Micro-Structured Two-Component 3D Metamaterials with Negative Thermal-Expansion Coefficient from Positive Constituents

Controlling the thermal expansion of materials is of great technological importance. Uncontrolled thermal expansion can lead to failure or irreversible destruction of structures and devices. In ordinary crystals, thermal expansion is governed by the asymmetry of the microscopic binding potential, which cannot be adjusted easily. In artificial crystals called metamaterials, thermal expansion can be controlled by structure. Here, following previous theoretical work, we fabricate three-dimensional two-component polymer microlattices by using gray-tone laser lithography. We perform cross-correlation analysis of optical microscopy images taken at different sample temperatures. The derived displacement-vector field reveals that the thermal expansion and resulting bending of the bi-material beams leads to a rotation of the 3D chiral crosses arranged onto a 3D checkerboard pattern within one metamaterial unit cell. These rotations can over-compensate the expansion and lead to an effectively negative thermal length-expansion coefficient for all positive constituents evidencing a striking level of thermal-expansion control.


Introduction
Three-dimensional (3D) printing of materials is a huge trend. It allows for individualizing products and for fabricating architectures that are very difficult if not impossible to make otherwise. Ultimately, one would like to 3D print any functional structure or device at the push of a button. Apart from 2 boosting spatial resolution and printing speed, achieving this goal requires the ability to obtain hundreds or thousands of different material properties with one 3D printer. Today's 2D graphical printers realize thousands of colors from only three cartridges (cyan, magenta, yellow). By analogy, future 3D material printers might be able to print thousands of different effective materials from only a few constituent-material "cartridges".
Physics is on our side: Upon 3D printing two constituent materials A and B to obtain a composite or metamaterial, one might naively think that its effective properties will always be in between those of A and B. Fortunately, this is not the case. [1][2][3][4] In some cases, the behavior is even conceptually unbounded, i.e., an effective material parameter can assume any value from minus infinity to plus infinity even if those of the constituents are all finite and, e.g., positive. Examples are the electric permittivity and the magnetic permeability in electromagnetism or the compressibility and the mass density in mechanics. [5][6][7][8][9][10][11][12][13] However, for the mentioned examples, sign reversal and unbounded effective parameters are only possible near resonances at finite frequency and not in the truly static regime for reasons of stability in mechanics and non-negative energy density in electromagnetism. 5,13 Static examples are rare. Theoretically, the thermal length-expansion coefficient and the Hall resistance have been discussed. [1][2][3][14][15][16][17][18][19][20][21] Regarding the Hall resistance, even one constituent material A and voids within suffices. 20 The situation is distinct for the thermal length-expansion coefficient.
Within the range of validity of the continuum approximation, any connected structure composed of one constituent material A and voids within will show exactly the same thermal length-expansion coefficient as the bulk constituent material A. In contrast, the work of Lakes and others has shown that the behavior of composites containing two components A and B plus voids within is principally unbounded. 1,3,[14][15][16] In regard to applications, thermal length-expansion is a small effect with huge consequences. A relative thermal length-expansion around 10 −4 to 10 −3 can lead to severe misalignment, failure, or cracks. Atomic-scale composites can provide near-zero or negative thermal-length expansion by 3 changing the microscopic binding potential. [22][23][24] More macroscopic composites with near-zero length expansion are based on one constituent material with positive and another one with negative thermal expansion. For example, CERAN® glass cooking fields are made like that and have led to considerable markets.

