## 3 Results and Discussion
### Trajectory of Side-by-Side Particles of Different Densities  
As discussed above, the path of a single particle settling in PAAm solution can be approximately regarded as a vertical line. In comparison, for the sedimentation of two similar side-by-side particles, a particle-particle interaction would occur and cause the particles to deviate from the vertical path at small \(L^{*}\)[7]. This interaction and deviation from the case of a single particle can be enhanced by the identical particle terminal velocity, which results in a highly similar elevation of the two particles. Therefore, within the critical distance \(L^{*}\), the drafting-kissing-tumbling phenomenon (DKT) had been widely observed for this initial particle configuration [7, 28]. In our study, we found that the difference of particles' density leads to great difference in settling velocity (Figure 12), and then the vertical distance between the particles is enlarged with time. Nevertheless, the wake of the heavy particle, in a leading position, can still affect the light particle settling into a non-linear path. The particle-particle interaction can also cause the transverse migration of the heavy particle, and the smaller the \(\alpha\), the greater the transverse migration. Notably, at the low value of \(\alpha\), the path of the heavy particle settling can be approximately regarded as a vertical line at \(L^{*}\) < 3. The clear transverse migration of the heavy particle occurred at \(3<L^{*}\) < 6.1 and disappeared at \(L^{*}\) > 7.1. Compared to the trajectory of heavy particles, the particle-particle interaction has a stronger impact on the light particle. There are three basic types of trajectories of light particles in the \(x\)-\(z\) plane as the light particle traces the heavy one:  
Type (i) in the horizontal direction (\(x\)), the light particle first is strongly attracted and then is repelled by the rectilinear path of the heavy one (for \(L^{*}\) = 1.2 and \(\alpha\) = 1.92, 2.79 and 4.71). Type (ii) the light particle approaches the heavy one in the \(x\) direction, and then travel a certain distance within the path of the heavy one (for \(L^{*}\) = 2.0 and \(\alpha\) = 1.92 and 2.79) without bouncing from the wake of the heavy particle in the limited field of view of the camera. Type (iii) the light particle is slightly away from the path of the heavy one at first in the horizontal direction, and then the light particle moves back to its original position in the \(x\) direction (for \(L^{*}\) = 3 and \(\alpha\) = 1.92, 2.79 and 4.71)  
Figure 13 shows three basic types of settling trajectories of the side-by-side particles. The line connecting the particles illustrates the instantaneous position of this pair of particles at a certain moment. In the cases of type (i), two particles settle vertically at first, and the heavy particle travels faster due to its larger density, causing the appearance of a corridor with reduced viscosity (CRV) due to the shear-thinning property of the fluid. Then, the light particle deviated from its initial position and was attracted by this corridor in the \(x\) direction. This phenomenon is similar to the drafting stage of the DKT pattern in the settling of two particle with different densities in a Newtonian fluid [11] and the attracting behavior reported in two particles settling in viscoelastic fluids [7, 20, 28]. In Newtonian fluids, the approach of two particle is mostly due to the lower pressure caused by the wake of the leading particle [42]. While in viscoelastic fluids (such as PAAm solution), a widely accepted explanation for particle-particle attraction is that the first normal stress difference of the viscoelastic fluid modifies the pressure distribution on the particle surface [7]. Hence in the current experiments, the viscoelasticity of the PAAm solution could have caused the light particle to approach the heavy one in the \(x\) direction. Further, the light particle would be sucked into the corridor with a reduced viscosity behind the heavy particle.