2.1 Physical problem

Figure 1.

Transport of elastic MPs in blood flow. (a) Computational model of margination and adhesion of elastic MPs in blood flow. The zoomed panels give the detailed adhesion behaviour of an elastic MP under a stochastic ligand–receptor binding effect. (b) Schematic of transport process of an elastic MP from the centre of the bloodstream (denoted (C)) to the cell-free layer (F), and then reaching the adhesion layer.

In blood flow, most parts of the vessel are occupied by a large number of RBCs. In a normal human blood vessel, the volume fraction of RBCs (haematocrit $Ht$ ) is approximately 20 %–45 %. Under the interplay effect of the flow and the vessel wall, RBCs move from the near-wall region to the centre of the vessel due to deformation-induced migration. This results in the formation of a cell-free layer (CFL). The CFL plays the role of a lubricant layer and reduces the blood flow resistance, which is also called the Fåhraeus–Lindqvist effect (Fåhraeus & Lindqvist Reference Fåhraeus and Lindqvist1931). When elastic MPs, acting as drug carriers, are injected into a vein, they move with the bulk flow as shown in figure 1(a). The elastic MPs deform under the shear stress and collide with RBCs. The deformation depends on the local flow environment. Additionally, the MPs may move in the cross-stream direction, migrating either towards the wall or to the centre of the channel. Once MPs migrate to the near-wall region, i.e. CFL, the ligands decorated on their surfaces have the chance to interact with the receptors on endothelial cells distributed on the vessel wall (figure 1 a). This ligand–receptor binding is required for the further release of drug molecules into tumour sites through a vascular targeting strategy (Schnitzer Reference Schnitzer1998; Neri & Bicknell Reference Neri and Bicknell2005). However, reaching the CFL cannot guarantee that such interactions will occur. Only when an MP reaches a closer distance to the vessel wall, in which ligands can interact with receptors, does the interaction occur. This distance is determined by the reaction distance between ligands and receptors. We name the layer within this distance as the adhesion layer ( $\unicode[STIX]{x1D712}$ ). Usually the thickness of the adhesion layer is in the range of tens to hundreds of nanometres (Decuzzi & Ferrari Reference Decuzzi and Ferrari2006; Müller et al. Reference Müller, Fedosov and Gompper2014, Reference Müller, Fedosov and Gompper2016). Here, it is set as $1.0~\unicode[STIX]{x03BC}\text{m}$ according to the reaction distance that we used in the computational model. This is reasonable compared to that used in the previous work of Müller et al. (Reference Müller, Fedosov and Gompper2014).

The numerical study is employed to study the transport of elastic MPs due to its flexibility in tuning the properties of MPs and adhesive interactions. Blood flow is considered as a suspension of RBCs. Owing to limited computational resources, a small part of the vessel is taken into account and modelled as a rectangular channel. The size of the channel is of height $36~\unicode[STIX]{x03BC}\text{m}$ , width $27~\unicode[STIX]{x03BC}\text{m}$ and length $54~\unicode[STIX]{x03BC}\text{m}$ . Periodic boundary conditions are applied in the width ( $x$ ) and length ( $y$ ) directions. The height ( $z$ ) direction is bounded by two flat plates. The bottom plate (vessel wall, also named substrate) is fixed and the flow is driven by the moving of the upper one with a constant velocity $U$ . In all of the simulations, shear rates stay at $200~\text{s}^{-1}$ . A total of 162 RBCs and 80 identical elastic MPs are placed inside the channel. The haematocrit is approximately 30 %. MPs are initially set to a spherical shape with radius $1~\unicode[STIX]{x03BC}\text{m}$ , and their total volume fraction is approximately 0.64 % in the channel. Additionally, the ligands and receptors are uniformly distributed on the surfaces of MPs and substrate, respectively. The densities of ligands and receptors are listed in table 1.

Table 1.

Parameters used in adhesive model for ligand–receptor binding.

2.2 Computational method

2.2.1 Lattice Boltzmann method for fluid flow

The RBCs are immersed within blood plasma in the blood flow. The other components, such as the white blood cells and platelets, are negligible due to their low volume fractions compared to that of RBCs. The plasma is usually considered as a Newtonian fluid. Its dynamics is described by the continuity equation and incompressible Navier–Stokes (NS) equation:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+F, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the plasma density, and $\boldsymbol{u}$ and $p$ represent the velocity and pressure of the flow, respectively. The term $F$ on the right-hand side of (2.2) is the external force; $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the plasma and is set as 1.2 cP. The LBM is employed to solve the NS equation due to its high efficiency and accuracy in handling incompressible Newtonian flow (Higuera et al. Reference Higuera, Succi and Benzi1989; Benzi et al. Reference Benzi, Succi and Vergassola1992; Chen & Doolen Reference Chen and Doolen1998). By discretizing the velocity of the linearized Boltzmann equation, a finite difference scheme is obtained:

(2.3) $$\begin{eqnarray}f_{i}(\boldsymbol{x}+\boldsymbol{e}_{i}\unicode[STIX]{x0394}t,~t+\unicode[STIX]{x0394}t)=\,f_{i}(\boldsymbol{x},t)-\frac{\unicode[STIX]{x0394}t}{\unicode[STIX]{x1D70F}}(f_{i}-f_{i}^{eq})+F_{i},\end{eqnarray}$$

where $f_{i}(\boldsymbol{x},t)$ is the distribution function and $\boldsymbol{e}_{i}$ is the discretized velocity. In the current simulation, the D3Q19 velocity model is used (Mackay, Ollila & Denniston Reference Mackay, Ollila and Denniston2013), and the fluid particles have the possible discrete velocities stated in Mackay et al. (Reference Mackay, Ollila and Denniston2013). In (2.3) $\unicode[STIX]{x1D70F}$ denotes the non-dimensional relaxation time, which is related to the dynamic viscosity in the NS equation as follows:

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70C}c_{s}^{2}(\unicode[STIX]{x1D70F}-{\textstyle \frac{1}{2}})\unicode[STIX]{x0394}t.\end{eqnarray}$$

Also in (2.3) $f_{i}^{eq}(\boldsymbol{x},t)$ is the equilibrium distribution function and $F_{i}$ is the discretized scheme of the external force. In the current simulation, the equilibrium distribution function adopts the form

(2.5) $$\begin{eqnarray}f_{i}^{eq}(\boldsymbol{x},t)=\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D70C}\left[1+\frac{\boldsymbol{e}_{i}\boldsymbol{\cdot }\boldsymbol{u}}{c_{s}^{2}}+\frac{(\boldsymbol{e}_{i}\boldsymbol{\cdot }\boldsymbol{u})^{2}}{2c_{s}^{4}}-\frac{(\boldsymbol{u})^{2}}{2c_{s}^{2}}\right],\end{eqnarray}$$

where the weighting coefficients are $\unicode[STIX]{x1D714}_{0}=1/3$ , $\unicode[STIX]{x1D714}_{i}=1/18~(i=1{-}6)$ and $\unicode[STIX]{x1D714}_{i}=1/36~(i=7{-}18)$ . The term $c_{s}$ represents the sound speed, which equals $\unicode[STIX]{x0394}x/(\sqrt{3}\unicode[STIX]{x0394}t)$ . The external forcing term can be discretized by the form (Guo, Zheng & Shi Reference Guo, Zheng and Shi2002):

