Designing quasicrystalline ground states of colloids: An inverse statistical-mechanics approach
Final Report Abstract
In this postdoctoral research project conducted in the lab of Prof. Salvatore Torquato at Princeton University, we have put the focus of our investigations on local density fluctuations and hyperuniformity in quasicrystals. During the funding period, I, the applicant, have gained broad knowledge on hyperuniformity and quasicrystals by working with the world-wide experts in both fields. Our project-related main achievements are shortly summarized below. Local fluctuations of constituents of a many-body system are of fundamental importance throughout various scientific disciplines, ranging from number theory to biology. In statistically homogeneous point patterns, local density fluctuations can be quantitatively studied by the local variance in the number of points contained within a regularly shaped window. Specifically, hyperuniform systems in d dimensions, i.e., point patterns that do not possess infinite-wavelength fluctuations, are characterized by a local variance that grows as the surface area rather than the volume of the window for large window sizes R: delta2(R) = lambda(R)R d-1 +O(R) d-2, with lambda(R) denoting the surface-area fluctuations. After having constructed quasicrystals with multiple incommensurate length scales and rotational symmetries in one and two dimensions, and after having provided a strict proof of the hyperuniformity of quasicrytals, we studied lambda(R) of the corresponding structures. Basing on these coefficients and their averaged value lambda (hyperuniformity metric), we find that quasicrystals' large-scale density fluctuations behave qualitatively in the same way as crystalline ones. Indeed, lambda is shown to serve as a useful metric to rank-order the systems according to the degree to which the large-scale density fluctuations are suppressed. All hitherto performed calculations confirm that the larger the structural disorder in a point pattern is, the larger the magnitude of normalized lambda becomes. Our results show that highly regular crystals possess lower lambda than quasicrystals which have again lower metric values than disordered hyperuniform systems. Moreover, as the number of involved length scales in quasicrystals increase, lambda increases as well, suggesting that adding complexity to the system in terms of multiple length scales increases the disorder by means of large-scale fluctuations. An alternative characterization of the hyperuniformity can be provided in the Fourier space. Hyperuniform systems' static structure factor vanishes in the limit of vanishing wavelength: S(k->0) = 0. Beyond that, the scaling of S(k) for small k becomes important. All hitherto known hyperuniform systems possess structure factors with a small-wavenumber scaling S(k << 1) ~ k gamma, where gamma > 1, with some exceptions like the disordered hyperuniform systems that can have an exponent 1 or less. On the one hand, it has been shown that in hyperuniform maximally random jammed hard-particle packings, the small-wavelength scaling is linear, i.e., gamma = 1. On the other hand, perfect crystals in all dimensions trivially possess an exponent of gamma = infinite. Our results show that quasicrystals exhibit a distinct scaling exponent different from disordered and crystalline hyperuniform systems. We identify S(k << 1) ~ k4 in one-dimensional Fibonacci quasicrystals, whereas in two dimensions, S(k << 1) ~ k8 is observed for the canonical Penrose quasicrystal. These results suggest that dimensionality (unlike for crystals) plays a crucial role for the scaling of the structure factor of quasicrystals. We further investigated the role of incommensurate length scales for the scaling behavior. As we generated three-, four- and five-lengthscale quasicrystals in one dimension, we obtained a decay in the scaling exponent. In the near future, we are willing to study the scaling behavior for various two-dimensional quasiperiodic structure as well as for three-dimensional icosahedral quasicrystal.
