bilayer system bending as a one-dimension plastic bending problem. We assume the plane sections remain planes during the entire bending process.
Additionally, we assume the polystyrene to be perfectly plastic once yield occurs. The zero point of the vertical y-axis is set to the bottom of the beam in the analysis below.
Starting with the compatibility condition, at every point in the bilayer system we should We assume the polyimide layer is always in the elastic region and its thermal expansion can be neglected, so the elastic strain is equal to the total strain for the polyimide layer.
 el =  tot for y £ h 1 . (S10) For the pre-stretched polystyrene layer, the elastic strain is given by (S11) The stress can be related to the elastic strain by multiplying the Young's modulus.
Explicitly, the stress is given by Since there is no external force applying onto the system, the total force and total bending moment should be 0, which leads to Naturally, any increment of stress should also obey these constraints: Notice for the polystyrene layer, once yield occurs, s is always equal to s Y since the polystyrene is assumed to be perfectly plastic.
An analytical solution for this problem is not attainable; in other words, the problem has to be solved numerically. We can discretize the [0, ] interval on the y-axis into n pieces so that the total force can be approximated by the sum of s (y i )Dy from i = 1 to i = n. In this way, a full solution of the system given at a certain temperature should include n stress s (y i )values from y 1 to y n , the curvature and a constant b . If we've already known the solution at one temperature T (for example, we can use the elastic solution derived in the previous section to find the solution for the temperature where the system is at the onset of yielding.), we are able to solve the system at the temperature by solving the following set of equations: The above are n+2 equations, from which the n+2 unknowns ( K , b and s (y i ) for i = 1, 2, ..., n ) can be solved. ( D pl is related to Ds (y i ) through Equation (S12), so it is not counted as an independent variable.) Equation (S19) indicates if the point y i does not reach the yield point, the plastic strain is 0, which automatically balance the equation. On the other hand, if the yield occurs at position , the plastic strain is no longer 0, which forces s (y i ) + Ds (y i ) -s Y = 0 to satisfy the equation. Thus, we can solve out Ds (y i ) and the corresponding  pl (y i ) using Equation (S12). Following the above steps, after the solutions at temperature are obtained, they can be used to solve for the next temperature state .    Table S1. Temperature reached on different voltages applied to a platinum heater.