Simultaneous measurement of refractive index, diameter and colloid concentration of a droplet using rainbow refractometry

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Simultaneous measurement of refractive index, diameter and colloid concentration of a droplet using rainbow refractometry Can Li InvestigationValidationData curationVisualizationWriting - original draft reviewing , Yingchun Wu MethodologySoftwareSupervisionReviewing , Xuecheng Wu ConceptualizationSupervisionFunding acquisitionreviewing , Cameron Tropea Project administrationResourcesSupervisionWriting - review & editing PII: DOI: Reference:

S0022-4073(19)30721-6 https://doi.org/10.1016/j.jqsrt.2020.106834 JQSRT 106834

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Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

7 October 2019 6 December 2019 3 January 2020

Please cite this article as: Can Li InvestigationValidationData curationVisualizationWriting - original draft reviewing , Yingchun Wu MethodologySoftwareSupervisionReviewing , Xuecheng Wu ConceptualizationSupervisionFunding a Cameron Tropea Project administrationResourcesSupervisionWriting - review & editing , Simultaneous measurement of refractive index, diameter and colloid concentration of a droplet using rainbow refractometry, Journal of Quantitative Spectroscopy & Radiative Transfer (2020), doi: https://doi.org/10.1016/j.jqsrt.2020.106834

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Highlights:   

Simultaneous measurement of refractive index, diameter and colloid concentration Intensity ratio of p=2 and 0 scattering is used to measure colloid concentration Measured ratio attenuations are verified by a Monte Carlo ray tracing simulation

Simultaneous measurement of refractive index, diameter and colloid concentration of a droplet using rainbow refractometry Can Lia, b, Yingchun Wua, Xuecheng Wua, *, Cameron Tropeab a

State Key Lab of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China

b

Institute of Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt, 64287, Darmstadt, Germany

*Corresponding author ([email protected])

Abstract In the present study, droplets of colloidal suspensions or emulsions are characterized using rainbow refractometry according to diameter, relative refractive index and concentration of the dispersed phase. The position and angular spacing of the rainbow are used to retrieve the relative refractive index and droplet size. For the measurement of colloid concentration, a novel method using the intensity ratio of the p=2 (second-order refraction) and p=0 (reflection) scattering components present in the rainbow signal is introduced. The experimental system, comprising a monodisperse droplet generation system and standard rainbow refractometry, has been used for validation of the technique. Distilled water mixed with monodisperse polystyrene nanoparticles at different volume concentrations (CV=0−0.3%) were tested, providing ground truth values of concentration. The measured relative refractive index and droplet diameter agree well with results obtained using the extended effective medium approximation for turbid media and with the known experimental values. The measured highly sensitive relation found to exist between intensity ratio attenuation and colloid concentration is numerically verified using a Monte Carlo ray tracing method. Keywords: Colloidal suspension droplet, Concentration measurement, Rainbow refractometry, Refractive index, Droplet sizing 1. Introduction Droplets containing dispersed insoluble particles, also known as colloidal droplets, occur in a wide variety of industrial applications such as spray coating (paints), spray drying, inkjet printing, scrubbers or injection of fuels with particulate additives. In designing and optimizing such systems, the characterization of droplets according to not only size, velocity and relative refractive index (temperature), but also according to the colloid concentration is desirable. For instance, in spray drying, the colloid concentration of the primary droplets will determine the final powder size of the product. For droplet characterization, optical techniques are preferred owing to their non-intrusiveness, insitu operation and high precision. Existing techniques are based either on exact theories and/or numerical simulations describing the light scattered from droplets, albeit many of these are applicable only to homogeneous, spherical droplets [1]. When confronted with more complex colloidal droplets, no exact solutions exist and methods such as effective-medium approximations [2, 3], and Monte Carlo ray tracing [4, 5] are used. These can become computationally demanding, at least for droplet sizes larger than some tens of microns [6]. Alternatively, purely experimental studies of light scattering from such droplets are possible, for example conducted for a droplet containing one inclusion [7]. In [8], the host droplet size was measured from the angular fringe spacing and the colloid concentration from the slope of the intensity distribution in the forward scattering domain. Norbert et al. [9] combined speckle image analysis and