(2.6) $$\begin{eqnarray}F_{i}=\left(1-\frac{1}{2\unicode[STIX]{x1D70F}}\right)\unicode[STIX]{x1D714}_{i}\left[\frac{\boldsymbol{e}_{i}-\boldsymbol{u}}{c_{s}^{2}}+\frac{(\boldsymbol{e}_{i}\boldsymbol{\cdot }\boldsymbol{u})}{c_{s}^{4}}\boldsymbol{e}_{i}\right]\boldsymbol{\cdot }\boldsymbol{F}.\end{eqnarray}$$

Equation (2.3) is advanced through the algorithm proposed by Ollila et al. (Reference Ollila, Denniston, Karttunen and Ala-Nissila2011). Here, the solver of the LBM is embedded in a large-scale atomic/molecular massively parallel simulator (LAMMPS) (Plimpton Reference Plimpton1995), which is implemented by Mackay et al. (Reference Mackay, Ollila and Denniston2013). After the particle density distributions are known in the whole fluid domain, the properties of the fluid, such as fluid density and velocity, can be calculated as

(2.7a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70C}=\mathop{\sum }_{i}f_{i},\quad \boldsymbol{u}=\frac{1}{\unicode[STIX]{x1D70C}}\mathop{\sum }_{i}f_{i}\boldsymbol{e}_{i}+\frac{1}{2\unicode[STIX]{x1D70C}}\boldsymbol{F}\unicode[STIX]{x0394}t.\end{eqnarray}$$

2.2.2 Coarse-grained models for RBC and MP

To capture the dynamics and deformation of RBCs and elastic MPs, we develop a coarse-grained model and implement it into LAMMPS (Ye, Shen & Li Reference Ye, Shen and Li2018b ). The RBC is modelled as a liquid-filled coarse-grained membrane, and its equilibrium shape is biconcave. The diameter of an RBC is $7.8~\unicode[STIX]{x03BC}\text{m}$ , and the thickness is approximately $2.1~\unicode[STIX]{x03BC}\text{m}$ . The surface area and volume of an RBC are $134.1~\unicode[STIX]{x03BC}\text{m}^{2}$ and $94.1~\unicode[STIX]{x03BC}\text{m}^{3}$ , respectively (Evans & Skalak Reference Evans and Skalak1980). In the simulation, the membrane is discretized into 3286 vertices and 6568 triangular elements.

To capture the in-plane shear property of an RBC, a stretching potential $U_{stretching}$ is used. It includes two parts, namely an attractive nonlinear spring potential (wormlike chain model, WLC) and a repulsive power potential (power function, POW) (Fedosov, Caswell & Karniadakis Reference Fedosov, Caswell and Karniadakis2010a ; Fedosov et al. Reference Fedosov, Pan, Caswell, Gompper and Karniadakis2011b ). They can be expressed as

(2.8a,b ) $$\begin{eqnarray}U_{WLC}=\frac{k_{B}Tl_{m}}{4p}\frac{3x^{2}-2x^{3}}{1-x},\quad U_{POW}=\frac{k_{p}}{l},\end{eqnarray}$$

where $k_{B}T$ is the basic energy unit, $x=l/l_{m}\in (0,1)$ , $l$ is the length of the spring, $l_{m}$ is the maximum spring extension, $p$ is the persistence length, and $k_{p}$ is the POW force coefficient. Applying the bending potential

(2.9) $$\begin{eqnarray}U_{bending}=\mathop{\sum }_{k\in 1,\ldots ,N_{s}}k_{b}[1-\cos (\unicode[STIX]{x1D703}_{k}-\unicode[STIX]{x1D703}_{0})],\end{eqnarray}$$

the out-of-plane bending property of an RBC is reflected. Here $k_{b}$ is the bending stiffness, $\unicode[STIX]{x1D703}_{k}$ is the dihedral angle between two adjacent triangular elements, and $\unicode[STIX]{x1D703}_{0}$ is the initial value of the dihedral angle. In the following, subscript $0$ represents the corresponding initial value; and $N_{s}$ denotes the total number of dihedral angles.

Besides the above, the bulk properties, such as surface area and volume conservation, are ensured by introducing the penalty forms:

(2.10) $$\begin{eqnarray}U_{area}=\mathop{\sum }_{k=1,\ldots ,N_{t}}\frac{k_{d}(A_{k}-A_{k0})^{2}}{2A_{k0}}+\frac{k_{a}(A_{t}-A_{t0})^{2}}{2A_{t}}\end{eqnarray}$$

and

(2.11) $$\begin{eqnarray}U_{volume}=\frac{k_{v}(V-V_{0})}{2V_{0}},\end{eqnarray}$$

where the first term in (2.10) represents the local area constraint, $A_{k}$ and $A_{k0}$ denote the area of the $k$ th element and its initial area, respectively, and $k_{d}$ is the corresponding spring constant. The second term in (2.10) is the global area constraint, $A_{t}$ is the total area, and $k_{a}$ is the spring constant. In (2.11), $k_{v}$ is the spring constant and $V$ is the total volume.

Then the total energy $U$ is

(2.12) $$\begin{eqnarray}U=U_{WLC}+U_{POW}+U_{bending}+U_{area}+U_{volume}.\end{eqnarray}$$

The nodal forces exerted on each vertex of the RBC membrane are derived from

(2.13) $$\begin{eqnarray}f_{i}=-\unicode[STIX]{x2202}U(X_{i})/\unicode[STIX]{x2202}X_{i},\end{eqnarray}$$

where $X_{i}$ denotes the vertex of the RBC membrane. Thus, if we know the positions of the membrane vertices, we can calculate the nodal force according to (2.13). The detailed derivation of the force formulae, such as two-point stretching force and three-point bending force, are presented in Ye et al. (Reference Ye, Shen and Li2018b ).

The elastic MPs adopt the same model as RBCs, but with 828 vertices and changeable in-plane shear strength. Before we choose the parameters for the coarse-grained model of RBCs and elastic MPs, we should know the corresponding macroscopic properties through experiments a priori. According to the relationships between coarse-grained model parameters and macroscopic properties (Allen & Tildesley Reference Allen and Tildesley1989; Dao, Li & Suresh Reference Dao, Li and Suresh2006; Fedosov et al. Reference Fedosov, Caswell and Karniadakis2010a ) we have

(2.14) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D707}_{0}={\displaystyle \frac{\sqrt{3}k_{B}T}{4pl_{m}x_{0}}}\left({\displaystyle \frac{x_{0}}{2(1-x_{0})^{3}}}-{\displaystyle \frac{1}{4(1-x_{0})^{2}}}+{\displaystyle \frac{1}{4}}\right)+{\displaystyle \frac{3\sqrt{3}k_{p}}{4l_{0}^{3}}},\\ K=2\unicode[STIX]{x1D707}_{0}+k_{a}+k_{d},\\ Y={\displaystyle \frac{4K\unicode[STIX]{x1D707}_{0}}{K+\unicode[STIX]{x1D707}_{0}}},\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{0}$ is the shear modulus, $K$ represents the area compression modulus and $Y$ denotes the Young’s modulus. Therefore, the potential parameters can be chosen on the basis of the physical quantities. The parameters used in the simulation are listed in table 2.

