Programmable and Surface‐Conformable Origami Design for Thermoelectric Devices

Abstract Thermoelectric devices (TEDs) show great potential for waste heat energy recycling and sensing. However, existing TEDs cannot be self‐adapted to the complex quadratic surface, leading to significant heat loss and restricting their working scenario. Here, surface‐conformable origami‐TEDs (o‐TEGs) are developed through programmable crease‐designed origami substrates and the screen‐printing TE legs. Compared with “π” structured TEDs, the origami design (with heat conductive materials) changed the heat‐transferring direction of the laminated TE legs, resulting in an enhancement in enlarging ΔT/T Hot and V out by 5.02 and 3.51 times. Four o‐TEDs with different creases designs are fabricated to verify the heat recycling ability on plane and central quadratic surfaces. Demonstrating a high V out density (up to 0.98 −2at ΔT of 50 K) and good surface conformability, o‐TEDs are further used in thermal touch panels attached to multiple surfaces, allowing information to be wirelessly transferred on a remote display via finger‐writing.

The basic folding element (BFE) of the traditional Miura-ori structure that conforms to the planer surface comprises four parallelograms.By modifying the BFE of Miura-ori structure as in Figure S1A, a different folding result can be obtained, called cylindrical origami.As shown in Figure S1B, the parallelogram is an element in the BFE of Figure S1A, in which the parallelogram has three intrinsic parameters: shorter edge a, long edge b, and angle gama.Note that, such intrinsic parameters determine the shape of the whole origami and will not change in the folding process.
Adding an extra intrinsic parameter gama_d to the parallelogram, as shown in Figure S1B, a deformed quadrangle can be obtained, whose intrinsic parameters are shorter edge a, long edge b, angle gama, and angle gama_d.Then, for the triangle Δ1 in Figure S1B, we can have the inner angles of Δ1, as listed in Eq. ( 1).
Then, using the law of sines, we can have the edge length of Δ1, as listed in Eq. ( 2).The folded cylindrical origami is shown in Figure S1C.Intuitively, the same as traditional origami.The cylinder origami is symmetric to its medial plane, i.e., the whole figure is symmetric to the plane p2p5p8.For better analyzing the folding property, the folded cylindrical BFE is analyzed in two parts, part A of p1p2p3p4p5p6 and part B of p4p5p6p7p8p9.

• Part A of BFE
Part A is the left part of BFE in Figure S1C, which p1p2p3p4p5p6 forms.Point p5 is taken as the origin to analyze the folding property, and the coordinate system associated with the origin (p5) is shown in Figure S1C.Then, the whole folded BFE is symmetric to the plane yoz.
The angle formed by quadrangle p1p2p5p4 and plane xoy, e.g., faA in Figure S1C, is the folding angle of BFE.With the change of folding angle faA, the geometry of BFE is changing.E.g., the spatial location of p1p2p3p4p6 is changing with folding angle faA.
In Δp2p5pd, we have: 2   =  2  5 *sin(A1), (A1 is from Eq.( 1)) In Δp2pdpf, we have: In Δp5pdpf, we have: Hence, from the shape of the above three triangles, the following geometry information can be obtained: Then, all points can be obtained: • Part B of BFE The spatial location of all quadrangle vertex points, e.g., p1p2p3p4p5p6, are obtained.
Then, based on the same coordinate system that is associated at p5, and with the shape of Part A, the folding geometry shape of Part B (p4p5p6p7p8p9) can be found.From the shape of Part A, we can find some geometry constraints to calculate the shape of Part B in BFE.
Since the BFE is symmetrical geometry, point p8 falls on the plane yoz, and its coordinates can be supposed as p8 = [0, (p8)y, (p8)z].Then the following equations can be obtained.
With two equations in Eq. ( 8), the coordinates of p8 can be solved.From Figure S1A, we can have  5  8 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ //  4  7 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ .Hence, the point p7 can be calculated as below.The final point p9 can then be calculated as The angle formed by plane p4p5p8p7 and plane xoy, is the resultant folding angle at the side of Part B, i.e., the faB in Fig. S1C.The faB can be calculated as below.
By now, all points of BFE (p1p2p3p4p5p6p7p8p9) are calculated and the shape of cylindrical origami BFE can be fully determined.Next, how the BFE is propagated to form the array of cylindrical origami will be introduced.