turbidity method to retrieve colloidal particle size, and employed the continuous wavelet transform to estimate the concentration of the monodisperse polystyrene particles within a droplet. Jakubczyk et al. [10] combined a weighting method and Mie scattering imaging to analyze the evaporation of levitated charged diethylene glycol droplets containing polystyrene nanoparticles. The time-shift technique [1, 11, 12], which separates the different scattering orders from a droplet in time, enables the measurement of colloid concentration [13, 14] from the amplitude ratio between reflection and refractive orders arising from the same illuminating beam. Despite this wide variety of techniques demonstrated, most yield information about droplet size, velocity and colloid concentration, but no technique simultaneously measures the relative refractive index. This is however desirable, since the relative refractive index can also be used as a measure for droplet temperature [15-18], concentration of liquid mixtures [19] or reactive mass transfer [20], which is essential in combustion systems [21] or emission control applications. This then motivates the present study based on rainbow refractometry in which the intensity ratio of the contributions to the rainbow of the two scattering orders p=2 (second-order refraction) and p=0 (reflection), i.e. Ip=2/Ip=0, is employed. Recently, rainbow refractometry has shown strong potential for such applications. The use of the p=2 scattering order allows the measurement of droplet size, relative refractive index, refractive index gradient [22-24], and non-sphericity [25-27]. Rainbow refractometry can also detect nanometer-size changes or the transient evaporation rate [15-18] of a droplet or monodisperse droplet stream from the phase shift of ripple fringes, which arise as the interference of the p=2 and p=0 rays. The present work differs from our previous study [28] in two respects: (a) the previous work focused on the effect of colloidal particles on the rainbow signal and on the parameters extracted from the signal, but the quantification of the colloid concentration failed; (b) the approach used in that study retrieves the colloid suspension concentration through the variation of the signal intensity of the primary rainbow peak, noting that the ripple structure has been strongly smoothed due to the polydisperse nature of the spray droplets. The present study proposes a different approach, namely the use of the intensity ratio Ip=2/Ip=0, to measure the colloid concentration. In the following sections the measurement principle of rainbow refractometry for relative refractive index and droplet diameter is briefly summarized, followed by a detailed introduction to the colloid concentration measurement from intensity attenuation of p=2 rays and extraction of the intensity ratio Ip=2/Ip=0 from a standard rainbow signal (section 2). Section 3 then describes the experimental set-up of a monodisperse droplet stream generator and the layout of the rainbow refractometry. Finally, results of the rainbow signal, measured diameter, relative refractive index and colloid concentration are presented with a comparison to analytic results given in Section 4. 2. Measurement principle In the following discussion, we consider a spherical droplet being illuminated by a plane wave of constant intensity I0 across the droplet. Nussenzveig [29] provides an excellent description of the resulting rainbows and their characteristics. Van Beeck and Riethmuller [30] summarize the use of the primary rainbow in optical devices for measuring droplet diameter and the real part of the relative refractive index. For the most part, the rainbow phenomena can be well understood in terms of geometrical optics [31] and in the present study we are interested in the primary rainbow, arising from the destructive/constructive interference of several parallel rays entering a droplet at different

intersection points (incident angles), but leaving the droplet with the same exit angle after one internal reflection. This contribution to the total light scattered is referred to as second-order refraction; and the minimum angle of deviation (deflection) is achieved and known as the geometrical rainbow angle, designated θrg. Such rays are depicted in Fig. 1(a), entering with the intensity I0 and exiting the droplet at the angle θrg with the intensity Ip=2. The interference of these rays lead to the well-known Airy fringes of the rainbow [32], their scattering angle being dependent on the relative refractive index and their frequency on the diameter of the droplet. Interference also occurs between reflected light (p=0) and the second-order refracted light. The reflected light is also shown in Fig. 1(a), leaving the droplet with intensity Ip=0. Thus, superimposed on the rainbow signal, are fringes of higher spatial frequency, arising from this second interference phenomenon. These fringes are known as ripples. A sample image of the primary rainbow is shown in Fig. 1(b), in which the main frequency and the superimposed ripples are evident. Also shown in Fig. 1(b) is the signal amplitude obtained by averaging the gray values of the image along the projected horizontal red line.