Results
In this work, by using 3D gray-tone two-photon laser lithography, we fabricate micro-structured twocomponent metamaterials using a single photoresist, leading to an effectively negative thermal length-expansion coefficient from all-positive constituents. Applying image cross-correlation analysis, we directly measure the temperature-induced displacement-vector field in different layers of the micro-lattice with sub-pixel precision and thereby visualize the underlying microscopic mechanism.
Figure 1a exhibits a single lattice constant of the micro-lattice blueprint we start from. This unit cell is placed onto a three-dimensional simple-cubic translational lattice. Apart from minor modifications, this blueprint has been taken from the literature. 16 The two components A and B shown in different colors have different positive thermal length-expansion coefficients. Intuitively, the operation principle is as follows (see Fig. 1b): The bi-material beams expand and bend upon heating. The bending leads to a rotation of the 3D crosses, the arms of which make them chiral. The chirality and hence the sense of rotation alternates between clockwise and counter-clockwise from one 3D cross to its neighbors, forming a 3D checkerboard pattern. The rotations counteract the length expansion of the beams. Thus, for sufficiently pronounced rotations, the crossing points move towards each other and the effective thermal length-expansion of the micro-lattice becomes negative, i.e., the lattice contracts. For somewhat less pronounced rotations, zero thermal length-expansion results (see Supplementary Fig. 1). If component B is left away, i.e., replaced by vacuum, the effective thermal length-expansion coefficient of the micro-lattice is identical to that of bulk A. The same is We mention in passing that the structure shown in Fig. 1a has an effective Poisson's ratio of = −0.41. However, in general, the sign of the thermal-length expansion coefficient is not necessarily the same as the sign of the Poisson's ratio. 15 How can we fabricate such complex 3D two-component metamaterials? Three-dimensional twophoton laser printing of a single polymer component is a well-established technology. Here, we use a commercial instrument (Photonics Professional, Nanoscribe GmbH). While characterizing the thermal length-expansion coefficients of various polymers in bulk cuboid form, we noticed that the thermal length-expansion coefficient does not only depend on the type of monomer we start from. For a given monomer (we use IP-Dip, Nanoscribe GmbH), it also depends on the light exposure dose during photo-polymerization. For example, when writing a bulk cube at a power scaling factor (see Experimental) of 60% (40%) at a fixed scan speed of 2 cm/s, we find a thermal length-expansion . The measurement procedure shall be described below. We interpret this behavior in that the exposure dose influences the polymer crosslinking density. A correlation between cross-linking density and thermal expansion has been established previously. 26 The bottom line of this finding is that we can realize two different components A and B plus voids within by using only a single photoresist and gray-tone optical lithography.
For the temperature-dependent measurements, the samples are fixed to a Peltier-element heater within a small encapsulated chamber. In this fashion, the air within the chamber is heated as well, such that, after a certain waiting time (we chose several hours here), we can safely assume that the polymer micro-lattice temperature is actually equal to the temperature measured by a calibrated thermo-resistor at the Peltier-element location (as well as by a second one at the upper end of the cell). The chamber has a glass window to allow for optical access via a home-built wide-field optical microscope. It is based on a single microscope lens (Zeiss LD Achroplan 20 × with a numerical aperture = 0.4) which images a sample plane directly onto the chip of a silicon-based chargecoupled-device (CCD) camera. We illuminate the sample by diffuse white light, for which we obtain the best image contrast for our micro-lattices. The entire chamber can be moved with respect to the microscope by using a 3D piezoelectric translation stage (Piezosystem Jena 1469), which is controlled by a computer.
How can we directly characterize the thermal length-expansion and the operation principle underlying the micro-lattices? For example, for temperature differences on the order of Δ = 20 K, we expect relative length changes of the samples on the order of |Δ / | = 10 −3 . If one images the entire sample onto a camera chip with 1000 pixels in one direction, this relative change corresponds to a movement of merely 1 pixel. Obviously, the measurement accuracy must be yet better than that. Image cross-correlation analysis can provide such sub-pixel sensitivity. 25 Our analysis starts from two optical images of a sample taken at two different temperatures. The images can refer to the sample's surface. For transparent samples, they can alternatively correspond to a plane below the surface within the volume of the sample (e.g., plane 1 or 2 in Fig. 1c). In one image taken at room temperature, regions of interest (ROI) are defined. These small regions (33 × 33 pixels) can be positioned onto characteristic points of the sample, e.g., onto the lattice points of a micro-lattice (compare Fig. 1c). A second image is taken at an elevated temperature. Next, the two-dimensional  Obviously, all yellow arrows in Fig. 3c roughly point towards the center of the sample. Furthermore, the length of the arrows roughly increases linearly from the center to the sides. These two observations indicate a nearly homogeneous and isotropic behavior of the 3D crossing points. It is 7 thus meaningful to describe the metamaterial by an effective thermal length-expansion coefficient.
In contrast, the single-component control sample depicted in Fig. 3a and b exhibits a positive value of = +4 × 10 −5 K −1 . All of the described aspects are reversible, i.e., we have observed them many times on a given sample. We have also seen them on several samples. Our analysis also allows for quantifying deviations from ideal isotropy. We find that the modulus of the expansion coefficient is 25% larger along the -direction as compared to the -direction for the two-component sample.
This anisotropy is assigned to remaining fabrication imperfections.
To further investigate whether the samples actually follow the behavior anticipated by the design as discussed above, we also track other points within the unit cell (compare plane 1 in Fig. 1c). The blue arrows in Fig. 3d show that the connecting beams actually expand upon heating. Here, for clarity, their average displacement vector has been subtracted. A similar behavior is found for all other beams as well (not depicted). The red arrows reveal the anticipated rotation of the crosses, with alternating clockwise and counter-clockwise sense of rotation. Again, for clarity, the average displacement vector for each cross has been subtracted from the corresponding set of four arrows.
Note that the red rotation vectors are generally longer than the blue ones. As explained above, this behavior leads to the negative thermal length-expansion: The shrinkage induced by the rotations over-compensates the expansions of the individual beams. For the control sample in Fig. 3b, no significant rotations are found within the noise (hence the red arrows are not depicted here) and the thermal expansion is positive -as for the bulk polymer constituents discussed above. The measured behavior in Fig. 3 is in good overall agreement with the calculated one shown in Fig. 4. 8 In summary, we have fabricated and characterized micrometer-scale two-component polymer-based metamaterials exhibiting an effectively negative thermal length-expansion coefficient from positive constituents. The necessary two components plus voids have been realized by 3D gray-tone laser lithography using only a single photoresist.

