A two-dimensional polymer (2DP) is a sheet-like monomolecular macromolecule consisting of laterally connected repeat units with end groups along all edges.  This recent definition of 2DP is based on Hermann Staudinger’s polymer concept from the 1920s. According to this, covalent long chain molecules (“Makromoleküle”) do exist and are composed of a sequence of linearly connected repeat units and end groups at both termini.
Topologically, 2DPs may thus be understood as covalent structures made up from regularly tessellated regular polygons (the repeat units). Figure 1 displays the key features of a linear and a 2DP according to this definition. For usage of the term “2D polymer” in a wider sense, see “History”.
There are several examples of 2DPs which include the individual layers or sheets of graphite (called graphenes), MoS2, (BN)x and layered silicates. As required by the above definition, these sheets have a periodic internal structure. A potential repeat units of, e.g., graphene is a sp2-hybridized carbon atom. Individual sheets can in principle be obtained by exfoliation procedures though in reality this is a non-trivial enterprise.
Some of the known 2DPs can be obtained by pyrolytic or related procedures which require forcing conditions. However, until recently there was no rational synthesis available that would operate under conditions mild enough to allow for implementing the repertoire of organic chemistry in terms of precise structure control including the exact positioning of functional groups on the sheets’ faces and edges.
The first rational organic synthesis of a molecular sheet meeting the above 2DP definition was achieved in 2012 and based on monomer structure with photoreactive anthracene and acetylene moieties. It rests upon monomers with three reactive sites that crystallize in layered single crystals, such that these sites of neighbouring monomers are directly opposing each other. Light irradiation allows connecting them through these sites, and subsequent exfoliation affords monolayer sheets. The internal periodicity is supported by electron microscopy imaging, electron diffraction and Raman-spectroscopic analysis.
2DPs should in principle also be obtainable by, e.g., an interfacial approach whereby proving the internal structure, however, is more challenging and has not yet been achieved. For a recent approach, see Ref.
2DPs as two dimensional sheet macromolecules have a crystal lattice, that is they consist of monomer units that repeat in two dimensions. Therefore, a clear diffraction pattern from their crystal lattice should be observed as a proof of crystallinity.
2DPs are expected to be superb membrane materials because of their defined pore sizes. Further they can serve as ultrasensitive pressure sensors, as precisely defined catalyst supports, for surface coatings and patterning, as ultrathin support for cryo-TEM, and many other applications. 2DPs are also important to polymer physics. It needs to be explored inasmuch the concepts which were developed for linear polymers can be applied to 2DPs.
First attempts to synthesize 2DPs date back to the 1930s when Gee reported interfacial polymerizations at the air/water interface in which a monolayer of an unsaturated fatty acid derivative was laterally polymerized to give a 2D cross-linked material. Since then a number of important attempts were reported in terms of cross-linking polymerization of monomers confined to layered templates or various interfaces. These approaches provide easy accesses to sheet-like polymers. However, the sheets’ internal network structures are intrinsically irregular and the term “repeat unit” is not applicable (See for example:). In organic chemistry, creation of 2D periodic network structures has been a dream for decades. Another noteworthy approach is "on-surface polymerization"  whereby 2DPs with lateral dimensions not exceeding some tens of nanometers were reported. Laminar crystals are readily available, each layer of which can ideally be regarded as latent 2DP. There have been a number of attempts to isolate the individual layers by exfoliation techniques (see for example:).
2D supramolecular polymer
Self-assembly may also be used to obtain 2D polymer. To realize periodicity, rigid and preorganized molecular unit needs to be used, as demonstrated by the formation of 2D monolayer honeycomb supramolecular polymer which was driven by strong encapsulation of CB for the linear aromatic units in water. The periodicity of the 2D supramolecular polymer was evidenced by synchrotron X-ray scattering in both solution and solid state. 
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