(a)

(b)

Fig. 1. (a) Light path of p=2 and p=0 scattered light exiting at the geometrical rainbow angle; (b) typical rainbow image and intensity distribution along a horizontal cut, illustrating the two main frequencies present in the signal.

The relative refractive index is given by the ratio of the refractive index of the droplet medium to that of the surroundings (n/n0), but assuming n0=1, the refractive index n can be considered in the following discussion. The geometrical rainbow angle is a function of the refractive index, given by [31] as

 1  4  n 2 3 2   rg  2cos1  2   ,  n  3  

(1)

The frequency of the Airy fringes depends on droplet size. Therefore, rainbow refractometry permits refractive index and droplet size measurement from the angular distribution of rainbow signal [30]. 2.1 Concentration measurement from intensity attenuation Considering the situation in which the droplet contains one monodisperse phase (as depicted in Fig. 1(a)), the intensity of the second-order refracted light p=2 exiting the colloidal droplet will decrease compared with that of a droplet without a dispersed phase. The intensity change due to the attenuation of the dilute dispersed phase can be approximated using the Beer-Lambert relation, leading to

 D 6 n2  1  I p 2 I p 2,CV 0  exp  N ext Lrg  exp  CVQext ,   D n 3 c  





(2)

where Ip=2 and Ip=2,Cv=0 are the intensity of the p=2 rays through the droplet with and without the dispersed phase. N is the number density of dispersed colloidal suspension particles, ζext and Qext (= 4 ext Dc ) are the extinction cross section and extinction efficiency of the colloidal suspension 2

particle with diameter of Dc. Lθrg is the optical path length at the rainbow angle, expressed as

4 Dn 1

n

2

 1 3 . CV is the volume concentration and D is the diameter of host droplet.

However, this intensity ratio cannot be measured directly, since either the droplet has a dispersed phase or does not and the intensity at the rainbow angle is equal to either the numerator or denominator of this ratio. Therefore, the ratio is measured indirectly, using the following relation: I  I p  2 I p  2,CV 0 =  p 2  I   p 0 

 I p  2,CV 0    ,  I p 0 

(3)

where Ip=0 is the intensity of the reflected light. In Eq. (3), the measured Ip=2 is normalized by the reflected light intensity Ip=0, which eliminates the effect of other sources of intensity variations, such as laser intensity variation or contamination of the optical access window. 2.2 Determination of intensity ratio Ip=2/Ip=0 from a rainbow signal Based on scalar diffraction theory and internal scattering, the superposition of the two rays p=2 and p=0 at a point s on a detector leads to an intensity given by

I  I p 2  I p 0  2 I p 2 I p 0 cos  ( s)  ,

(4)

where δ(s) is the optical path length difference between these two rays and ∆ is the left side baseline intensity from light scattered by internal colloidal particles (if CV=0, ∆=0). The term 2(Ip=2Ip=0)0.5cosδ(s) is the interference term of the rays. The term Ip=0 in this equation is negligible compared with the other terms and can be omitted.

(a)

(b)

Fig. 2. (a) A calculated rainbow signal and its decomposition, (b) calculated p=2 and p=0 ray intensity (perpendicular polarization) of a droplet (D=135 μm, n=1.3350) without colloidal particles under plane wave illumination.

Fig. 2(a) illustrates the intensity distribution of scattered light across the primary rainbow under plane wave illumination (perpendicular polarization). This signal was computed using a Mie program [33] for a pure liquid droplet (D=135 μm, n=1.3350). Filtering the signal in spectral domain, the low frequency Airy fringe component can be separated from the higher frequency ripple structure, indicating the contributions coming separately from the p=2 and p=0 rays. Using Eq. (4), the intensity information of the two components can be retrieved from their maximum oscillation amplitude, expressed as  Aripple 2 =2 I p  2 I p 0  A  I  R  p 2  16  airy   A  I p 0  ripple   Aairy  I p  2

2

(5)