The accuracy of this model for RBCs and elastic MPs has been validated in our previous works (Ye et al. Reference Ye, Shen, Yu, Wei and Li2017a ; Ye, Shen & Li Reference Ye, Shen and Li2018a ; Ye et al. Reference Ye, Shen and Li2018b ). In § 3, we will show two more validations to confirm the convergence of both the fluid and membrane meshes and modelling of the rheology of blood flow. Details about the computational efficiency and cost are discussed in § 5 and Ye et al. (Reference Ye, Shen and Li2018b ). In addition to the above potentials, it is necessary to employ intermolecular interactions between RBCs to characterize their interactions. Here we use the Morse potential as intermolecular interaction (Liu & Liu Reference Liu and Liu2006; Fedosov et al. Reference Fedosov, Pan, Caswell, Gompper and Karniadakis2011b ; Tan, Thomas & Liu Reference Tan, Thomas and Liu2012), in the form

(2.15) $$\begin{eqnarray}U_{morse}=D_{0}[\text{e}^{-2\unicode[STIX]{x1D6FD}(r-r_{0})}-2\text{e}^{-\unicode[STIX]{x1D6FD}(r-r_{0})}],\quad r<r_{c},\end{eqnarray}$$

where $D_{0}$ represents the depth of the potential energy well and $\unicode[STIX]{x1D6FD}$ controls its width, $r$ is the distance between the two particles, $r_{0}$ is the equilibrium distance, and $r_{c}$ is the cutoff distance. Additionally, a short-range and purely repulsive Lennard-Jones potential is applied to prevent overlap between RBCs and MPs (Ye et al. Reference Ye, Shen and Li2018b ).

Table 2.

Coarse-grained potential parameters for RBCs and elastic MPs, and their corresponding physical values.

2.2.3 Immersed boundary method for fluid–structure interaction

The IBM is used to couple the LBM with LAMMPS to account for fluid–structure interaction (Peskin Reference Peskin2002; Krüger, Varnik & Raabe Reference Krüger, Varnik and Raabe2011; Krüger, Kaoui & Harting Reference Krüger, Kaoui and Harting2014; Ye et al. Reference Ye, Wei, Huang and Lu2017b ). We use the Lagrangian ( $X$ ) and Eulerian ( $x$ ) mesh points in the computational domain to represent RBCs (or MPs) and fluid particles, respectively. The Eulerian fluid mesh is uniform and the resolution is $\unicode[STIX]{x0394}x=250~\text{nm}$ in all directions. The Lagrangian mesh for RBCs or MPs is generated by MATLAB code (Persson & Strang Reference Persson and Strang2004; Persson Reference Persson2005). The mesh is approximately uniform and the mesh size is set at approximately $\unicode[STIX]{x0394}X=(0.6{-}0.8)\unicode[STIX]{x0394}x$ . Then there are approximately 32 Eulerian points across one RBC in diametral direction. That is sufficient to resolve the deformation and motion of an RBC (MacMeccan et al. Reference MacMeccan, Clausen, Neitzel and Aidun2009; Vahidkhah & Bagchi Reference Vahidkhah and Bagchi2015). The coupling is fulfilled by the interpolation of velocity and force distributions between Lagrangian and Eulerian mesh points (Mittal & Iaccarino Reference Mittal and Iaccarino2005).

To ensure a no-slip boundary condition, the membrane vertex $X$ with Lagrangian coordinate $s$ should move at the same velocity as the fluid around it, that is

(2.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}X(s,t)}{\unicode[STIX]{x2202}t}=u(X(s,t)).\end{eqnarray}$$

The velocity can be interpolated by the fluid velocity through a smoothed Dirac delta function $\unicode[STIX]{x1D6FF}$ :

(2.17) $$\begin{eqnarray}u(X,t)=\int _{\unicode[STIX]{x1D6FA}}u(x,t)\unicode[STIX]{x1D6FF}(x-x(X,t))\,\text{d}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

This condition will cause the membrane to move and deform. The membrane force density $F(s,t)$ is obtained by derivation of potential functions such as (2.13), and is distributed to the surrounding fluid mesh points by

(2.18) $$\begin{eqnarray}f^{fsi}(x,t)=\int _{\unicode[STIX]{x1D6FA}}F^{fsi}(X,t)\unicode[STIX]{x1D6FF}(x-x(X,t))\,\text{d}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

2.2.4 Adhesive model for ligand–receptor binding

The ligand–receptor binding is described by the association and dissociation of biological bonds, and it is governed by the probabilistic adhesion model (Hammer & Lauffenburger Reference Hammer and Lauffenburger1987). Figure 1(a) gives a schematic of the adhesive model. When the ligands on the MP approach the receptors on the vessel wall, they have the chance to bind together, which is determined by the probability $P_{on}$ . In the reverse situation, the existing bond has a probability $P_{off}$ to break. They can be expressed as

(2.19a,b ) $$\begin{eqnarray}P_{on}=\left\{\begin{array}{@{}ll@{}}1-\text{e}^{-k_{on}\unicode[STIX]{x0394}t},\quad & l<d_{on},\\ 0,\quad & l\geqslant d_{on},\end{array}\right.\quad \text{and}\quad P_{off}=\left\{\begin{array}{@{}ll@{}}1-\text{e}^{-k_{off}\unicode[STIX]{x0394}t},\quad & l<d_{off},\\ 0,\quad & l\geqslant d_{off}.\end{array}\right.\end{eqnarray}$$

Here $\unicode[STIX]{x0394}t$ is the time step in the simulation, $d_{on}$ and $d_{off}$ are the cutoffs for bond creation and breakup, respectively, and $k_{on}$ and $k_{off}$ are the association and dissociation rates with the forms

(2.20a,b ) $$\begin{eqnarray}k_{on}=k_{on}^{0}\exp \left(-\frac{\unicode[STIX]{x1D70E}_{on}(l-l_{0})^{2}}{2k_{B}T}\right)\quad \text{and}\quad k_{off}=k_{off}^{0}\exp \left(\frac{\unicode[STIX]{x1D70E}_{off}(l-l_{0})^{2}}{2k_{B}T}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{on}$ and $\unicode[STIX]{x1D70E}_{off}$ are the effective on and off strengths, representing a decrease and increase of the corresponding rates within $d_{on}$ and $d_{off}$ , respectively, and $k_{on}^{0}$ and $k_{off}^{0}$ are the reaction rates at the equilibrium length $l=l_{0}$ between ligand and receptor. The mechanical properties of the biological bond are described by a harmonic spring. The equilibrium length is $l_{0}$ , and the force exerted on the receptor and ligand is $F_{b}=k_{s}(l-l_{0})$ . Here, $k_{s}$ represents the adhesive strength. This model and the relevant parameters (cf. table 1) are chosen according to the previous works of Fedosov (Reference Fedosov2010) and Fedosov, Caswell & Karniadakis (Reference Fedosov, Caswell and Karniadakis2011a ).

