Evaluation of photon migration using a two speed model for characterization of packed powder beds and dense particulate suspensions

Tianshu Pan; Sarabjyot S. Dali; Eva M. Sevick-Muraca

doi:10.1364/OPEX.13.003600

1. Introduction

Measurements of time-dependent propagation of light in particulate and powder media may enable their characterization of the multiply scattering media in situ. Recently, our group has focused upon employing frequency domain photon migration (FDPM) measurements and the optical diffusion equation for monitoring the blend homogeneity of active pharmaceutical ingredient (API) within pharmaceutical blending operations by relating the measured absorption coefficient of the powder bed to the API concentration [1–5].

However, the applicability of diffusion approximation in a powder bed or a dense particulate suspension may be more complicated than that in tissue or colloidal suspensions, in which the transit of photons can be approximated by an average constant velocity, C/n̄_med , where c is the light speed in vacuum and n̄_med is the average refractive index of scattering medium. For dilute systems, n̄_med≈n_med , where n_med is the refractive index of the suspending medium. In complex media whereby the difference of the refractive index between the scattering particles and suspending fluid can be significant, (such as the case of metal oxides, lactose, or other powders), the relative refractive index and the structure resulting from the packing of scatters may dramatically impact time-dependent measurements of multiply scattered light. As illustrated in Fig. 1, propagating photons in dense powder beds have two velocities. Outside particles, photons travel at a velocity of c/n_med , whereas they travel within particles at velocity, c/n_p , where n_p is the refractive index within the particle of the powder or particulate suspension. While a one-speed diffusion equation may describe photon migration in tissue media, [6] a two-speed diffusion equation may be applicable in dense systems with disparate refractive indices.

Fig. 1. Schematic of light trajectory in ensemble of large dense, scatters (d_p≫λ) with significant refractive index difference with surroundings. The trajectory of photons within particles occurs at speed c_p=c/n_p , while outside the particles, photons travel with speed c_med=c/n_med , where c is the light speed in vacuum and n_p and n_med are refractive indices of particle and surrounding medium, respectively.

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In this contribution, we present a two-speed diffusion equation for prediction of time-dependent light propagation through powders and dense particulate suspensions and develop a simulation method, which consists of (i) dynamic simulation for generating structures of powder bed and dense particulate suspensions and (ii) Monte Carlo for tracking photon trajectories through media with generated structures. We then compare the transport coefficients predicted from both two-speed diffusion equation and Monte Carlo simulation results with experimental measurements of time-dependent light propagation in order to confirm the validity of a two-speed model.

The organization of this paper is as follows: In Section 2, we describe (i) the two-speed diffusion equation based on the hypothesis of photon equilibrium in the two-speed radiative transfer equation; (ii) dynamic simulation of powder particle sedimentation; and (iii) Monte Carlo simulation of photon migration within the simulated powder bed. We detail the experimental measurements of time-dependent light propagation made in resin and lactose powders and particulate suspensions using frequency domain photon migration (FDPM) techniques. In Section 3, we present results which (i) validate the hypothesis of photon equilibrium in two-speed diffusion equation through Monte Carlo simulations of photon density distributions; (ii) confirm the two-speed diffusion equation and the scattering coefficient predicted by Monte Carlo simulation through experimental measurements in resin and lactose powder beds; and (iii) demonstrate the impact of volume fraction upon absorption coefficients through FDPM experiments in resin-water and lactose-ethanol suspensions.

2. Methods and approach

2.1 Two-speed photon migration diffusion equation

Under the illustrated conditions of a two-phase medium (consisting of the suspending medium and particle phases with a significant relative refractive index difference) one can represent the diffusion propagation of photons with disparate speeds in terms of a two-group model in neutron transport [7].

An expression for the density of photons, ϕ_p , propagating at speed c/n_p within the particle phase with particle absorption, µ_a,p , can be written as

\frac{\partial ϕ_{p}}{\partial t} + \frac{c}{n_{p}} \vec{\nabla} \cdot {\vec{J}}_{p} + \frac{c}{n_{p}} \cdot μ_{a, p} \cdot ϕ_{p} + r_{loss, p} \cdot ϕ_{p} - r_{loss, med} \cdot ϕ_{med} = 0 .

The density of photons, ϕ_med , propagating at speed c/n_med within the medium of absorption cross section, on, µ_a,med , can be expressed from the solution of

\frac{\partial ϕ_{med}}{\partial t} + \frac{c}{n_{med}} \vec{\nabla} \cdot {\vec{J}}_{med} + \frac{c}{n_{med}} \cdot μ_{a, med} \cdot ϕ_{med} - r_{loss, p} \cdot ϕ_{p} + r_{loss, med} \cdot ϕ_{med} = 0 .

In these expressions, the term r_loss,p·ϕ_p represents the loss of photons from the particle phase into the suspending medium phase; and r_loss,med·ϕ_med represents the loss of photons from the suspending medium phase into the particle phase. We assume that the photon loss comes only from the absorption processes and that photons travel between the particle and surrounding medium phases in a purely elastic scattering process. The terms $\vec{J}$ _med and $\vec{J}$ _p are current flux of photons within the suspending medium and particle phase, respectively.

Furthermore, we introduce a hypothesis, which will be confirmed by Monte Carlo simulation later in the contribution, that at late times when the diffusion approximation is valid (i.e., $∣ \frac{\partial \overset{̅}{J}}{\partial t} ∣ ≪ \frac{c}{3 D} ∣ \vec{J} ∣$ , where D is a diffusion coefficient), there is a local equilibrium between photon densities within each phase such that:

\frac{ϕ_{p} (r, t)}{ϕ_{med} (r, t)} = K_{e}

where K_e is the equilibrium constant, which depends upon the available volume fraction for photon transit inside particles and their surrounding media.

