## 1 Introduction  
The separation of particles is an important process in the medical laboratories and biomedical industries. Some traditional separation technologies have been developed for centuries, such as filtration, centrifugation, chromatography, and electrophoresis [1]. In recent decades, with the rapid development of micro-electromechanical systems (MEMS), several continuous flow separation methods have been proposed in microfluidic devices. [PERSON] [2] proposed a size-based continuous separation technique, termed as \"deterministic lateral displacement\" (referred to as DLD). A schematic diagram of the DLD device is shown in Figure 1. The DLD device is mainly composed of an ordered array of obstacles such as cylindrical posts. The array has a fixed lateral shift between the neighboring columns. Small particles are able to largely follow the undisturbed streamlines through the array along a zigzag path (referred to as zigzag mode), whereas large particles travel across the flow lane or the \"separation\" streamlines which go through the stagnation points of the flow past the posts due to the interaction with the posts, being laterally displaced (referred to as lateral-displacement mode), as shown in Figure 1.  
The DLD separation technique is attractive due to its advantages of easy implementation, fast sorting speed, and high size-resolution, and has been widely used to separate bioparticles [3; 4; 5], such as separation of the blood cell components [6; 7] and separation of living parasites [8] and circulating tumor cells [9] from blood. The critical particle diameter for the separation or two motion modes is a key fundamental issue for the DLD separation technique, and has been investigated theoretically,numerically, and experimentally. [PERSON] [10] proposed a theoretical model for the critical diameter of rigid particles, which was assumed to be equal to twice the width of the first flow lane above the post surface, as shown in Figure 1. Although the theory generally underestimates the critical diameters [10], it has been widely used as a convenient tool to predict the critical separation size by using the single-phase flow field without solving the particle-fluid interactions [11; 12; 13]. The effects of the post shape, the fluid property, and the particle property on the DLD separation performance have been receiving much attention. Regarding the post shape, the triangular post [11; 12], the diamond and airfoil post [14], the I-shaped post [15], and the airfoil post with various angle-of-attacks [13] have been investigated. Regarding the fluid property, [PERSON] [16] investigated the effects of the shear-thinning property of the non-Newtonian fluids on the particle critical diameter with a distributed-Lagrange-multiplier-based fictitious domain method and observed that fluid shear-thinning contributed to decrease the critical particle diameter compared to the Newtonian case by altering the flow field between the posts. [PERSON] [17] further demonstrated experimentally that the critical diameter was affected by not only shear-thinning but also elastic effects of the viscoelastic medium. The fluid inertial effects on the separation performance were studied by [PERSON] [18] and [PERSON] [19]. Regarding the particle property, [PERSON] [20] experimentally examined the effects of the deformation of erythrocytes in the DLD device, and found that the motion mode could be changed from the lateral displacement to the zigzag, as the particle deformability was increased. The effects of the particle flexibility on the separation has also been investigated by two-dimensional direct numerical simulations [21; 22; 23; 24; 25] and three-dimensional simulations [26; 27]. The reader is referred to [PERSON] [3], [PERSON] [4], and [PERSON] [5] for a nice review of the progress in the DLD field. Despite wide applications of the DLD technique, there are still various fundamental issues that have not been well resolved [5].