There are three dimensionless parameters:

(2.21) $$\begin{eqnarray}\displaystyle \text{Reynolds number} & \quad & \displaystyle Re=\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}R^{2}/\unicode[STIX]{x1D707},\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle \text{capillary number} & \quad & \displaystyle Ca=\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FE}}R/\unicode[STIX]{x1D707}_{0},\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle \text{adhesion number} & \quad & \displaystyle Ad=k_{s}/\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FE}}R.\end{eqnarray}$$

Considering the physiological environment surrounding the cell, the fluid flow is considered as a Stokes flow. Thus, $Re$ is very small, and we fix its value as $Re=0.0134$ to approximately represent the Stokes regime. The capillary number represents the ratio of the shear stress exerted on the surface of an elastic MP to the elastic force induced by the deformation of the elastic MP; and $\unicode[STIX]{x1D707}_{0}$ is the shear modulus of the MP. The higher the $Ca$ , the softer the particle. The adhesion number denotes the ratio of adhesive strength to shear stress of the flow. Thus, the higher the $Ad$ , the stronger the adhesion strength. In our simulations, $Ca$ is tuned by varying the shear modulus $\unicode[STIX]{x1D707}_{0}$ , and $Ad$ is varied by changing the adhesive strength  $k_{s}$ .

4.1 Margination of elastic MPs without adhesion

The margination of MPs without adhesion is first investigated. The margination process of a typical case of MPs with $Ca=0.0037$ is shown in figure 4. In these snapshots, at $t=0~\text{s}$ , we can see that MPs are randomly distributed as well as RBCs. The RBCs and MPs are considered at their equilibrium states. The shapes of RBCs and MPs are biconcave and spherical, respectively. At time $t=1.0~\text{s}$ , the fluid flow is developed. A large deformation has been observed for RBCs. Under the shear flow, we find that RBCs align their major axes along the flow direction. Though the deformation of MPs is small due to their high stiffness (small $Ca=0.0037$ ), they should deform under the shear stress; and the deformation will be significant for the case with high $Ca$ . In addition to the deformation, RBCs and MPs both demonstrate cross-stream migration, but towards the opposite directions. RBCs migrate from the near-wall region to the centre of the channel, while MPs move towards the wall. We also find that the CFL becomes clear and that some MPs reach the CFL quickly. As simulation time further advances, at $t=2.0~\text{s}$ , the CFL is fully developed and MPs start to accumulate at the CFL.

Localization of MPs at the wall is characterized by margination probability $\unicode[STIX]{x1D6F7}(t)$ , which is defined as

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(t)=\frac{n_{f}(t)}{N},\end{eqnarray}$$

where $n_{f}(t)$ represents the number of MPs with centres locating in the CFL at time $t$ , and $N$ denotes the total number of MPs in the channel. Before quantifying the margination probability, the thickness of the CFL is estimated in the absence of MPs. We use the same method as proposed by Fedosov et al. (Reference Fedosov, Caswell, Popel and Karniadakis2010b ); the thickness of the CFL is approximately $2.8~\unicode[STIX]{x03BC}\text{m}$ for current blood flow with $Ht=30\,\%$ . This is consistent with previous numerical studies (Lee et al. Reference Lee, Choi, Kopacz, Yun, Liu and Decuzzi2013; Müller et al. Reference Müller, Fedosov and Gompper2014). Figure 5(a) gives the evolution of margination probabilities $\unicode[STIX]{x1D6F7}$ for three different stiffnesses ( $Ca=0.00037$ , 0.037 and 3.7). We find that the margination process can be split into two stages. In the first stage, the margination probability increases very fast, which signifies that there are more and more particles moving from the centre to the CFL. We note that, in this stage, the margination probabilities of softer particles increase faster ( $Ca=0.037$ and 3.7) than that of stiff particles ( $Ca=0.00037$ ). However, the duration of this stage for stiff particles is longer than those of soft particles. Therefore, when the first stage ends, the margination probability of stiff particles is higher than those of soft ones. In the second stage, the margination probabilities of both stiff and soft particles increase slower than those in the first stage. Also, the growth rates for these particles are almost the same.

Figure 5.

Margination behaviour of elastic MPs with different stiffnesses. (a) Evolution of margination probabilities for elastic MPs. (b) Mean-square displacement of elastic MPs during margination. (c) Probabilities of two types of motion: centre to cell-free layer (C–F) and cell-free layer to centre (F–C). (d) Time-averaged margination probability in the steady-state regime.

To investigate this stiffness dependence of margination behaviour, the mean-square displacements (MSDs) for MPs with different stiffnesses are calculated. The deformation of RBCs in the blood flow induces the fluctuation of flow around them. This is considered as the root cause of migration of rigid particles such as platelets in blood flow (Zhao, Shaqfeh & Narsimhan Reference Zhao, Shaqfeh and Narsimhan2012). From figure 5(b), we find that there are no obvious differences among MSDs for all of the MPs. At the initial stage ( $t<1.0~\text{s}$ ), the MSDs are almost the same. After that, the MSDs for MPs become different, but with only small variations. We calculated the diffusivities, defined as $D=\langle \unicode[STIX]{x0394}z^{2}\rangle /2t$ , and they range from approximately 0.9 to $1.2\times 10^{-7}~\text{cm}^{-2}~\text{s}^{-1}$ for these MPs. This is in good agreement with previous studies (Zhao & Shaqfeh Reference Zhao and Shaqfeh2011; Vahidkhah & Bagchi Reference Vahidkhah and Bagchi2015). The diffusivity is approximately two orders of magnitude higher than the Brownian diffusivity, which means that the existence of RBCs augments the diffusion of MPs. However, from these results, RBC augmentation of diffusion is stiffness-independent. Thus, diffusion alone cannot explain the observed stiffness dependence of margination behaviour.