• Propagation of cylindrical origami BFE
The folded origami geometry will remain a planar shape in traditional origami, where the BFE consists of four parallelograms.However, with the added intrinsic parameter gama_d, the folded origami is no longer in a planar shape, and the BFE array will be cylindrical.
Based on one BFE, the following BFEs can be propagated.Since the BFE is symmetrical geometry, we can assume that the coordinate of np5 in Hence, the following equations can be obtained.
With two equations in Eq. ( 12), the two unknowns (np5)y and (np5)z can be solved, thus, the coordinate of np5 in xyz is obtained.
Hereafter, we can obtain that: From Eq. ( 14), the transformation from xyz to xnynzn can be obtained.We refer to the method of Axisymmetric Origami Design proposed by Hu et al. 1 for designing the origami structure to cover the axisymmetric surface, in which the designed origami structure will also be axisymmetric.As shown in Figure S2A, the basic folding element (BFE) is composed of nine points and symmetrically to the line of p2-p5-p8.Before generating the entire origami structure, one column of origami BFE will be designed first to cover the section profile of the axisymmetric surface, as shown in Figure S2B.Note that, in the folding process, a column of origami BFE will always be symmetrical to its symmetry plane in any folding status of the origami structure.
Then, by circular arraying such a column of origami BFE, we can obtain an origami structure to cover the axial symmetry surface, as shown in Figure S2B.The section profile and the cap profile are used to design the column BFEs.The user can select the cap profile.For the case in Figure S3A., which is a sphere surface, the cap profile (Figure S3C) is selected as the offset curve of the section profile.Then, two series of sample points will be selected on the section profile and the cap profile, respectively, i.e., {a1, a2, a3, …} from the section profile and {b1, b2, b3, …} from the cap profile, as shown in Figure S3D.{a1, a2, a3, …} and {b1, b2, b3, …} will then be the control points for designing the origami structure.While for designing the axisymmetric origami structure to cover the axisymmetric surface, the selected sample points should satisfy the geometry condition that k(ai, bi) = -k(ai+1, bi), wherein k(a, b) indicates the slope of the line that goes through point a and b.
Another parameter to determine is the spanned angle of BFE, as the angle α shown in  With the above specifications, the structure of BFE can now be determined.For the first BFE, i.e., BFE 1 , three sample points a1, b1, and a2 are, respectively, the structure points p2, p5, and p8, with the folding angle being θ, recalling Figure S2A.Hereafter, with the assigned spanned angle α, other structure points p1, p4, p7, p3, p6, and p9 can be calculated accordingly.
Then for the second one, i.e., BFE 2 , sample points a2, b2, and a3 will be the structure points p2, p5, and p8 of BFE 2 .Note that, the structure points p1, p2, and p3 of BFE 2 , are exactly the structure points p7, p8, and p9 of BFE 1 .Thus, only the structure points p4, p6, p7, and p9 of BFE 2 need to be determined, which can be similarly calculated following the design procedure of BFE 1 .
Therefore, step by step, one column of BFE can be obtained, which is sandwiched by the section profile and the cap profile when the folding angle is θ. Figure S4A and B show the example of designing one column of the BFE, wherein the element number is 4, and the folding angle is 60°.By duplicating the column of BFE in a circular pattern, we can obtain an origami structure to fit the surface, as shown in Figure S4C.We also display its other folding status for the designed origami structure, i.e., folding angle 120° (Figure S4D and 180° (Figure S4E).
Based on the aforementioned origami design, other quadrics surface conformable designs, including hemisphere, conical, paraboloid, and ellipsoid surfaces with specific parameters, are listed in Figure S5.Another device application is the infrared light image display.Here, thick paper masks with "H", "K", "U", "S", and "T" holes that could block part of the light were prepared and packaged around the curve TTP.When the infrared light is on, the corresponding letters can be shown on the remote display (Figure S6D and Figure S6E).The cycling finger-touching test is demonstrated in Figure S11A, showing good stability.In Figure S11B, the response time of 2.34s is calculated.When the system is in a steady state, with fixed temperatures at the hot and cold ends are TH and TC, respectively.The external environment inputs energy from the hot end to maintain the temperature at the hot end, and all the input energy is absorbed by the device at the hot end.To establish a thermal balance equation at the hot end, the received heat consists of the external input heat QH and half of the Joule heat, which is 0.5I²R.