Given the equilibrium between ϕ_p and ϕ_med and given Fick’s law for multi-component diffusion,[8]

{\vec{J}}_{med} = D_{med, med} \vec{\nabla} ϕ_{med} + D_{med, p} \vec{\nabla} ϕ_{p}

{\vec{J}}_{p} = D_{p, med} \vec{\nabla} ϕ_{med} + D_{p, p} \vec{\nabla} ϕ_{p}

where D_ij(i, j=med, p) is an element of the multi-component diffusion coefficient tensor, Equations (1)–(5) can be combined to obtain

\frac{n_{med}}{c} \frac{\partial ϕ_{med}}{\partial t} - D_{bed} {\vec{\nabla}}^{2} ϕ_{med} + μ_{a, bed} \cdot ϕ_{med} = 0

where the diffusion coefficient for the bed is given by:

D_{bed} = \frac{(D_{med, med} + K_{e} \cdot D_{med, p}) + \frac{n_{med}}{n_{p}} (D_{p, med} + K_{e} \cdot D_{p, p})}{(1 + K_{e})}

and the absorption coefficient of the powder bed is given by:

μ_{a, bed} = \frac{μ_{a, med} + \frac{n_{med}}{n_{p}} \cdot K_{e} \cdot μ_{a, p}}{(1 + K_{e})} .

Furthermore, the weak absorption coefficient in powder bed allows the approximation of the isotropic scattering coefficient as ${μ'}_{s, bed} = \frac{1}{(3 D_{bed})}$ .

Equations (6)–(8) constitute the diffusion equations for a general two-speed photon migration in powder bed or dense particulate suspension. They differ from a typical one-speed diffusion equation in that the scattering and absorption coefficients are dependent upon the refractive indices of the two phase system. In addition, the two speed equation suggests that the absorption coefficient is also dependent upon the volume fraction of particles which act as scatters.

2.2 Dynamic simulation of sedimentation process

In order to simulate particle positions (or powder structure), we determine the final state through dynamic simulation as described by Yang et al. [9] The dynamic simulation numerically first solves the Newtonian equations of motion for up to 50,000 interacting particles, which fall and collide with each other under gravity, friction, and interaction forces, and then records the evolution of spatial particle positions and velocities with time.

The forces acting on a particle govern both translational and rotational motions within the ensemble. The Newtonian equations describing the evolution of particle positions and rotational and translational velocities within the ensemble at time t are:

{\vec{F}}_{i} = m_{i} \vec{g} + {\vec{F}}_{i, vdW} + {\vec{F}}_{i, normal} + {\vec{F}}_{i, friction} + {\vec{F}}_{i, drag} = m_{i} \frac{{d \vec{v}}_{i}}{dt}

{\vec{T}}_{i} = \sum_{i = 1}^{N_{i, contact}} {\vec{T}}_{ij, friction} + {\vec{T}}_{ij, rolling} = I_{i} \frac{{d \vec{ω}}_{i}}{dt}

where $\vec{v}$ _i is the velocity of mass center of particle i and $\vec{ω}$ _i is its rotational velocity. The total force on particle i, $\vec{F}$ _i , arises from (i) gravity, m_i $\vec{g}$ ; (ii) van der Waals force, $\vec{F}$ _i,vdW ;(iii) the normal and frictional forces due to particle contacting, $\vec{F}$ _i,normal and $\vec{F}$ _i,friction ; and (iv) drag forces, $\vec{F}$ _i,drag . $\vec{T}$ _i is the total torque, which arises from $\vec{T}$ _ij , friction and $\vec{T}$ _ij,rolling; I_i is the inertial moment; and N_i,contact is the number of neighboring particles that contact particle i. Each force in Eqs. (9) and (10) and associated parameter values used in the simulation are described in Appendix I.

At the beginning of the simulation, the vectors of particle positions are set so that particles are separated and do not contact each other. The algorithm then progresses in time in steps, h, of 2×10^-8 seconds and computes particle position and translational and rotational velocities at each time step:[10]

{\vec{r}}_{i} (t + h) = {\vec{r}}_{i} (t) + h \times {\vec{v}}_{i} (t) + \frac{h^{2}}{2} \frac{{\vec{F}}_{i} (t)}{m_{i}}

{\vec{v}}_{i} (t + h) = {\vec{v}}_{i} (t) + \frac{h}{m_{i}} \times [\frac{3}{2} {\vec{F}}_{i} (t) - \frac{1}{2} {\vec{F}}_{i} (t - h)]

{\vec{ω}}_{i} (t + h) = {\vec{ω}}_{i} (t) + \frac{h}{I_{i}} \times [\frac{3}{2} {\vec{T}}_{i} (t) - \frac{1}{2} {\vec{T}}_{i} (t - h)]

In the dynamic simulation, periodic boundary conditions along horizontal x and y directions are used to approximate semi-infinite settling of a bed of particles. The computation required 5×10⁶ seconds for the simulation of 50,000 particles on a Pentium 4, 2.8 GHz machine with a Linux operating system. The calculation flowchart is given in Appendix II.

In order to investigate the impact of volume fraction upon bed D_bed and µ_a,bed , the magnitude of forces acting on the particles can be adjusted in order to generate various particle ensembles. However, it is difficult to experimentally manipulate properties which govern these forces. Consequently, in order to investigate the influence of volume fraction, we simulated fluid particulate suspension by uniformly expanding the final simulated resin and lactose powder beds at f_v =0.64 and conducting Monte Carlo simulation of photon trajectories (as described below) on particulate arrangements at lower volume fractions. This enables comparison of Monte Carlo simulation results (described below) to experimental measurements in dense resin and lactose suspensions.

2.3 Monte Carlo simulation of photon migration in particulate suspensions

The powder structure generated from dynamic simulation was employed as the scattering medium in which the statistics of time-dependent photon propagation was simulated with Monte Carlo techniques. An infinite medium of particles with diameter, d_p , was simulated based upon the generated powder structure within a 10d_p ×10d_p ×32d_p cubic with periodic boundary conditions along x, y, and z directions. Photon trajectories were computed using Monte Carlo technique.

In contrast to the use of Mie scattering for characterizing dense colloidal suspensions, where d_p≤λ (wavelength), [11,12] geometric optics can be employed for simulating trajectories of photon propagation when particle diameters are larger than the wavelength of near-infrared light employed [13]. Geometric optics dictates the propagation direction and scattering length in the simulated photon trajectories.

