### Forces analysis

The Reynolds number for a spherical particle moving through a viscous fluid is as follows:

\mathrm{Re}=2{\mathit{\rho}}_{\mathit{f}}\mathit{vr}/\mathit{\eta},

(1)

where *ρ*_{
f
} is the density of the viscous fluid, *v* is the velocity of the particle, *r* is the radius of the particle, and *η* is the viscosity of the viscous fluid. Figure 2 shows the microstructure photograph of the phosphor particles. Since the radius of the phosphor particle is at micron level and silicone is a high-viscosity fluid, the Reynolds number for phosphor particle sedimentation in silicone is quite small, so the inertia force is much smaller than the resistance force, which means the inertia force can be ignored.

According to the Stokes’ law, the resistance force for a spherical particle moving through a viscous fluid can be written as follows, ignoring the inertia force [8]:

{\mathit{F}}_{\mathit{Stokes}}=6\mathit{\pi \eta rv}.

(2)

Figure 3 shows the forces acting on the phosphor particle. Taking the gravitational and the buoyancy forces into account, *F*_{
grav
} is written as follows:

{\mathit{F}}_{\mathit{grav}}=\left(4/3\right)\mathit{\pi}{\mathit{r}}^{3}\mathit{g}\left({\mathit{\rho}}_{\mathit{p}}-{\mathit{\rho}}_{\mathit{f}}\right),

(3)

where *g* is the acceleration of gravity, and *ρ*_{
p
} is the density of the particle.

Hence, the kinetic equation for the phosphor particle sedimentation in the silicone is as follows:

{\mathit{F}}_{\mathit{grav}}-{\mathit{F}}_{\mathit{stokes}}=\mathit{ma}=\left(4/3\right)\mathit{\pi}{\mathit{r}}^{3}{\mathit{\rho}}_{\mathit{p}}\frac{\mathit{dv}}{\mathit{dt}}.

(4)

Substituting Eq. (2) and Eq. (3) into Eq. (4), and the velocity of the particles could be described as this:

\mathit{v}\left(\mathit{t}\right)=\left(\mathit{A}-\mathit{A}{\mathit{e}}^{-\mathit{Bt}/\mathit{C}}\right)/\mathit{B},

(5)

where *A* = (4/3)*πr*^{3}*g*(*ρ*_{
p
} - *ρ*_{
f
}), *B* = 6*πηr*, and *C* = (4/3)*πr*^{3}*ρ*_{
p
}. Thus, the size distribution of the phosphor particles and the viscosity-time function of silicone during curing are the key to get the velocity of any phosphor particle at any time during the sedimentation process, according to A, B and C.

### Size distribution of phosphor

Figure 4 shows the size distribution of phosphor particles provided by the phosphor manufacturer named Intematix. Fitting the distribution curve of the particles gives the size distribution function as follows:

\mathit{f}\left(\mathit{x}\right)=\left\{\begin{array}{l}0.124{\mathit{e}}^{0.391\mathit{x}}-1.040\phantom{\rule{23.5em}{0ex}}\mathit{x}<12.5\\ -56.741+5.802\mathit{x}\phantom{\rule{22em}{0ex}}12.5\le \mathit{x}<21.2\\ -504.968+27.245\left(1-{\mathit{e}}^{-0.041\mathit{x}}\right)+577.003\left(1-{\mathit{e}}^{-0.152\mathit{x}}\right)\phantom{\rule{1em}{0ex}}\mathit{x}\ge 21.2\end{array}\right.\phantom{\rule{3em}{0ex}},

(6)

where *f*(*x*) means the probability that the particle size is smaller than *x*.

### Viscosity-time function of silicone

In order to find the variation of viscosity with time during isothermal curing, CPA2000+ viscometer was used. Keeping the temperature of the heating plate at 85°C, a series of data was measured with the lapse of time. Figure 5 shows the relationship between the viscosity of silicone and time during isothermal curing at 85°C. The viscosity increases exponentially with time [9]. Fitting the curve gives the viscosity-time function as follows:

\mathit{\eta}\left(\mathit{t}\right)=0.131+{\mathit{e}}^{0.024\mathit{t}-3.587}.

(7)

Substituting Eq. (6) and Eq. (7) into Eq. (5), the velocity of any phosphor particle at any time during the sedimentation is obtained, and the displacement of the particle can be calculated by integration as a result.