What is spontaneous symmetry breaking? There is an interesting math problem that illustrates this concept: how would you build the shortest roads that connect the four houses on the corners of a square?
Our intuition might tell us to use the diagonal lines, like this:
But a careful examination will show us that the shortest path is actually like this:
Why does a four-fold symmetry problem have a two-fold symmetry solution? This is somewhat counter-intuitive, but it reveals the essence of spontaneous symmetry breaking:
The basic idea is that, while one single solution spontaneously “breaks” the system’s symmetry, when we consider the ensemble of all solutions, we can recover the original high symmetry. In our shortest road problem, the solutions are indeed like this:
Spontaneous symmetry breaking is a fundamental phenomenon tied to a myriad of physical processes including phase transformation, degenerate quantum states, and the Higgs mechanism. Our work is an example of spontaneous symmetry breaking in directed self-assembly of block copolymers.
Block copolymers are a type of very important soft materials. People have been using the self-assembly property of block copolymers to create nanopatterns on surfaces. In the last two decades, different groups have combined lithography-generated template to guide block copolymer self-assembly for creating large-scale, high-resolution (sub-10 nm) nanopatterns of complex surface geometries.
Our work focuses on how block copolymers behave when the template symmetry differs from their innate symmetry. In a typical experiment, we deposit the block copolymer thin film onto a template consisting of standing pillars matrices of different symmetries (each pillar is ~15 nm in diameter, ~40 nm in height), and see how they adjust their assembly accordingly (Fig. 1a).
Figure 1 | a, fabrication procedure of block copolymer in epitaxial template. b, square symmetric post array template. c, block copolymer in post array template, exhibiting broken symmetry and supperlattice structure; d, highlights of the two-fold symmetric block copolymer domains; e, geometric illustration of the superlattice structure.
Our original intuition is that the block copolymer, being a soft material, should follow the symmetry of the template. Yet what we have found was surprising: in a template that has square symmetry (white dots in the Fig. 1b and c, with symmetry group p4mm), the block copolymer structures (e.g., blue and green dots in Fig. 1d and e) can only retain two-fold symmetry (symmetry group p4gm). And they gather together forming larger superlattice domains.
By using both theoretical model and self-consistent field simulation, we eventually came to a full understanding of the system: the symmetry breaking phenomenon is driven by the trade-off between entropy (polymer molecular configuration) and enthalpy (di-block interface). It is only by adopting this broken symmetry geometry can the block copolymers minimize the free energy of the system.
We have further applied this finding to more complicated template geometries (Fig. 2 is an example of the Ammann-Beenker eight-fold quasicrystal tilings) and have found that this phenomenon of spontaneous symmetry breaking persists.
Figure 2 | a, block copolymer in octagonal quasicrystalline template (Ammann-Beenker tilings). b, post array without block copolymers; the dashed lines indicate the local 8-fold rotational symmetry, composed of square and triangular geometries. c, Fast Fourier Transformation of the original image shown in a. d–g, block copolymer domain contours, showing different ways of interaction between neighboring domains.
Scientifically, this research provides an intriguing example of spontaneous symmetry breaking at nanoscale. Technologically, the finding unravels the potentials and limits of directed self-assembly of block copolymers, which is considered to be a very promising nanofabrication method. This shows that new morphologies might be accessed by controlling the design of the template; and demonstrates that, while we can achieve higher resolution by using block copolymers, we should keep in mind that the achieved pattern might not maintain the template’s symmetry.
We hope this study can provide new insights for other researchers in the field of directed self-assembly.
Learn more about our experiments and simulations here.
Yi Ding, Karim R. Gadelrab, Katherine Mizrahi Rodriguez, Hejin Huang, Caroline A. Ross, Alfredo Alexander-Katz, Emergent symmetries in block copolymer epitaxy. Nature Communications 10, 2974, doi:10.1038/s41467-019-10896-5 (2019).