The simulation is conducted as follows: a photon package, with unity weight, is launched at a point of illumination in the suspending medium. When the photon encounters a particle, it is either reflected or refracted, depending upon (i) a randomly generated number, (ii) the relative refractive index of the particle, and (iii) the probability for reflection and refraction, which is calculated according to Fresnel’s formula [14].

If the photon package is reflected, it leaves the particle surface in the reflection direction and travels until it is re-directed upon encountering the next particle. The photon weight is reduced due to the absorption coefficient of the non-particle phase.

If the photon package is refracted, it transits inside the particle with an updated propagation direction and the photon weight is reduced due to the non-zero particle absorbance. Once the photon package arrives at the next particle interface, the process is repeated and the package is either reflected or refracted as described before.

The photon weight, w, after each i^th scattering step is attenuated by

{(Δ \ln w)}_{i} = - μ_{a, j} \cdot \frac{c}{n_{j}} \cdot t_{i}

where the subscript j denotes the medium in which the step has occurred (i.e., either within the suspending, med, or particle, p, phases) and t_i , represents the “time-of-flight” associated with the photon scattering step length.

The photon position, propagating direction, photon weight, and residing phase (i.e., whether in suspending medium or particle) are recorded with propagation time. At each recording time, a photon contributes to the photon density in suspending medium, ϕ_med , if it is within the medium, and contributes to the photon density within particles, ϕ_p , if it is located within the particle.

Photon trajectories are counted for propagation times less than 2.3×10^-8 seconds. We simulate 10⁶ photon trajectories with a computational time of 4×10⁵ seconds on a Pentium 4, 2.8 GHz machine with a Linux operating system. The details of the Monte Carlo simulation of geometric optics are provided in Appendix III.

2.4 Determination of scattering and absorption coefficients from Monte Carlo simulation

From the square of the mean displacement at time t, <r(t)>, derived from Monte Carlo statistics, the effective diffusion coefficient,D_bed , in powder bed can be determined [15]

{< r (t) >}^{2} = \frac{16 \cdot c \cdot D_{bed}}{π \cdot n_{med}} \cdot t

whereby D_bed is a measurable quantity from time-dependent measurements, as described by Eq. (7).

The effective absorption coefficient, µ_a,bed , in Eq. (8) for powder bed or dense particulate suspension is a function of the photon equilibrium constant, K_e , which can be related to the ratio of mean of the total time of collected photons spent within the particle phase, <t_total,p >, to that within the suspending medium, <t_total,med >:

K_{e} = \frac{ϕ_{p}}{ϕ_{med}} = \frac{< t_{total, p} >}{< t_{total, med} >} .

Furthermore, <t_total,p > and <t_total,med > can be computed from the mean scattering length within particle phase, scat, <l_scat,p >, and the mean scattering length within the suspension medium, <l_scat,med >,

< t_{total, j} > = \frac{< l_{scat, j} > \cdot N_{scat, j}}{(c ⁄ n_{j})} (j = p, med)

where N_{scat, j} is the number of total scattering events within suspending medium or particle phase.

In terms of Fresnel’s formula, [14] the reflection probability for a photon, which propagates in the particle phase and interacts with the interface, is equal to the reflection probability for a photon, which propagates in the suspending2 phase and interacts with the interface. Consequently, N_scat,med=N_scat,p (see Appendix IV for proof). Therefore, the photon equilibrium constant and associated absorption coefficient can be expressed as:

K_{e} = \frac{< t_{total, p} >}{< t_{total, med} >} = \frac{n_{p} \cdot < l_{scat, p} >}{n_{med} \cdot < l_{scat, med} >}

μ_{a, bed} = \frac{n_{med} (μ_{a, med} < l_{scat, med} > + μ_{a, p} < l_{scat, p} >)}{(n_{med} < l_{scat, med} > + n_{p} < l_{scat, p} >)} .

Again, it is noteworthy that the absorption coefficient of the bed is influenced by the particle configuration as indicated by its dependence upon <l_scat,med >, which is a function of volume fraction, f_v , and particles diameter, d_p . In this contribution, we us Monte Carlo simulation to investigate the dependence of <l_scat,med > upon f_v , which varies from 0.05 to 0.64. For a system of monodisperse particles, we provide a mathematical expression for <l_scat,med > as a function of f_v and d_p .

2.5 Quantitative experimental time-dependent measurements

To experimentally confirm the dynamic simulation, Monte Carlo, and two-speed diffusion equation analysis, time-dependent measurements were made using frequency domain system established in the Photon Migration Laboratories. In the following, the measured samples are described, the experimental approaches are briefly presented, and the data analysis is provided.

2.5.1 Samples

Measurements were conducted on resin and lactose powder beds, and in order to investigate the influence of varying volume fraction of particles, suspensions were interrogated. A uniform spatial distribution of particles was achieved by suspending the resin and lactose particles in water and liquid ethanol with use of an external agitator (H3770, Sigma-Aldrich, St. Louis, MO). The volume fraction, f_v , of lactose-ethanol suspensions was varied from 0.17 to 0.48, and f_v of resin-water suspension was varied from 0.20 to 0.64 by adding desired amount of resin and lactose particles into water and ethanol.

Monodisperse particles of AG 1-X8 Resin (Bio-Rad, Hercules, California), shown in the micrographs in Fig. 2(a), were employed. Resin particles have a regular spherical shape and narrow particle size distribution with a mean particle diameter of 77 µm and a standard deviation of 8 µm as determined from image analysis (Image Pro 3.0; Cybernetics, Silver Spring, MD). The polydisperse pharmaceutical lactose particles (Le-Pro^™ Lac 455; Leprino Food, Denver, CO), shown in Fig. 2(b), have irregular shapes and a broad particle size distribution with a mean value of around 50 µm and a standard deviation of 30 µm.

The refractive index for resin particles, which are made up of polystyrene divinylbenzene, is 1.59 at 650 nm [16]. The refractive index of lactose particles in crystal form is 1.58 at 650 nm [17]. The refractive indices for purified water and liquid ethanol (A406^P-4; Fisher Scientific, Fair Lawn, NJ) are both 1.33 at 650 nm [18].