To gain a better insight into the margination behaviour, the types of motion of MPs in blood flow are studied. Compared to rigid particles in blood flow, elastic MPs may experience deformation-induced migration, which can drive them to move away from the vessel wall (Coupier et al. Reference Coupier, Kaoui, Podgorski and Misbah2008; Kumar & Graham Reference Kumar and Graham2012b ). This is the essential mechanism for CFL formation in a blood vessel. Here the motion of elastic MPs can be classified into four types: (i) staying in the centre; (ii) staying in the CFL; (iii) moving from the centre to the CFL (C–F); and (iv) moving from the CFL to the centre (F–C). Obviously, the first two types cannot contribute to the localization of MPs at the wall. The margination probability is attributed to the difference between the last two types as shown in figure 5(c). We find that the probabilities of F–C motion for all of the elastic MPs are the same and the values are almost 0. This indicates that there are only a few particles migrating from the CFL to the centre region. We also observe that the C–F motion has the same tendency as the margination probability $\unicode[STIX]{x1D6F7}$ . All these results lead to the conclusion that localization of MPs at the wall in the current study is determined by the C–F motion. This is different from our previous study in Ye et al. (Reference Ye, Shen, Yu, Wei and Li2017a ), in which F–C motion at some time can dominate the margination behaviour of particles. The reason for this difference mainly lies in the size of the particle and the haematocrit of blood flow. If the size of the particle is large ( $2~\unicode[STIX]{x03BC}\text{m}$ in diameter), and the haematocrit is high (30 %), there is no available space for the particle to stay in the centre of the channel, because, under shear flow, most parts of the centre region are occupied by RBCs. Hence, F–C motion is not significant in the present study.

To quantify the stiffness effect on margination probability, the mean margination probabilities $\langle \unicode[STIX]{x1D6F7}\rangle$ are calculated and are given in figure 5(d). The mean value takes the time-averaged value of margination probability, which is estimated in a time interval within the steady-state regime. We can see that the margination process of MPs reaches a steady state after approximately $t=2.5~\text{s}$ . The localization of MPs at the wall decreases dramatically when the particles are very stiff (small $Ca$ ). With further decrease of stiffness (increase of $Ca$ ), there is no obvious change of the margination probability.

The underlying mechanism of this stiffness dependence of margination behaviour relies on the interplay of collision with RBCs and deformation-induced lateral migration of elastic MPs (Qi & Shaqfeh Reference Qi and Shaqfeh2017). At the beginning of the simulation, the RBCs near the wall of the channel sense the shear flow, and then deform under the shear stress. The existence of the wall makes RBCs move away from the wall, and then the CFL forms. According to the previous study (Katanov, Gompper & Fedosov Reference Katanov, Gompper and Fedosov2015), the time needed to fully form the CFL is approximately $0.8~\text{s}$ under the conditions (channel size and haematocrit) in the current study (Ye et al. Reference Ye, Shen, Yu, Wei and Li2017a ). This signals that the first stage of margination probability corresponds to the development of the CFL (cf. figure 5 a). In this stage, a large number of RBCs move from the near-wall region to the centre of the channel. The migration of RBCs should induce a reverse flow moving from the centre to the CFL in the regions around RBCs, due to the mass conservation of the fluid. Hence, if the MPs are located in these reverse flow regions, they will move along with the flow from the centre to the CFL. This phenomenon looks like the exclusion effect that particles are excluded by RBCs from the near-wall region to the centre of the channel (Crowl & Fogelson Reference Crowl and Fogelson2011). Specifically, the exclusion effect appears more significant for soft particles than for stiff particles. Here soft MPs have stronger alignment with the flow due to deformation. This is the reason why, in the first stage, the soft MPs marginate faster than stiff ones. In the second stage, the CFL is fully formed and the flow is fully developed. The soft MPs in the near-wall region may experience deformation-induced migration due to the existence of the wall. This results in low accumulation of soft MPs in the CFL. However, when the stiffness of the MPs decreases to a critical value (approximately $Ca=0.037$ ), the deformation-induced migration dominates the motion of MPs. Therefore, under this circumstance, changing stiffness of MPs will not result in an obvious difference of the margination probability.

4.2 Adhesion effect on localization of elastic MPs at wall

Figure 6.

Snapshots for the margination behaviour of elastic MPs ( $Ca=0.37$ ) under the influence of adhesion: (a) $Ad=0.07$ and (b) $Ad=32.8$ .

In figure 5(a), it is obvious that the evolution of margination probability oscillates. In some time intervals, the oscillation amplitude can reach approximately 20 % of the margination probability. This indicates that many MPs are travelling between the centre of the channel and the CFL. Under this circumstance, MPs near the CFL have a chance to interact with the vessel wall through ligand–receptor binding. Since the diameter of an MP is $2.0~\unicode[STIX]{x03BC}\text{m}$ , when it moves near the CFL, part of its surface will be located inside the adhesion layer according to the thicknesses of the CFL ( $2.8~\unicode[STIX]{x03BC}\text{m}$ ) and adhesion layer ( $1.0~\unicode[STIX]{x03BC}\text{m}$ ).

To have a direct comparison, figure 6 presents the adhesion effect on the localization of elastic MPs. In figure 6(a,b), the stiffnesses of MPs are the same $Ca=0.37$ , while the adhesion strengths are different: (a)  $Ad=0.07$ and (b)  $Ad=32.8$ . We find that there are more MPs entering and staying inside the CFL when increasing the adhesion strength $Ad$ . With small $Ad=0.07$ , when MPs move into the CFL, only a small contact area forms between the MP and the substrate. The ligand–receptor binding is not strong, and these MPs move freely near the substrate. However, with strong adhesion $Ad=32.8$ , the MPs collapse on the substrate like a droplet on the ground. It should be emphasized that this collapse phenomenon only happens for soft MPs. If the MP is stiff or rigid, it cannot deform any more. They can only roll on the substrate (King & Hammer Reference King and Hammer2001; Decuzzi & Ferrari Reference Decuzzi and Ferrari2006; Coclite et al. Reference Coclite, Mollica, Ranaldo, Pascazio, de Tullio and Decuzzi2017). However, elastic MPs can either roll or firmly adhere on the substrate, depending on the adhesion strength  $Ad$ .

Figure 7.

Margination probabilities of elastic MPs with adhesion effect. (a) Margination probability against adhesion strength with different MP stiffnesses. (b) Margination probability against MP stiffness with different adhesion strengths.

To differentiate the margination probability of MPs with and without adhesion, we use $\unicode[STIX]{x1D6F1}$ rather than $\unicode[STIX]{x1D6F7}$ to represent the margination probability with the adhesion effect. The interplay of stiffness and adhesion strength effects is isolated in figure 7. Figure 7(a) gives the relationship between the margination probability and the adhesion strength for MPs with different stiffnesses. We use $\langle \,\cdot \,\rangle$ to denote the mean value over a time interval, and subscript $m$ represents margination. We find that the margination probabilities have the same tendencies with increment of adhesion strength for MPs with different stiffness. Under relatively low adhesion strength ( $Ad<5$ ), the margination probability dramatically increases with increasing adhesion strength; while further increment of adhesion strength makes the margination probability slowly decrease ( $5<Ad<23$ ). But when the adhesion strength exceeds a critical value, the margination probability will increase with the increment of adhesion strength again. The critical value differs among MPs with different stiffnesses. The margination probability against stiffness is given in figure 7(b) for MPs with different adhesion strengths. The margination probability result of MPs without adhesion ( $Ad=0$ ) is also presented to allow a comparison. We find that, with the same adhesion strength, the margination probabilities increase with the increment of $Ca$ when MPs are stiff (relatively small $Ca$ ); while a further increase of $Ca$ results in the decrease of margination probability. Though the margination probability has a decrement when the MP is soft (high $Ca$ ), it is still higher than that of an MP without adhesion. The difference of margination probability between the cases with and without adhesion is determined by the adhesion strength. These relationships remain to be discussed in detail later.

Figure 8.

Contours of margination probability on the $Ca{-}Ad$ plane.

Furthermore, we summarize the results of margination probabilities in the contours on the $Ca{-}Ad$ plane as shown in figure 8. We find that two peaks exist in the contours for the margination probability. One is in the region with high adhesion strength $Ad$ and moderate stiffness (moderate $Ca$ ), namely I; the other one is located in the region with moderate adhesion strength $Ad$ and large stiffness (small $Ca$ ), denoted as II. These two regions, which favour margination behaviour, are determined by the interplay of the adhesion effect and deformability. To investigate the underlying mechanisms, the adhesion behaviour of elastic MPs is first examined.

4.3 Adhesion behaviour of elastic MPs

Figure 9.

Identification of motion types of elastic MPs. (a) Snapshots for firm adhesion (FA), stop-and-go motion (SG), stable rolling (SR) and free motion (FM). (b) Corresponding trajectories of the four different types of motion for elastic MPs along the flow direction.

The adhesion behaviour should be influenced by the deformability according to previous studies (Ndri, Shyy & Tran-Son-Tay Reference Ndri, Shyy and Tran-Son-Tay2003; Khismatullin & Truskey Reference Khismatullin and Truskey2005; Balsara, Banton & Eggleton Reference Balsara, Banton and Eggleton2016; Luo & Bai Reference Luo and Bai2016; Ye et al. Reference Ye, Shen and Li2018a ). The deformability of MPs can affect the hydrodynamics, which balances the spring force exerted by the biological bonds. It is revealed that deformation of an MP can promote the adhesion of the MP to the substrate. Previous studies (Ndri et al. Reference Ndri, Shyy and Tran-Son-Tay2003; Khismatullin & Truskey Reference Khismatullin and Truskey2005) demonstrated that when an elastic MP moved near the substrate, the bottom of the MP was flattened. This resulted in a large contact area between the MP and the substrate. Then the adhesion became strong. Before we present the adhesion behaviour of elastic MPs in blood flow, the classification of the types of motion of elastic MPs is shown first. On the basis of our adhesive model, the probabilistic model (Hammer & Lauffenburger Reference Hammer and Lauffenburger1987), there are a total of four motion types of elastic MPs, which are presented in figure 9(a). They are characterized by the snapshots at $t=0.01$ , 0.02, 0.03 and 0.04 s along the flow ( $y$ ) direction. In the case of firm adhesion (FA), the MP collapses on the substrate like a droplet and cannot move any more. The MP can slowly move at some time intervals despite collapsing on to the substrate in stop-and-go motion (SG). However, in stable rolling (SR), the MP moves on the substrate. Additionally, the MP deforms like an ellipsoid under shear stress with a flattened contact area between MP and substrate. In the free motion (FM), the MP totally becomes an ellipsoid, and it moves freely near the substrate. Under FM, there is no obvious contact between the MP and the substrate. These motions are distinguished by calculating the velocity of the MP’s centre (cf. figure 9 b) along the flow direction ( $y$ direction). In FA, the velocity is nearly zero through the simulation time. When the velocity is non-zero at some time intervals and zero at other time intervals, it is referred to as SG. As for the SR and FM, there is no difference in terms of trajectories of the MP’s centre, while their velocities are not identical. If the velocity of the MP’s centre is the same as the fluid velocity at the same location, it is defined as FM. Otherwise, it is SR motion. The detailed classification of these adhesion types is discussed in the supporting information (supplementary material is available at https://doi.org/10.1017/jfm.2018.890).

The adhesion probability is used to quantify the behaviours of MPs near the substrate; it is defined as

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{a}(t)=\frac{n_{a}(t)}{N},\end{eqnarray}$$

where $n_{a}(t)$ represents the number of elastic MPs that have interactions with the substrate at time $t$ , $N$ is the total number of elastic MPs, and subscript $a$ is adopted to distinguish it from margination probability. The formation of a biological bond between the ligand and the receptor is an indication of interaction between the MP and the substrate. Additionally, the adhesion probability of the four motion types of MPs on the substrate is defined as the number fraction of MPs with definite motion types. In the following, for simplicity, the type names of the MPs represent the corresponding adhesion probability. Figure 10 presents the adhesion probabilities for elastic MPs and the motion types. In figure 10(a), the adhesion strength is weak ( $Ad=0.7$ ). We find that when the MPs are stiff (low $Ca$ ), FM and SR dominate the motion of MPs compared to SG and FA; almost no MP has FA. But when the MPs become soft (increasing $Ca$ ), FM and SR decrease; and the FM can even vanish when $Ca$ is large enough. When MPs are very soft (high $Ca$ ), SG and FA start to increase, and SG can exceed SR and FM. One thing that should be noted is that the sum of the adhesion probabilities of all four motion types should be equal to the total adhesion probability. The tendency is consistent with previous studies that deformability can promote the firm adhesion of particles on the substrate (Ndri et al. Reference Ndri, Shyy and Tran-Son-Tay2003; Khismatullin & Truskey Reference Khismatullin and Truskey2005; Shen et al. Reference Shen, Ye, Kröger and Li2018). However, when the adhesion strength increases (cf. figure 10 b), the adhesion probabilities have the opposite trend compared to that under weak adhesion. FA and SG dominate the motion of MPs when MPs are stiff. When MPs become soft (increasing $Ca$ ), SG and FA start to decrease, but SR and FM increase. When MPs are soft enough (high $Ca$ ), SR can outperform FA and SG. This results in the opposite conclusion that a stiff MP demonstrates superior adhesion compared to soft particles when adhesion strength is strong.

Figure 10.

Adhesion probabilities for elastic MPs and corresponding adhesion probabilities of different motion types: (a) $Ad=0.7$ and (b) $Ad=32.8$ .

Figure 11.

(a) Phase diagram for motion type of elastic MPs on $Ca{-}Ad$ plane. (b) Adhesion probability contours of elastic MPs on $Ca{-}Ad$ plane.

To have detailed motion type distributions with different adhesion strengths and stiffnesses, we give the phase diagram of motion types on the $Ca{-}Ad$ plane in figure 11(a). The type is chosen as follows: for example, when $Ca=0.037$ and $Ad=32.8$ , the adhesion probability of FA dominates compared to the other three motion types, and then we use FA to represent the motion type of MPs under the specific adhesion strength and stiffness. Comparing figure 11(a) with the margination probability contour under the adhesion effect (cf. figure 8), we find that the two regions favouring margination are just the FA and SG regions corresponding to the motion type phase diagram. This means that the adhesion-favouring region is also the margination-favouring region. Additionally, the adhesion probability contour is provided in figure 11(b). We find that in the regions where the margination probabilities are high, the adhesion probabilities are also high. Thus high margination should be a prerequisite for high adhesion, but the intrinsic relationship between margination and adhesion is not clear. We will discuss it in detail below.