This equation shows that the amplitude ratio of the ripple interference fringes to the primary rainbow fringes is directly related to the intensity ratio of the p=2 ray to the p=0 ray. In order to verify this method, the perpendicular intensities of p=2 and p=0 rays of a pure water droplet are calculated using the Debye series decomposition, also provided by the Mie program, shown in Fig. 2(b). The intensity ratio based on this method is 103.8, which is very close to the value 104.03 obtained directly from the signal in Fig. 2(b). This value represents the denominator on the right-hand side of Eq. (4). This example verifies the method for the extraction of the intensity ratio. 3. Experimental set up The experimental setup consists of a vibrating orifice monodisperse droplet stream generator [34] and the rainbow measurement system as depicted in Fig. 3. The main components of the droplet stream generator are a monodisperse droplet generator (MTG-01-G1, FMP Inc) with an orifice diameter of 75 μm, a signal generator (TOE 7404) and a syringe pump (NE-1010, Pump Systems Inc). The droplet generator can generate a droplet stream with well-controlled and known uniform diameter and droplet spacing via control of the liquid flow rate and frequency of the driving signal. The droplet stream is generated based on the Plateau–Rayleigh instability of a laminar fluid jet [35, 36]. The relation between droplet size D, flow rate Q and excitation frequency f of the orifice is given by

1/3

 6Q  D  . πf 

(6)

Polystyrene (PS) particles with a diameter of 320±30 nm are dispersed in distilled water with different concentrations by controlled dilution. The polystyrene particles have almost the same density as water, which hinders settling and maintains the dispersion over a long time [28]. A flow rate of Q=2.472 ml/min is used and the droplets exiting from the orifice exhibit a velocity of about V=7.0 m/s at the measurement position. The exit velocity of the droplets from the orifice is higher, but the droplets are decelerated due to aerodynamic drag, which is stronger than gravitational acceleration.

Fig. 3. Schematic of experimental set up.

The droplet stream is illuminated by a vertically polarized laser beam (λ=532 nm, 200 mW) with a diameter of 1.8 mm. The backward scattered light is collected by two spherical plano-convex lenses (D=50 mm, f=75 mm) and recorded by a CMOS camera (2048×2592 pixels, 4.8 μm/pixel, 73.0 fps). The measurement volume is about 150 mm from the first lens. The aperture, locating at the image plane of measurement volume, could eliminate stray environmental light. The two spherical lenses and one aperture, together with the camera, form a standard rainbow measurement system. 4. Results and discussions Rainbow signals from colloidal suspension droplets of different concentrations (CV=0%, 0.05%, 0.1%, 0.14%, 0.2% and 0.3%) were recorded under the same experimental conditions. The experiments were conducted at 24 °C in ambient air. The mean free path length l between the polystyrene particles in the droplet can be computed according to Eq. (7), using an extinction efficiency Qext(Dc, λ, n) of 0.5 (calculated by the Mie program [33]). The different volume concentrations lead to mean free path length values of ∞, 853, 427, 304, 213, 142 μm. l

2Dc . 3CVQext  Dc ,  , n 

(7)

The laser is operated at 190 mW to obtain a signal of the high signal-to-noise ratio at a rather low exposure time of the camera (0.2 ms). The number of recorded droplets during each exposure is between 4 and 8. It is known that the ripple fringes are very sensitive to droplet size changes [15-17]. This becomes a problem at higher suspension concentrations, because the dispersed phase begins to disturb the Rayleigh-Plateau breakup and leads to small variations of the droplet size. Reducing the

number of droplets captured in each rainbow image is desirable, on the other hand, with fewer droplets per image the signal-to-noise ratio decreases. The 4 to 8 droplets chosen for this study is a compromise.

(a)

(b)

Fig. 4. (a) Calibration relation between scattering angle and pixel number of the camera, (b) comparison of measured and fitted rainbow signal.