The volume fraction, f_v , of the powder bed was determined by the ratio of the mass density of powder bed, ρ_bed , to the particle mass density, ρ_p . In a densely packed state, ρ_bed of resin bed was determined to be 0.65 gram/cm³ by measuring the weight of powder bed and the associated bed volume, while ρ_p of resin particle was determined to be 1.02 gram/cm³ by measuring the weight of particle bed and the associated net volume occupied by particles, which was measured by settling the resin particles in water and reading the volume increment. We found f_v for the monodisperse resin powder bed in air to be 0.64. For polydisperse lactose powders, the same measurements gave ρ_p =1.46gram/cm³, 0.95 ρ_bed =gram/cm³, and f_v =0.65.

Fig. 2. (a) Micrograph of resin particles in packed powder bed. The resin particles have regular spherical shape with a mean diameter of 77µm. In a densely packed powder bed containing such resin particles, the volume fraction, fv, is 0.64; (b) Micrograph of multi-disperse lactose particles with a mean diameter of around 50µm.

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2.5.2 Frequency domain photon migration measurements

FDPM measurements were made using the system detailed elsewhere [1]. Briefly, modulated light from a 650 nm, 35 mW laser diode was split into two beams. One beam was directed into a 1-mm-diameter optical fibers (FT-035 mm; Thorlabs, Newton, NJ) and was delivered to a reference photomultiplier tube (PMT) (H6357, Hamamatsu, Japan). The second light beam was introduced into a second optical fiber whose end was embedded within the powder bed. The light propagated through the scattering medium and was collected with a 1 mm optical fiber located (i) 5–8 mm away from the point of illumination in increments of 0.5 mm for the resin powder bed; (ii) 10–18 mm away from the point of illumination in increments of 2 mm in the lactose powder bed; and (iii) 15–22 mm away from the point of illumination in increments of 0.5 mm for lactose-water and resin-ethanol suspensions. The two PMTs were modulated at the same modulation frequencies as the laser diode (40–125 MHz), but with an additional offset frequency of 100 Hz. The resulting mixed signals from the PMTs were converted into voltage; the high frequency components were filtered by transimpedance amplifiers (70710; Oriel Co., Stratford, CT); and the heterodyned signals were acquired by LabView data acquisition software (National Instruments, Austin, TX). The phase shift (PS), amplitude (AC), and mean intensity (DC) of the signal from the sample PMT relative to the reference PMT were recovered for data analysis.

2.5.3 Analysis of phase shift, amplitude, and mean-intensity of photon density wave

FDPM measurements of phase shift (PS), amplitude (AC), and mean intensity (DC) for (i) the resin and lactose powder beds with a volume of 150 cm³ and (ii) the resin-water and lactose-ethanol suspensions with volumes of more than 400 ml were evaluated from the solution of the standard diffusion equation to provide the absorption coefficient, µ_a,bed and scattering coefficient, µ′_s,bed [19]. For simplicity, we denote the optical properties of the powder bed as well as the dense suspension in terms of µ_a,bed and µ′_s,bed .

From the experimental quantities of PS, AC, and DC, we compute a parameter, Y:

Y (PS, AC, DC, r) = {{[{(PS - {PS}_{0})}^{2} + \ln^{2} (\frac{r \cdot AC}{r_{0} \cdot {AC}_{0}})]}^{2} - \ln^{4} (\frac{r \cdot DC}{r_{0} \cdot {DC}_{0}})}^{1 ⁄ 4}

from which the bed scattering coefficient can be determined by

Y (PS, AC, DC, r) = \sqrt{3 {μ'}_{s, bed} \cdot ω \cdot n_{med} ⁄ c} \times (r - r_{0}) .

Here ω is the modulation frequency, r is the source-detector separation, and the subscript, 0, denotes the values at the smallest source-detector separation of (i) 5 mm in resin powder bed and (ii) 10 mm in lactose powder bed. Equation (21) was employed to extract the FDPM measured scattering coefficients, µ′_s,bed , from the measurements of Y(PS,AC,DC,r) when ω was varied (i) from 65 to 125 MHz in the resin powder bed and (ii) from 40 to 120 MHz in lactose powder bed.

3. Results and discussion

3.1 Powder structure from dynamic simulation of sedimentation processes

In order to illustrate the evolution of powder structure with time, the sedimentation of a powder bed, which contained 2,000 particles of 50 p_d =50µm in a (5×10^-4 m).(5×10^-4 m).(2×10^-3 m) cube with periodic boundary conditions along horizontal x and y directions, is shown in Fig. 3.

Fig. 3. (Movie) Powder structure evolution computed from dynamic simulation for a sedimentation process with 2,000 particles of d_p =50µm confined by a plate (consisting of orderly arranged particles and positioned at the top of simulated particles) in a (5×10^-4 m).(5×10^-4 m).(2×10^-3 m) cube with periodic boundary conditions along horizontal directions. [Media 1]

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The powder bed becomes more dense with the sedimentation time. At sedimentation times of 0, 0.012, 0.015, 0.024, and 0.032 seconds, the volume fractions of the powder bed, f_v , are 0.21, 0.28, 0.32, 0.53, and 0.59, respectively. It is assumed that a mechanically balanced state is achieved when the volume fraction no longer changes with sedimentation time.

When the number of simulated lactose particles was increased to 4,000 and 50,000, the corresponding volume fraction, f_v , grew to 0.61 and 0.64, respectively, due to the increase of vertical pressure caused by bed weight.