Figure 12.

(a) Interaction modes of elastic MPs in blood flow. (b) Three mechanisms dominating the motion of elastic MPs near the wall of a channel.

4.4 Mechanism of localization of elastic MPs under adhesion

When the elastic MP is located in the blood flow, it can interact with other objects, such as RBCs, the wall of the channel and other MPs. Particularly, considering the adhesion effect, the elastic MPs can also interact with the wall through ligand–receptor binding. We give a simple schematic to show the interaction modes of an elastic MP with other objects in figure 12(a). Here, the interaction mode between MP and MP is negligible due to the low volume fraction (less than 1 %). When the MP is located in the centre region of the channel, it can only interact with RBCs through hydrodynamic collision, which is denoted as interaction mode $A$ in figure 12(a). We define a near-wall region $\unicode[STIX]{x1D6E5}$ , in which the existence of the wall will influence the motion of objects within it. The thickness of $\unicode[STIX]{x1D6E5}$ is approximately three times the radius of the object (Singh et al. Reference Singh, Li and Sarkar2014). If an MP enters the region $\unicode[STIX]{x1D6E5}$ , it can experience deformation-induced migration besides a hydrodynamic collision with an RBC. We name this mode as  $B$ . Further moving towards the wall makes an elastic MP locate around the interface of the CFL ( $\unicode[STIX]{x1D6FF}$ ). An MP in this region has complicated interactions with its surroundings. It not only collides with an RBC and experiences deformation-induced migration, but also starts to interact with the wall through ligand–receptor binding. The interaction mode in this region is symbolized as  $C$ . After the MP moves into the CFL and adheres on the substrate, it experiences both adhesive interaction and deformation-induced migration. We call this interaction mode $D$ . We focus on mode $C$ and isolate it in figure 12(b). Here we believe that the motion of an elastic MP with mode $C$ is complex but crucial to the margination and adhesion process. The region where this mode happens is located around the interface of the CFL and adhesion layer. If the MP moves away from the wall, it will not be counted as an MP having localization. While, when it moves towards the wall, it will be regarded as an MP owing to margination and adhesion. The moving direction is attributed to the competition among three mechanisms: (i) deformation-induced migration; (ii) adhesion effect; and (iii) near-wall hydrodynamic collision with an RBC. The deformation-induced migration makes an MP move away from the wall, and thus hampers localization. The adhesion effect plays a role through biological bond, and it facilitates localization. As for the near-wall hydrodynamic collision with an RBC, because there is no RBC within the CFL, then the collision should be a one-sided collision. The RBCs are always located on one side of the MP. Furthermore, it is confirmed that three-body and higher-order collision schemes can be negligible under the current circumstances ( $Ht=30\,\%$ ) (Kumar & Graham Reference Kumar and Graham2012a ; Rivera, Zhang & Graham Reference Rivera, Zhang and Graham2016; Qi & Shaqfeh Reference Qi and Shaqfeh2017). Therefore, only a side pair collision is considered here. According to the locations of the RBC and MP, the pair collision hinders the penetration of the MP into the centre of the channel, and thus promotes localization.

Figure 13.

(a) Numerical experiment for pair collision between an RBC and an elastic MP. (b) Collision displacement of an RBC and an elastic MP with different stiffnesses. (c) Comparison of collision displacement and deformation-induced migration displacement of elastic MPs. (d) Number of biological bonds established when elastic MPs adhere on a substrate with different adhesion strengths against stiffness of elastic MP.

The side pair collision is first examined systematically for elastic MPs with different stiffnesses. Figure 13(a) gives the side pair collision illustration. In this numerical experiment, the channel and the flow condition are the same as in the above simulations for margination of elastic MPs. A single RBC and an elastic MP are placed in the centre of the channel to eliminate the wall effect. The centre distance between their initial positions in the height direction ( $z$ direction) is $\unicode[STIX]{x1D70E}=2~\unicode[STIX]{x03BC}\text{m}$ . During the simulation, the trajectories of the centres of the RBC and the MP are tracked. The displacement of the centres of the RBC and the MP in the z direction refers to the collision displacement. The evolution of collision displacements for the RBC ( $\unicode[STIX]{x1D6E5}_{R}$ ) and the elastic MP ( $\unicode[STIX]{x1D6E5}_{S}$ ) are presented in figure 13(b). As for the RBC, we find that the collision displacements are almost the same; and they are much smaller compared to all of the elastic MPs. They have no dependence on the stiffness of MPs. This is attributed to the small size of an MP compared to the RBC. The trajectories of MPs with different stiffnesses have the same trend. The first approach between the RBC and the MP makes the MP abruptly migrate towards the wall. After collision ends, it can partially restore towards its initial position. It is located in a specific equilibrium position between the initial and maximum migration positions. We denote the distance between this equilibrium position and the initial position as the collision displacement, which is based on the definition in Loewenberg & Hinch (Reference Loewenberg and Hinch1997), Kumar & Graham (Reference Kumar and Graham2011, Reference Kumar and Graham2012a ,Reference Kumar and Graham b ) and Zhao & Shaqfeh (Reference Zhao and Shaqfeh2013). From the zoom in figure 13(b), we can see that the difference of collision displacements among MPs with different stiffnesses is small. But this is an individual collision between an RBC and an MP. Repeated collisions between RBCs and MPs will distinguish the collision displacement with a large value for MPs with different stiffnesses. The result is shown in figure 13(c), $L_{p}$ represents the collision displacement. We find that when $Ca<0.037$ , the collision displacement increases with the increment of $Ca$ , while when $Ca>0.037$ , it decreases slightly with the increment of $Ca$ . Hence, there is an optimal stiffness for the pair collision of an elastic MP and an RBC (results in the supporting information point out that this optimal $Ca$ is a bit larger than 0.037).

As far as we know, this is the first time that a hydrodynamic collision between an RBC and an elastic MP has been presented. There are also a number of previous studies showing the pair collision between particles with either the same volume or the same shape (Kumar & Graham Reference Kumar and Graham2011, Reference Kumar and Graham2012b ; Sinha & Graham Reference Sinha and Graham2016). The collision result can be explained as follows. Owing to the size difference between an RBC (diameter $8~\unicode[STIX]{x03BC}\text{m}$ ) and an elastic MP (diameter $2~\unicode[STIX]{x03BC}\text{m}$ ), the motion of the elastic MP is probably governed by the fluctuation of the flow field near the RBC induced by its deformation. The RBC should perform a tank-treading motion under shear flow, and the flow field around it is presented in figure 13(a). When $Ca<0.037$ , with the increment of $Ca$ , it is easier for a soft MP to align itself with the flow field, thus the collision displacement increases. However, further increase of $Ca$ makes the tank-treading motion of a soft MP become significant (Fedosov et al. Reference Fedosov, Caswell and Karniadakis2010a ). The tank-treading motion of a soft MP can also induce the fluctuation of flow around it, but with the opposite direction to the flow field induced by RBC motion. It acts as resistance to the alignment of the MP with the flow field. Therefore, further increase of $Ca$ will lead to the decrease of the collision displacement of an MP.