For a quantitative evaluation of the rainbow signal, a calibration of the scattering angle is necessary, since the position and angular spacing of the rainbow will be related to the refractive index and size. In this measurement system, a mirror is used on the detection side of the system; hence, a calibration curve between camera pixel and scattering angle is required, as shown in Fig. 4(a). The calibration relation is highly linear. To analyze the rainbow image, the intensity distributions of the middle 100 pixel rows are averaged across the rainbow. The modified Nussenzveig theory put forward by Sawitree [37] is then used to fit a theoretical rainbow signal to the measured signal. The modified Nussenzveig theory is a compromise between the accuracy of Lorenz-Mie theory and the computational speed of Airy theory for rainbows. Fig. 4(b) shows a typical fitted signal and the measured signal of a droplet (D=134 μm, n=1.3360); the fit is very good for the position, shape, and amplitude of the rainbow. Once a theoretical signal is fit to the measured signal, the size and refractive index used for the theoretical signal are extracted as the measured values. We call this the inversion method. 4.1 Measured size and refractive index Following this signal analysis, measured results of refractive indices and sizes of colloidal suspension droplets at different concentrations are shown in Fig. 5. Each data point in these figures is obtained by averaging over 300 images and each image is captured over 4-8 droplets.

Fig. 5. Measured results of refractive index and diameter of colloidal suspension droplets at different concentrations and as a function of the droplet generator excitation frequency.

For the conditions CV=0%, 0.05%, 0.1%, 0.14%, the size decreases with increased orifice excitation frequency, in good accordance with the theoretical value predicted by Eq. (6). For CV=0.2% and 0.3%, the measured size begins to deviate from the prediction and the deviation increases with excitation frequency, reflecting the less stable Rayleigh-Plateau breakup for higher colloid particle concentrations. For the measured refractive index, the measurement is very accurate when CV is less than 0.05%. In the last four diagrams in Fig. 5, the measured refractive index becomes larger than the true refractive index of water (nhm=1.3350), calculated according to the temperature-dependent correlations given in [38]. The effective medium approximations [3], takes the dielectric constant of the medium into account. As shown in Eq. (8) [39], a formulation allowing for larger size parameters of colloidal particles are available and known as extended effective medium approximations (EEMA):

 eff   hm 

3CV hm  r  1   r  1  r  10 2 2  r  1 3   1   xc  i xc  , r  2 3 r  2    r  2 10

(8)

where εeff is the effective dielectric constant of the colloidal suspension; εr=εc/εhm, εc is the dielectric constant of the colloidal particles, εhm is the dielectric constant of the host medium (distilled water), i=(-1)0.5 and xc=πDc/λ is the size (Mie) parameter of colloidal particle (equal to 1.9 in our case). The refractive index of the medium, directly related to its electromagnetic properties, is expressed as n=(εμ)0.5 where ε and μ are the relative permittivity and permeability (about 1 for most substances) of the medium. Using Eq. (8), the theoretical relationship between the effective refractive index of the colloidal suspension droplet and the volume concentration is computed and shown in Fig. 6. The real and

imaginary

parts

2 2 neff, real  0.5  real   imag 

of



1/2

the

effective

refractive

index

2 2   real  and neff, imag  0.5  real   imag  



1/2

are

calculated

as

  real  . 

Fig. 6. Effective refractive index and its deviation of colloidal suspension droplets as a function of the volume concentration.

The refractive index of polystyrene is nc=1.5979 (24℃, 532 nm) [40], considerably higher than that of water. The deviation ∆n is defined as the difference between the real part of the effective refractive index neff,real and the refractive index of the host medium nhm. It is evident that when ∆n is more than 2×10-4 , then it predominates over the measurement accuracy achievable with rainbow refractometry [41]. This explains the measured deviation of the refractive index shown in Fig. 5 at higher volume concentrations. Furthermore, due to the less stable droplet generation at higher volume concentrations, also the standard deviation of the measured refractive index increases. Nevertheless, using the EEMA, a good measurement of the refractive index of the liquid phase nhm can be retrieved. 4.2 Measurement of the concentration The volume concentration is obtained from the intensity ratio Ip=2/Ip=0 and in Fig. 7 typical rainbow images, their decomposed compositions of p=2 and p=0 rays and the corresponding computed intensity ratios are shown for various volume concentrations. As non-sphericity of the droplet will cause distorted rainbow signals with strong curvature or strange shapes, it is easy to detect whether a droplet is spherical or not [27, 30]. In the present study the measured droplets remain spherical, as inferred from the rather straight rainbow fringes seen in the images of Fig. 7. The four decomposed figures exhibit the expected decrease of amplitude ratio with increasing volume concentration. In addition, the obvious overall decrease in intensity with increasing volume

