How to Quantify and Avoid Finite Size Effects in Computational Studies of Crystal Nucleation Sarwar Hussain 1 and Amir Haji-Akbari 1, * 1 Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06520 (Dated: November 5, 2021) Computational studies of crystal nucleation can be impacted by finite size effects, primarily due to unphysical interactions between crystalline nuclei and their periodic images. It is, however, not always feasible to systematically investigate the sensitivity of nucleation kinetics and mechanism to system size due to large computational costs of nucleation studies. Here, we use jumpy forward flux sampling to accurately compute the rates of heterogeneous ice nucleation in the vicinity of square-shaped model structureless ice nucleating particles (INPs) of different sizes, and identify three distinct regimes for the dependence of rate on the INP dimension, L. For small INPs, the rate is a strong and non-monotonic function of L due to artificial spanning of critical nuclei across the periodic boundary. Intermediate-sized INPs, however, give rise to the emergence of non-spanning ’proximal‘ nuclei that are close enough to their periodic images to fully structure the intermediary liquid. While such proximity can facilitate nucleation, its effect is offset by the compression of the intermediary liquid by the growing non-proximal nuclei, leading to artificially small nucleation rates overall. The critical nuclei formed at large INPs are neither spanning nor proximal. Yet, the rate is a weak function of L, with its logarithm scaling linearly with 1/L. The key heuristic emerging from these observations is that finite size effects will be minimal if critical nuclei are neither spanning nor proximal, and if the density of the unstructured part of the intermediary liquid is statistically indistinguishable from the supercooled liquid density under the same conditions. I. INTRODUCTION The main premise of molecular simulations is to use the information obtained from simulating finite-sized systems to predict their behavior in the thermodynamic limit. The accuracy of such predictions, however, can depend strongly on the size of the simulated system, as estimates of thermodynamic, 1–8 structural, 9 and transport 10 prop- erties and nucleation rates 11 in small systems can devi- ate from those in the thermodynamic limit in a statisti- cally significant manner. Such a dependence on system size is typically referred to as finite size effects, which can be fairly strong for very small systems, while be- ing mostly unnoticeable for larger systems. Therefore, finite size effects can, in principle, be mitigated by sim- ulating very large systems, a task that is only compu- tationally feasible for simple model systems. 12,13 For- tunately, this is not always necessary as similar con- clusions can usually be obtained from simulating ”suf- ficiently large“ computationally tractable systems. 12,14 What constitutes ”sufficiently large“, however, is sub- ject to the property that is being estimated or the process that is being studied. For instance, a sys- tem comprised of several hundred molecules is usually large enough for accurately estimating thermodynamic, structural and transport properties of liquids, 1–3,9,10,15 but might be too small for studying collective phe- nomena such as cavitation, 16 condensation 17 and crys- tal nucleation. 12,18,19 It is therefore critical to develop heuristics for determining what qualifies as ”sufficiently * Electronic address: amir.hajiakbaribalou@yale.edu large“ for studying such collective phenomena, in order to ensure the accuracy and reliability of the conducted simulations. One such collective phenomenon that has been exten- sively studied using molecular simulations is crystal nu- cleation. As such, understanding the role of finite size effects on the thermodynamics and kinetics of crystal nu- cleation has been a topic of interest for decades. 11 Nu- cleation is a process in which a sufficiently large nucleus of the new phase forms within the old metastable phase, and is usually the rate-limiting step of a first-order phase transition when the underlying thermodynamic driving force is small. 20 Finite size effects in nucleation primarily arise due to periodic boundary conditions, which can re- sult in an unphysical confinement of the metastable phase between the nucleus and its periodic images, 15,18,19,21 or the formation of nuclei that span across the periodic boundary. 16,22 In the case of crystal nucleation, the ef- fect of periodic boundaries might be stronger due the extension of the diffuse crystal-liquid interface beyond the nucleus. 12 However, finite size effects can also arise due to other factors such as solute depletion in multi- component systems, 23,24 or peculiarities of the employed ensemble. 14,17 These effects can collectively lead to un- physical nucleation rates in both homogeneous and het- erogeneous nucleation, and has also been found to impact crystal growth. 25–27 Indeed, the findings of several high- profile computational studies of nucleation are believed to have been strongly impacted by finite size effects. For instance, Matsumoto et al.’s observation 28 of homoge- neous ice nucleation in a system of 512 water molecules represented using the fully atomistic TIP4P 29 model has never been reproduced in larger systems, and was later shown to be an artifact of strong finite size effects. 30 Ear- arXiv:2008.10179v1 [cond-mat.stat-mech] 24 Aug 2020