When an elastic MP is placed in the flow, the shape cannot be analytically captured, especially for an MP with large deformation. Therefore, the deformation-induced migration, which is highly dependent on the shape of the MP, can hardly be determined. In the literature, numerical simulations are employed to study the deformation-induced migration of an elastic MP in a flow (Doddi & Bagchi Reference Doddi and Bagchi2008; Kaoui et al. Reference Kaoui, Ristow, Cantat, Misbah and Zimmermann2008; Nix et al. Reference Nix, Imai, Matsunaga, Yamaguchi and Ishikawa2014; Singh et al. Reference Singh, Li and Sarkar2014; Qi & Shaqfeh Reference Qi and Shaqfeh2017). Here, an empirical relationship in Singh et al. (Reference Singh, Li and Sarkar2014) is adopted, due to its systematics. In their study, the lateral migration velocity of the deformable capsule is a function of its stiffness ( $Ca$ ) and distance away from the wall ( $h$ ). A phenomenological formula for migration velocity is given as

(4.3) $$\begin{eqnarray}\frac{V_{d}}{\dot{\unicode[STIX]{x1D6FE}}a}=\left\{\begin{array}{@{}ll@{}}(0.65Ca+0.021)\left({\displaystyle \frac{a}{h}}\right)^{2},\quad & Ca\leqslant Ca_{cr},\\ V_{cr}^{\ast }+0.02(Ca-Ca_{cr})^{0.6}\left({\displaystyle \frac{a}{h}}\right)^{1.35},\quad & Ca>Ca_{cr},\end{array}\right.\end{eqnarray}$$

where $V_{cr}^{\ast }=(V_{d}/\dot{\unicode[STIX]{x1D6FE}}a)|_{Ca=Ca_{cr}}$ . Their simulation results pointed out a power-law relation for the capsule velocity. There exists a critical stiffness, $Ca_{cr}$ , of the capsule; here we choose it as $Ca_{cr}=0.15$ according to the proposed regime in Singh et al. (Reference Singh, Li and Sarkar2014). When $Ca\leqslant Ca_{cr}$ , the migration velocity is linearly proportional to $Ca$ and related to $h^{-2}$ , while when $Ca>Ca_{cr}$ , the velocity has 0.6 and $-1.35$ power scalings with $Ca$ and $h$ , respectively. On the basis of this relation, we set $h=2~\unicode[STIX]{x03BC}\text{m}$ , which corresponds to a position between the adhesion layer and the CFL, and then integrate (4.3) from $t=0$ to $t=0.1$ . The deformation-induced migration displacement $L_{d}$ against stiffness of the elastic MP is obtained and is shown in figure 13(c), denoted as a blue line.

To study the adhesion effect, we also conduct simulation experiments to investigate the adhesion behaviour of a single elastic MP on a substrate under shear flow. The flow and the channel size are the same as in the above margination study. The interaction between an elastic MP and the substrate is established by the biological bonds formed in the adhesion process, and the bond is modelled as a linear spring. The number of bonds can be used to quantify the adhesion effect. Figure 13(d) shows the relationship between the number of bonds and the stiffness of the elastic MP under different adhesion strengths. We find that the number of bonds increases with the increment of $Ca$ when $Ca$ is not large ( $Ca<0.037$ ). However, further increase of $Ca$ does not significantly affect the number of bonds. When the adhesion strength is very strong ( $Ad=32.8$ ), the number of bonds may slightly decrease with increment of  $Ca$ .

The results given above demonstrate the strength of three mechanisms against the stiffness of MPs. They are combined to explain the margination results in figure 7(b). When the MP is relatively stiff (low $Ca<0.037$ ), with the increment of $Ca$ , the collision displacement increases, the adhesion effect increases, and the deformation-induced migration displacement remains almost unchanged. Two promoting factors of localization increase and one impeding factor remains unchanged. Thus, the margination probability increases with the increment of $Ca$ . However, when $Ca$ exceeds the critical value $Ca=0.037$ , the MP becomes relatively soft. With the increment of $Ca$ , the collision displacement decreases, the adhesion effect stays almost unchanged, and the deformation-induced migration drastically increases. The impeding factor of localization dominates compared to the other two promoting factors. Hence, in this circumstance, the localization of the MP decreases with the increment of  $Ca$ .

Figure 14.

(a) Deformation of particles with $Ca=0.037$ under different adhesion strengths. Dashed lines are used as a guide to show the deformation of particles. (b) Number of biological bonds established when elastic MPs adhere on a substrate against adhesion strengths for particles with different stiffnesses.

Furthermore, the relationship between margination behaviour and adhesion strength is isolated for investigation. With the same $Ca$ , the side pair collision displacements for MPs under different adhesion strengths should be the same, because the collision displacement is independent of the adhesion, while the deformation-induced migration displacement should be influenced by the adhesion strength. From (4.3), it seems that the deformation-induced migration velocity is only relevant to $Ca$ . But this is not true. The root cause of deformation-induced migration is the deformation of the MP under shear flow. Here, considering the adhesion effect, the deformation of the MP is affected not only by $Ca$ , but also by the adhesion strength $Ad$ . Figure 14(a) presents the configurations of the MP with the same $Ca=0.037$ , but under different adhesion strengths. We find that, when the adhesion strength is weak ( $Ad=0.07$ ), the deformation of the MP is small. With the increment of adhesion strength ( $Ad=0.7{-}13.2$ ), the deformation becomes significant and increases, while further increase of $Ad$ will not cause any further increment of MP deformation. This result reveals that the adhesion effect plays a role in the localization of MPs through influencing the deformation of the MP. Additionally, the relationship between the number of bonds and the adhesion strength for MPs with different stiffnesses is displayed in figure 14(b). We find that, when the adhesion strength is small ( $Ad<3.3$ ), the number of bonds dramatically increases with the increment of $Ad$ ; while further increment of $Ad$ also results in the increase of number of bonds, but with a slow growth rate.

Apart from the collision effect, the adhesion effect and deformation-induced migration are combined to reveal the underlying mechanism of margination probability against adhesion strength for MPs with different stiffnesses in figure 7(a). When $Ad$ is small ( $Ad\sim 0.7$ ), the deformation-induced migration displacements are almost the same, but the adhesion effect dramatically increases, and thus the margination probability grows rapidly with the increment of adhesion strength. When $Ad$ becomes relatively large ( $Ad\sim 0.7{-}13.2$ ), the adhesion effect slowly increases, while the deformation of particles is significant, leading to large deformation-induced migration displacement. Therefore, in this regime, the margination probability decreases with the increment of adhesion strength. With the further increase of $Ad$ , the deformation-induced migration displacements are almost constant, but adhesion effects still slowly increase. Hence, the margination probability should increase with the increment of adhesion strength in this regime.