Sample design
For the metamaterial design, a simple cubic unit cell was analyzed by a finite-element approach using the commercially available software package COMSOL Multiphysics and the MUMPS solver within.
For the mechanical part of the problem, the thermal expansion was introduced as a volumetric stress. The constituent materials A (and B) were modeled with a Young's modulus of A = 4 GPa 1b, we used periodic boundary conditions. 28 To assess the influence of finite sample size, we also performed calculations for samples containing 4 × 4 × 2 unit cells fixed to a rigid substrate (Fig. 4).
Based on the two mirror planes cutting the structure through the middle, one parallel to the and one parallel to the -plane, we have reduced the computational domain to 2 × 2 × 2 unit cells fixed to a rigid substrate. The other sides are left free to move (stress-free boundary conditions). The derived effective thermal length-expansion coefficients were different from those for the periodic boundary conditions by only a few percent. Typically, each unit cell (compare Fig. 1a) was discretized into 5 × 10 5 tetrahedral elements.

Sample fabrication
The structures composed of 4 × 4 × 2 unit cells were written using a liquid photoresist (IP-Dip, Nanoscribe GmbH, Germany) and a commercial three-dimensional laser lithography system (Photonics Professional, Nanoscribe GmbH). The objective lens (63 ×, = 1.4, Carl Zeiss) was dipped directly into the photoresist. An average power of 50 mW measured at the backfocal aperture of the objective lens with a diameter of 7.3 mm was defined as the reference power. The actual power was given by this reference power times the power scaling factor, which was varied locally in the spirit of gray-tone lithography. The component with low (high) thermal lengthexpansion coefficient was written with a power scaling factor of 65% (37%). The line spacing in the -and -direction was 200 nm, that in the -direction was 500 nm. Small parts were written in the galvo-scanning mode with a scan speed of 2 cm/s and stitched by using micro-positioning stages. 28 The exposed samples were developed in mr-Dev 600 for 60 minutes, transferred to acetone, followed by supercritical-point-drying in acetone.