3.2 Photon Equilibrium

Figure 4(a) is a plot of the time-dependent photon densities, ϕ_med and ϕ_p , statistically computed from Monte Carlo trajectories within a simulated powder bed (f_v =0.61, 50d_p =50µm, n_med =1.0, and n_p =1.6), which was generated based on the dynamic simulation of 4,000 particles. The squares, crosses, and triangles represent ϕ_p and ϕ_med at positions of 75d_p , 165d_p , and 255d_p from the light source. While ϕ_med and ϕ_p both change with propagation time at all positions, they are related by a factor,K_e , as hypothesized in Eq. (3), found in this case to be 2.44. To illustrate photon equilibrium, Fig. 4(b) shows ϕ_p and (K_e ×ϕ_med ) at a series of source-detector separations for migration times varying from 6×10^-12 to 1×10^-8 second. Figure 4(b) indicates that at all times, there is an equilibrium between the photon density within particles and photon density in medium phase, regardless of source-detector separation in the simulated powder bed. We found that the equilibrium constant, K_e , varies with f_v between values of 0.05 and 0.64 (see Appendix V). These results confirm the photon equilibrium relationship between ϕ_med and ϕ_p , used to develop the two-speed diffusion approximation of time-dependent photon migration for characterization of powder beds and dense particulate suspensions.

Fig. 4. (a) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕ_p (blue symbols) and ϕ_med (red symbols) as a function of time and source-detector separation, r, of 75d_p , 165d_p , and 255d_p in a powder bed with volume fraction of 0.61, n_p =1.6, and d_p =50µm ; (b) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕ_p (blue symbols) and (2.44×ϕ_med ) (red symbols) as a function of time and r of 45d_p , 75d_p , 105d_p , 135d_p , 165d_p , 195d_p , 225d_p , 255d_p , and 285d_p in a powder bed with volume fraction of 0.61, n_p =1.6, and d_p =50µm.

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The value of K_e at each volume fraction can be computed by Eq. (18) based on the statistical quantities of <l_scat,med > and <l_{scat, p} >. Figure 5(a) illustrates the values of <l_scat,med >/d_p computed from Monte Carlo as a function of f_v , which varies from 0.05 to 0.64, the relative refractive index, n_p/n_med , which varies from 1.15 to 1.8, and particle diameter, d_p , which varies from 20 to 50µm. As expected, the relative refractive index of the particle has little impact on <l_scat,med >. The reduction of <l_scat,med >/d_p is consistent with the decrease of the void space between particles when the volume fraction is increased. Quantitatively, in the ranges of f_v measured for the resin-water and lactose-ethanol suspensions, the dependence of <l_scat,med >upon fv in Fig. 5(a) can be described empirically by a power form fit:

\frac{< l_{scat, med} >}{d_{p}} = \frac{0.395}{f_{v}^{1.26}} (0.15 \leq f_{v} \leq 0.64) .

Figure 5(b) shows that <l_scat,p >/d_p increases with the increase of n_p/n_med , which varies from 1.1 to 1.8 and that the volume fraction, which varies from 0.1 to 0.64, has little impact upon <l_scat,p >d_p . Therefore, <l_scat,p > becomes a constant when both d_p and n_p/n_med are given. From Fig. 5, the mean scattering lengths were determined to be, <l_scat,p >=0.895d_p and <l_scat,med >=0.575d_p for the powder bed of f_v =0.61 and n_p/n_med =1.6. For this powder bed, Eq. (18) predicts 2.49 Ke=2.49, which agrees well with K_e =2.44 obtained from the ratio of ϕ_p to ϕ_med computed from Monte Carlo.

Fig. 5. (a) Ratio of mean scattering length in suspending medium to particle diameter versus volume fraction varying from 0.05 to 0.64 at various relative refractive indices, n_med/n_p , and particle diameters, d_p , varying from 20 to 50µm with points representing Monte Carlo simulation results and the curve denoting a fitting relationship; (b) Ratio of mean scattering length within particles to particle diameter versus relative refractive index, n_med/n_p , with triangles (f_v =0.61, d_p =50µm), squares (f_v =0.30, d_p =40µm) and diamonds (f_v =0.10, d_p =30µm) representing Monte Carlo simulation results and the curve denoting a fitting relationship.

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3.3 Measurements in powder beds for verifications of two-speed diffusion equation and simulated scattering coefficient

FDPM measurements of phase shift (PS), amplitude (AC), and mean intensity (DC) were made in the resin and lactose powder beds. Figure 6(a) shows the FDPM measurements for Y(PS,AC,DC,r) (i.e., Eq. (20)) versus the source-detector separation, r, at 65 MHz (diamonds), 95 MHz (triangles), and 125 MHz (squares) for the monodisperse resin powder bed with d_p =77µm and 0.64 f_v =0.64. Fig. 6(b) shows Y(PS,AC,DC,r) versus r at 40 MHz (diamonds), 80 MHz (triangles), and 120 MHz (squares) for the polydisperse lactose powder bed with a mean diameter of 50 µm and f_v =0.65. In both cases, Y(PS,AC,DC,r) changes linearly with r at each modulation frequency, as predicted by Eq. (21) derived from the diffusion equation. From Eq. (21), we found the values of FDPM measured scattering coefficients to be µ′_s,bed =47.8 cm^-1 and µ′_s,bed =243cm^-1 for resin and lactose powder beds, respectively. The Monte Carlo predicted scattering coefficient, µ′_s,bed , for a simulated monodisperse resin powder bed with particle diameter of 77µm, refractive index of 1.6, and volume fraction of 0.64 was found to be µ′_s,bed =48.9 cm^-1. In the monodisperse resin powder bed, the scattering coefficient predicted by Monte Carlo simulation is close to the actual FDPM measured scattering coefficient and correspondingly, the predictions of Y(PS,AC,DC,r) with use of simulated µ′_s,bed in Eq. (21) are also close to the measured values, as shown in Fig. 6(a).

Fig. 6. (a) Y(PS,AC,DC,r) versus the source-detector separation at 65 MHz (diamonds), 95 MHz (triangles), and 125 MHz (squares) in frequency domain photon migration measurements for resin powder bed of d_p =77µm and f_v =0.64 with the solid lines computed from two-speed diffusion equation using scattering coefficient predicted from Monte Carlo simulation. (b) Y(PS,AC,DC,r) versus the source-detector separation at 40 MHz (diamonds), 80 MHz (triangles), and 120 MHz (squares) in frequency domain photon migration measurements for polydisperse lactose powder bed with f_v =0.65 and mean diameter of around of 50 µm with the solid lines denoting fitting curves of the two-speed diffusion equation.