concentration seen in the rainbow images indicates that less light propagates through the droplet and reaches the camera. This decrease in intensity is inherently coupled with an increase of internal scattering events for the p=2 rays. This internal scattering adds randomness to the path length and frequency of the light rays, the latter due to Brownian motion of the nanoparticles in the droplet. This internal scattering adds a baseline component to all signals, recognized by the higher starting amplitude of each signal with increasing volume concentration. The signal for CV=0 starts at zero amplitude, since the background is first removed from each image. However, for larger CV values the signal starts above zero. Together with the increase in baseline intensity, these effects will result in a smaller angular frequency, thus a smaller measured diameter, which is shown by the red signals in Fig. 7. This can be considered responsible for the discrepancy between measured and theoretical diameter at high volume concentrations in Fig. 5.

Fig. 7. Some typical rainbow images, the extracted intensity signals and their corresponding decompositions into low and high frequency components.

The measured intensity ratio as a function of volume concentration is shown in Fig. 8(a). The two measured ratios Rr for pure water, highlighted by the dashed circle, are slightly lower than the theoretical value (104.03) shown in Fig. 2(b), which can be explained by the non-uniform illumination arising from the Gaussian beam intensity distribution (shaped beam). This effect, also known as the aperture effect, is illustrated in Fig. 8(b). The receiving aperture admits scattered light from a probe volume about the same size as the Gaussian beam width. The effect of this shaped beam on the measured intensity ratio will now be estimated quantitatively.

(a)

(b)

Fig. 8. (a) Measured intensity ratio as a function of volume concentration; (b) Trajectory of a droplet moving through the illuminated volume.

The laser beam intensity along the X axis (coordinate system given in Fig. 8(b)) is given by a Gaussian distribution, 2 I  x   I 0 exp  2  x wx   ,  

(9)

where wx is the radius of the laser beam waist and I0 is the maximum intensity at the center of the beam. The real intensity of the second-order refracted light Ip=2, r and zero-order reflected light Ip=0, r must therefore be expressed as a function of droplet position 2 I  I p  2 exp  2  x2 wx    p  2, r   ,  2    I p 0, r  I p 0 exp 2  x0 wx    

(10)

where the subscript r denotes real, x2 and x0 are the incident points of the two rays on the surface of the droplet. The real intensity ratio Ip=2, r/Ip=0, r then reads I p 2, r

I p 2



I p 0, r Gausian wave

I p 0

 2  exp   2  x2 2  x0 2  ,  wx 

(11)

Plane wave

This ratio is then integrated and averaged over the droplet passage through the Gaussian beam to be Rr Rr 

1 xu  xl  

xu

I p  2, r

xl 

I p 0, r



dx2 

I p 2

1 I p 0 xu  xl  



xu

xl 

I  2  exp   2  2 x2     dx2  p 2  F , (12) w I p 0 x   Plane wave

where xu and xl are the upper and lower bounds on the X axis for which significant incident light is received on the aperture. The quantity δ=x2-x0 is related to the diameter and refractive index of the droplet and varies little in our case; hence it is assumed constant. The factor F is the integral result and can be simplified as b b2 b3 1     4  exp  2  xu  xl     1 2! 3! 4! wx exp  b   1  4  xu    omitted F  exp    exp  a    2 4 a a3 wx2  b  x  xl    1+a    2  u wx 2! 3!

, (13)

omitted

where a=4δxu/wx2, b=4δ(xu-xl-δ)/wx2 and therefore b<2a<<1 in our cases. Using a Taylor series expansion, the value of F can be approximated as (1+b/2)/(1+a), which is definitely smaller than 1, as seen by substituting the condition (b<2a<<1) into Eq. (13). This explains the observed deviation of measured intensity ratios Rr of pure water. According to the measured intensity ratio Rr and the theoretical value for the pure water, the factor F equals 0.88. On the other hand, the factor F, caused by the aperture effect, is present for all volume concentrations and will therefore cancel out when calculating the attenuation of the intensity ratio Rr, Cv/Rr, Cv=0 using Eq. (12).