As shown in Fig. S14A, the heat transferred downwards includes the Peltier heat πI and the Fourier heat KΔT.Here, R represents the total resistance of the thermoelectric leg, I represents the current generated by the Seebeck voltage in the circuit, and K represents the total thermal conductivity of the thermoelectric leg, which signifies the heat flow through the unit temperature difference.The equation expression is as follows: When the load resistance is RL, the current and output power and be expressed as: As the heat transferring coefficient η can be defined as the ratio of output power and input heat.Here, the Peltier coefficient can be replaced by the Seebeck coefficient (π = S * TH) using the Kelvin relationship.Then the η can be expressed as: , where the total resistance and total thermal conductivity of a pair of thermoelectric arms are used to define the thermoelectric value Z of the device.Therefore, capital letter Z ̅ is directly point to thermoelectric device, which can be expressed as: When the direction of the heat flow current is the same, the ratio of the material's conductivity to thermal conductivity (or the ratio of thermal resistance to resistance) is equal to the ratio of electrical conductivity to thermal conductivity.At the same time, considering the Seebeck coefficient, electrical conductivity, thermal conductivity is temperature dependent, so it is used to use lowercase z times T as a dimensionless thermoelectric value to describe the thermoelectric performance of the material at a certain temperature.So the lowercase zT can be expressed as: Then the relationship between   and  can be depicted in Fig. 3B.
And Z can be further elaborated as As we can see from the above equation, the value of capital Z is related to the 1) Seebeck coefficient of the thermoelectric material, and 2) size of the TE Legs.
Therefore, for all three types of structures, capital Z value for all three types can be estimated by the given value   ,   ,   ,     and   all at the T of ~30 °C, and these values are summarized below.
As capital Z value for all three structure are at the same order of magnitudes, when taken the Z value back to the Fig. S14B, the   for all π structure, plane o-TEG with and without HCM still demonstrated the same rising characteristic as the one we mentioned in the manuscript.When T is around 30 °C, the   are 0.12%, 0.43%, and 0.60%, respectively, and the same rising trend still demonstrated as the T grows, demonstrating the superiority of the origami structure design and add of the HCM materials.
Movie S1-S2.Demonstration for capital letters "H" and "K" on plane thermal panel (TP).Table S1.Comparison on the output performance of our o-TEG with previously reported flexible or foldable TEGs.
Table S2.Comparison of the materials and output performance of o-TEG with previously reported origami or kirigami-designed TEG.
Figure S1.Programmable designs for cylindrical origami.(A) BFE in cylindrical a1 = 0.5*a b1 = a1*sin(B1)/sin(A1) c1 = a1*sin(C1)/sin(A1) Figure S1D shows the current BFE with the Part A of the next BFE.The points of the next BFE are np1np2np3np4np5np6np7np8np9.The coordinate system (xnynzn) associated with point np5, is the coordinate system to indicate the folding process of the next BFE.Now, we need to figure out the spatial relation between xyz and xnynzn, and how the folding process is propagated.

2)
Figure S2.(A) BFE in cylindrical origami.(B) One column and a circular array of

Figure
Figure S3.(A) axisymmetric surface to be covered.(B) section profile.(C) the cap

Figure S3E .
Figure S3E.The spanned angle α of BFE changes with the folding of the origami

Figure S4 .
Figure S4.(A) front view and (B) oblique view of one column of BFE.(C) Designed

Figure S5 .
Figure S5.Programmable designs for origami substrates conformable to (A)

Figure S7 .
Figure S7.Viscosity and thermal conductivity of HCM under different AlN concentrations.

Figure S10 .
Figure S10.(A) Testing scene of the handwriting TTP.(B) The Vout-∆T relations for

Figure
FigureS10Ashows the testing scene of the plane TTP.Four operational amplifiers

Figure S11 .
Figure S11.(A) Cycling finger touching test on one single sensing pixel.(B) Response

Figure S12 .
Figure S12.Electrical design for one array of TTP.

Figure S13 .
Figure S13.The output performance of o-TEG in this work compared with several literature data.

Figure S14 .
Figure S14.(A) The energy conservation inside a single pair of TE when the system is