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For a simulated monodisperse lactose powder bed of d_p =50µm and f_v =0.64, the scattering coefficient predicted by Monte Carlo simulation was found to be µ′_s,bed =75 cm^-1. Contrary to the results for the resin powder bed, the scattering coefficient predicted by Monte Carlo simulation for the monodisperse lactose powder bed of d_p =50µm differs significantly from the actual FDPM measured scattering coefficient for the polydisperse powder bed with a mean diameter of d̄_p.poly =50µm. The prediction of scattering coefficient in the polydisperse powder bed may require more considerations of the particle size distribution and particle shapes.

3.4 Particulate suspension experiments for confirmation of the dependence of absorption coefficient upon volume fraction

Figure 7 shows the experimentally derived absorption coefficient of the suspensions, µ_a,bed , versus volume fraction, f_v . In the resin-water particulate suspension, µ_a,bed decreased from 0.0193 cm^-1 to 0.0098 cm^-1 when f_v was increased from 0.20 to 0.64. In the lactose-ethanol particulate suspension, µ_a,bed decreased from 0.0143 cm^-1 to 0.0082 cm^-1 when f_v was increased from 0.17 to 0.48.

The dependence of µ_a,bed upon f_v can be evaluated explicitly based on the combination of Eq. (19) and the empirical relationship between <l_scat,med > and f_v (Eq. (22)):

μ_{a, bed} = \frac{μ_{a, med} + a_{1} \cdot f_{v}^{1.26}}{1 + a_{2} \cdot f_{v}^{1.26}} (0.15 \leq f_{v} \leq 0.64)

where a ₁ and a ₂ are constants defined as $a_{1} = \frac{μ_{a, p} < l_{scat, p} >}{0.395 d_{p}}$ and $a_{1} = \frac{< l_{scat, p} >}{0.395 d_{p}} . \frac{n_{p}}{n_{med}}$ .

Fig. 7. Absorption coefficients of particulate suspensions, µ_a,bed , versus volume fraction, f_v , for lactose-ethanol (triangles) and resin-water (squares) suspensions with points representing the FDPM measurement results and the lines denoting the fitting curves of two-speed diffusion equation.

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Equation (23) was employed to fit the FDPM measured absorption coefficients in lactose-ethanol and resin-water suspensions with adjustable parameters, µ_a,med ,a ₁ and a ₂. The values of µ_a,med at 650 nm were thus found to be 0.028 and 0.022 cm^-1 for suspending water and ethanol, respectively, and the fitting curves (solid lines) are also shown in Fig. 7. The determined µ_a,med for suspending water is higher than the corresponding absorption coefficient for pure water reported in literature[20] possibly because of the release of chloride ions from the resin particles into the suspending water, which became slightly dark in color compared with the pure water.

Finally, Eq. (19) indicates that the particle diameter, d_p , has little impact on the absorption coefficient of powder bed, µ_a,bed , since both <l_scat,med > and <l_{scat, p} > are proportional to d_p (as shown in Fig. 5.), which is then cancelled as a common factor in the denominator and the numerator of Eq. (19). The conclusion of the independence of µ_a,bed upon d_p is consistent with our past experimental results.^[4]

4. Conclusions

In the powder beds and dense particulate suspensions, where (i) particle size is much larger than operating near-infrared wavelength; (ii) refractive index of particle is significantly different from that of the suspending medium (i.e. air or liquid); and (iii) the volume fraction of the particles is of the same order of that of the suspending medium, we find that, (i) there is a local photon equilibrium between photon density in the suspending medium and the photon density within particles; (ii) the photon equilibrium validates the two-speed diffusion equation; and (iii) the absorption coefficient is a function of the volume fraction, but independent of particle diameter. These results suggest that for light propagation in disparate media, the diffusion approximation still applies, but that the transport coefficients can be predicted from accurate representation of a two-speed model of time-dependent light propagation.

5. Appendix

5.1 Calculation of forces on particles

(i) van der Waals force, $\vec{F}$ _i,vdW [21]

{\vec{F}}_{i, vdW} = \sum_{j = 1, j \neq i}^{N_{m}} {\vec{F}}_{ij, vdW}

{\vec{F}}_{ij, vdW} = - \frac{H_{vdW}}{6} \cdot \frac{64 R_{i}^{3} R_{j}^{3} (h_{vdW} + R_{i} + R_{j}) {\vec{n}}_{ij}}{{(h_{vdW}^{2} + 2 R_{i} h_{vdW} + 2 R_{j} h_{vdW})}^{2} {(h_{vdW}^{2} + 2 R_{i} h_{vdW} + 2 R_{j} h_{vdW} + 4 R_{i} R_{j})}^{2}}

h_{vdW} = max [(∣ {\vec{R}}_{i} - {\vec{R}}_{j} ∣ - R_{i} - R_{j}), h_{vdW}^{0}]

{\vec{F}}_{i, normal} = \sum_{j = 1}^{N_{i, contact}} {\vec{F}}_{ij, normal}

{\vec{F}}_{ij, normal} = [\frac{2}{3} E \sqrt{\overset{̅}{R}} η^{3 ⁄ 2} - γ_{n} E \sqrt{\overset{̅}{R} η} ({\vec{v}}_{ij} \cdot {\vec{n}}_{ij})] {\vec{n}}_{ij}

{\vec{F}}_{i, friction} = - γ_{s} \cdot \sum_{j = 1}^{N_{i, contact}} ∣ {\vec{F}}_{ij, normal} ∣ {\vec{t}}_{ij}

{\vec{F}}_{i, dragging} = - 6 π μ R_{i} {\vec{v}}_{i}

{\vec{T}}_{ij, friction} = (- R_{i} {\vec{n}}_{ij}) \times {\vec{F}}_{ij, friction}

{\vec{T}}_{ij, rolling} = - γ_{r} R_{i} ∣ {\vec{F}}_{ij, normal} ∣ {\vec{ω}}_{i} ⁄ ∣ {\vec{ω}}_{i} ∣

The parameters are listed in the following table.

View Table | View all tables in this article

5.2 Calculation flow-chart for dynamic simulation

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* The mechanical properties are included in the table in Appendix 1.

** See reference [29]

5.3 Algorithm for Monte Carlo of geometric optics

The number of photon trajectories is set to be 10⁶ and the migration time, t_end , for a photon in Monte Carlo is 2.3×10^-8 second. The photon trajectories are simulated one by one. A photon package includes time, t, propagating direction, $\vec{v}$ , position vector, $\vec{r}$ , photon weight, w, and refractive index for the surrounding phase, n_med or n_p .

When the simulation of a photon begins, the photon is launched into the powder bed at t=0. The initial propagating direction, $\vec{v}$ ₀ , at the origin, $\vec{r}$ ₀, is randomly chosen from the isotropic distribution and the photon weight, w, is set to be 1. The photon package at the origin is [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[0, $\vec{v}$ 0, $\vec{r}$ 0,1,n_med ].

The particle, which the photon encounters, is identified from the photon moving direction, $\vec{v}$ ₀, and the particle positions, which are obtained from the dynamic simulation of sedimentation process for powder bed. The time for a photon propagating from the origin, $\vec{r}$ ₀, to the encountering point, $\vec{r}$ _ext , on the external surface of the interacting particle with a speed,c/n_med , is Δt. The attenuation of photon weight, w, is exp(-µ_a,med ·| $\vec{r}$ _ext- $\vec{r}$ ₀|). Correspondingly, the photon package at $\vec{r}$ _ext , is updated [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[(t+Δt), $\vec{v}$ $\vec{r}$ _ext ,(w.exp(-µ_a,med ·| $\vec{r}$ _ext- $\vec{r}$ ₀|)),n].

At the encountering point, $\vec{r}$ _ext , the incident angle, α_i , between the photon propagating direction, $\vec{v}$ ₀, and the normal, $\vec{L}$ , of the external surface is employed to calculated the reflection probability in terms of Fresnel’s formulas[11]

P (α_{i}) = \frac{1}{2} [\frac{\sin^{2} (α_{i} - α_{t})}{\sin^{2} (α_{i} + a_{t})} + \frac{\tan^{2} (α_{i} - α_{t})}{\tan^{2} (α_{i} + a_{t})}]

where α_t is the transmission angle determined by the Snel’s law^[10]

n_{med} \sin α_{i} = n_{p} \sin α_{t} .

A random number, ξ, from the uniform distribution in the range of [0,1] is generated. The photon undergoes an external reflection if ξ<P. Otherwise, it undergoes a refraction from outside to inside of the particle.

If the photon undergoes an external reflection, the photon propagating direction is updated to $\vec{v}$ in terms of Snel’s law and the photon package becomes [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[t, $\vec{v}$ ₁, $\vec{r}$ ,w,n].

In terms of $\vec{v}$ ₁ and the particle positions, the next interacting particle is determined and similar updating for propagating outside particles is conducted.

If the photon undergoes a refraction from outside to inside of the particle, the photon propagating direction is updated to $\vec{v}$ ₂ in terms of the Snel’s law. The refractive index in the photon package is changed from n_med to n_p since the photon transmits inside the particle. The photon package is updated [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[t, $\vec{v}$ ₂, $\vec{r}$ ,w,np].

In terms of the propagating direction, $\vec{v}$ ₂, the position of the interacting particle, and the particle radius, the ending point for the photon’s internal travel is determined as $\vec{r}$ _int . The propagating time, Δ_t, is calculated from the distance between $\vec{r}$ _ext and int $\vec{r}$ _int and the particle refractive index, n_p . The attenuation of the photon weight is ext (-µ_a.p ·| $\vec{r}$ _int- $\vec{r}$ _ext|). At position, $\vec{r}$ _int, the photon package is updated [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[(t+Δt), $\vec{v}$ $\vec{r}$ _int, (w.exp(-µ_a.p ·| $\vec{r}$ _int- $\vec{r}$ _ext|)), n].

At the interacting point, $\vec{r}$ _int, the incident angle, α_i , between $\vec{v}$ ₂ and the normal of the particle surface, $\vec{L}$ , is calculated. The reflection probability, P, is calculated in terms of Fresnel’s formulas. Then a random number, ξ, is generated from the uniform distribution in the range of [0,1].

The photon undergoes an internal reflection if ξ<P. Otherwise, it undergoes a refraction from inside to outside of the particle.

When the photon undergoes an internal reflection, the propagating direction becomes $\vec{v}$ ₃ in terms of Snel’s law and the photon package is updated 3 [t, $\vec{v}$ , $\vec{r}$ ,w,n]⃖[t, $\vec{v}$ ₃, $\vec{r}$ ,w,n].

Similar updating for propagation within the particle is conducted.

When ξ≥P, the photon undergoes an refraction from inside to outside of the particle. The propagating direction is changed to $\vec{v}$ ₄ in terms of the Snel’s law and the associated refractive index is changed from n_p to n_med . Then the photon package is updated [t, $\vec{v}$ $\vec{r}$ ,w,n]⃖[t, $\vec{v}$ ₄, $\vec{r}$ ,w,n_p ].

Similar updating for photon propagating outside particles is conducted.

In the simulation, when the photon package is updated, t and t_end is compared. When t<t_end , the simulation for the photon continues. Otherwise, the simulation for the photon terminates and another photon is launched into the powder bed. When the total simulated photon trajectories reaches 10⁶, the simulation is finished.

The simulation flowchart is shown below.

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where, $\vec{v}$ is the photon propagating direction; $\vec{v}$ _reflection is the propagating direction after reflection at an interface; and $\vec{v}$ _transmission is the propagating direction after refraction at an interface.

5.4 Monte Carlo simulation of scattering numbers

The verification of <N_scat,p >=<N_scat,med > has been conducted under various conditions of f_v . The Monte Carlo simulation results are shown in the table.

View Table | View all tables in this article

5.5 Photon equilibrium for powder beds with volume fraction of 0.45 and 0.38

Fig. a1. 1. (a) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕ_p (blue symbols) and (1.46×ϕ_med ) (red symbols) as a function of time and source-detector separation, r, in a powder bed with volume fraction of 0.45 and d_p =45µm. (b) The time-dependent photon densities computed from time-domain Monte Carlo simulation for ϕ_p (blue symbols) and (1.13×ϕ_med ) (red symbols) as a function of time and source-detector separation, r, in a powder bed with volume fraction of 0.38 and d_p =42µm.

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References and links

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5. S. E. Torrance, Z. Sun, and E. M. Sevick-Muraca, “Impact of excipient particle size on measurement of active pharmaceutical ingredient absorbance in mixtures using frequency domain photon migration,” J. Pharm. Sci. 93, 1879–1889 (2004). [CrossRef] [PubMed]

6. V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing medium and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998). [CrossRef]

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17. A. N. Winchell, “The Optical Properties of Organic Compounds,” (Academic Press, 1954), Chapter 4.

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20. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. 32, 3513–3540 (1993). [CrossRef]

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E	Y/(1-σ̃ ²)			$\vec{t}$ _ij	Unit vector on the tangential plane perpendicular to the vector connecting particle i and particle j
H_vdW	Hamaker constant			$\vec{v}$ _ij	Relative velocity between the center of particle i and the center of particle j
$h_{vdW}^{0}$	Minimal calculation separation in			Y	Young modulus
N_i,contact	Number of particles in contact with particle i			γ_n	Normal dissipative coefficient in collision
Nm	Number of particles in the simulation system			γ_s	Friction coefficient
$\vec{n}$ _ij	Unit vector pointing from particle j to particle i			γ_r	Friction coefficient with respect to rolling
R̄	\|R_iR_j /(R_i+R_j )			η	R_i+R_j -\| $\vec{R}$ _i- $\vec{R}$ _j\|
R_i	Radius of particle i			µ	Viscosity of the suspending medium
$\vec{R}$ _i	Vector pointing from coordinate origin to the mass center i			$\tilde{σ}$	Poisson’s ratio
The parameter values in simulation are as follows
Quantity	Y ^[26]	$\tilde{σ}$ ^[27]	γ_n ^[9]		γ_s [9]	γ_r ^[9]	H_vdW ^[28]	$h_{vdW}^{0}$ ^[29]
Unit	Pa	none	none		none	none	J	m
Value	8×2.10⁹	0.29	3×10^-7		0.3	0.002	6×4.10^-20	10^-9

n_p	n_med	d_p(µm)	f_v	t_end(ns)	<N_scat,med >	<N_scat,p >	$\frac{< N_{scat, med} >}{< N_{scat, p} >}$
1.6	1.0	100	0.554	22.5	3371±2	3372±2	1.0000±0.0003
1.6	1.0	100	0.515	22.5	3214±2	3213±2	1.0000±0.0003
1.6	1.0	100	0.476	22.5	3012±2	3013±2	1.0000±0.0003

E	Y/(1-σ̃ ²)			$\vec{t}$ _ij	Unit vector on the tangential plane perpendicular to the vector connecting particle i and particle j
H_vdW	Hamaker constant			$\vec{v}$ _ij	Relative velocity between the center of particle i and the center of particle j
$h_{vdW}^{0}$	Minimal calculation separation in			Y	Young modulus
N_i,contact	Number of particles in contact with particle i			γ_n	Normal dissipative coefficient in collision
Nm	Number of particles in the simulation system			γ_s	Friction coefficient
$\vec{n}$ _ij	Unit vector pointing from particle j to particle i			γ_r	Friction coefficient with respect to rolling
R̄	\|R_iR_j /(R_i+R_j )			η	R_i+R_j -\| $\vec{R}$ _i- $\vec{R}$ _j\|
R_i	Radius of particle i			µ	Viscosity of the suspending medium
$\vec{R}$ _i	Vector pointing from coordinate origin to the mass center i			$\tilde{σ}$	Poisson’s ratio
The parameter values in simulation are as follows
Quantity	Y ^[26]	$\tilde{σ}$ ^[27]	γ_n ^[9]		γ_s [9]	γ_r ^[9]	H_vdW ^[28]	$h_{vdW}^{0}$ ^[29]
Unit	Pa	none	none		none	none	J	m
Value	8×2.10⁹	0.29	3×10^-7		0.3	0.002	6×4.10^-20	10^-9

n_p	n_med	d_p(µm)	f_v	t_end(ns)	<N_scat,med >	<N_scat,p >	$\frac{< N_{scat, med} >}{< N_{scat, p} >}$
1.6	1.0	100	0.554	22.5	3371±2	3372±2	1.0000±0.0003
1.6	1.0	100	0.515	22.5	3214±2	3213±2	1.0000±0.0003
1.6	1.0	100	0.476	22.5	3012±2	3013±2	1.0000±0.0003

Evaluation of photon migration using a two speed model for characterization of packed powder beds and dense particulate suspensions

Abstract

1. Introduction

2. Methods and approach

2.1 Two-speed photon migration diffusion equation

2.2 Dynamic simulation of sedimentation process

2.3 Monte Carlo simulation of photon migration in particulate suspensions

2.4 Determination of scattering and absorption coefficients from Monte Carlo simulation

2.5 Quantitative experimental time-dependent measurements

2.5.1 Samples

2.5.2 Frequency domain photon migration measurements

2.5.3 Analysis of phase shift, amplitude, and mean-intensity of photon density wave

3. Results and discussion

3.1 Powder structure from dynamic simulation of sedimentation processes

3.2 Photon Equilibrium

3.3 Measurements in powder beds for verifications of two-speed diffusion equation and simulated scattering coefficient

3.4 Particulate suspension experiments for confirmation of the dependence of absorption coefficient upon volume fraction

4. Conclusions

5. Appendix

5.1 Calculation of forces on particles

5.2 Calculation flow-chart for dynamic simulation

5.3 Algorithm for Monte Carlo of geometric optics

5.4 Monte Carlo simulation of scattering numbers

5.5 Photon equilibrium for powder beds with volume fraction of 0.45 and 0.38

References and links

Supplementary Material (1)

Cited By

Figures (10)

Tables (2)

Equations (34)

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