Fig. 9. Intensity distribution of colloidal suspension droplets (D=135 μm, nhm=1.3350) at different concentrations around the geometrical rainbow angle calculated by Monte Carlo ray tracing [42].

The behavior of the p=2 ray intensity with increasing colloidal volume concentration can also be computed using Monte Carlo ray tracing [42]. Fig. 9 displays the overall decrease of scattered intensity and the increase of baseline intensity from such droplets (D=135 μm, nhm=1.3350) near the rainbow angle with increasing volume concentration, albeit without capturing the interference phenomena. It is noted that the internal scattering phase function of colloidal particles is pre-calculated by the Mie program [33] and made available as a look-up table. Changes in polarization by scattering from colloidal particles is not accounted for.

Fig. 10. Measured attenuation ratios of different concentrations and comparisons with the Monte Carlo simulation for different size ranges.

Fig. 10 shows the measured attenuation ratio at different concentrations and a comparison with the Monte Carlo simulation for different size ranges. The attenuation values at different concentrations are expressed as (1-Rr/Rr, Cv=0)×100%. As expected, the attenuation ratio increases with the concentration for both measured and simulated values. For concentrations as high as 0.3%, the measured and simulated values are in excellent agreement, exhibiting a maximum deviation of about 2.7%. This confirms the feasibility of measuring colloid concentration of droplets using rainbow refractometry. Two further issues will now be addressed which are relevant for practical measurement systems.

It is known that the phase of ripple fringes is very sensitive to droplet size variations and this has been used for studies of droplet evaporation [15-18]. Under the present experimental conditions evaporation is expected to be negligibly small. Furthermore, the intensity ratio technique does not use phase information directly, thus the effect of droplet size variation in the nanometer range is not expected to influence the measurement results. A limit for the upper measurable colloid concentration CV is primarily determined by the host droplet diameter D and the extinction efficiency Qext of the suspended colloidal particles, as seen in Eq. (2). With either a smaller droplet diameter or lower extinction efficiency, a higher volume concentration can be measured. For the colloidal particles in the present study with a droplet diameter of approx. 60 μm, the upper limit of volume concentration is estimated to be about 0.7%. Although this upper limit will increase with smaller droplet sizes, for such particles it becomes increasingly difficult to separate the ripple structure from the primary rainbow signal. Moreover, the signal-to-noise ratio will decrease with decreasing droplet size, decreasing the measurement accuracy. The smallest measurable colloidal droplet size is estimated to be about 30−40 μm. Producing a monodispersed droplet stream at this diameter also poses challenges, especially when colloid suspension particles are added. The pinhole of the monodispersed droplet generator rapidly becomes clogged and ceases to function. 5. Conclusion In this study, the simultaneous measurement of refractive index, host droplet size and colloid concentration of suspension droplets has been demonstrated using rainbow refractometry. This has been achieved by identifying separately the contributions of the p=0 and p=2 rays in the rainbow signal. By using an intensity ratio approach, the technique is not sensitive to overall intensity variations in the optical system, arising for instance from laser intensity fluctuations, containment wall contamination or other obscuration sources. The technique is dependent on the droplets being spherical, thus limiting the application to small droplets, in which the surface tension will insure sphericity. The refractive index of the host droplet must not necessarily be known beforehand, since this is one of the measured quantities. Several issues must be investigated in future studies. The present concentration measurement relies on the extinction efficiency, which in turn requires a prior information about the size of the colloid particles in the droplet. One possible approach to circumvent this need would be to employ additional light sources at different wavelengths, a technique common in extinction spectroscopy. Another aim in future studies is to conduct measurements on individual droplets, rather than averaging over the 4-8 monodisperse droplets in the present study. This will require a triggering strategy, such that the camera shutter or the illumination will be activated only when a droplet is in the illuminating beam. Funding Training Program of the Major Research Plan of the National Natural Science Foundation of China (91741129); National Natural Science Foundation of China (51576177).

Can Li: Investigation, Validation, Data curation, Visualization, writing – original draft, Reviewing

Yingchun Wu: Methodology, Software, Supervision, Reviewing

Xuecheng Wu: Conceptualization, Supervision, Funding acquisition, Reviewing

Cameron Tropea: Project administration, Resources, Supervision, Writing – review & editing

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: