New experimental method to measure pure and crossdiffusion coefficients of transparent ternary mixturesusing Mach–Zehnder interferometry

Amirhossein Ahadi, M. Ziad Saghir n

Department of Mechanical & Industrial Engineering, EPH-322, 87 Gerrard St. East, Ryerson University, Toronto, Ontario, Canada M5B 2K3

a r t i c l e i n f o

Article history:Received 19 December 2013Received in revised form9 February 2014Accepted 22 March 2014Available online 12 April 2014

Keywords:Mach–Zehnder InterferometryTernary mixtureDiffusion coefficient measurementImage processingSpace experiment

a b s t r a c t

In this study, a Mach–Zehnder interferometer that is equipped with two lasers of different wavelengthswas used to conduct high resolution measurements of concentration profiles of a ternary mixture insidea diffusion cell. Windowed Fourier transform along with an advanced unwrapping procedure wasemployed to extract the phase image from fringe images. Then the phase difference was obtained for aspatial resolution of 1920�1240. According to the measured refractive index profile, concentrationcontours of two components (out of three) were measured. Consequently, the concentration profile ofthe third components was calculated. Previously, the analytical solution for binary mixtures was used toestimate only the pure diffusion coefficients. In this study, for the first time, the refractive indicesmeasured by two lasers along with the analytical solution for the ternary system, based on Fick's law,and an evolutionary algorithm (EA) known as a genetic algorithm (GA) were employed to measure thepure and cross diffusion coefficients of a transparent ternary mixture simultaneously. The optimizationmethod to estimate diffusion coefficients was tested against various objective functions, and the bestapproach was that which was proposed herein. In order to validate the proposed measurement method,the experimental results of the Selectable Optical Diagnostics Instrument-Diffusion Coefficients inMixtures (SODI-DCMIX1 project) on board the International Space Station (ISS) were analyzed using thistechnique and the obtained results were compared with previous techniques.

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1. Introduction

Diffusion phenomena play a vital role in many applications ofmechanical and chemical engineering, oil discovery, oil reservoirsimulation, in pollution control, and biological systems [1–3].Scientists in these fields require accurate measurements of thediffusion coefficients of the components in mixtures in order toproperly design new products such as drugs or to accuratelysimulate the mass transfer in multi-component phenomena suchas species transport in oil reservoirs [4]. The diffusion coefficient ofa component in a mixture can be predicted theoretically orestimated from various correlations [1]. In addition to that, thesecoefficients might also be determined experimentally. Opticalmethods are the most popular among the wide variety of techni-ques for this measurement.

The most important advantages of optical methods, whichhave been utilized for measurement of the diffusion coefficients,are their non-intrusiveness and high sensitivity [5–7]. As these

methods are contactless, their measurements do not disturb theprocess under research; consequently, more accurate measure-ment results [8,9]. Optical methods suggest the qualitative and thequantitative data. The phenomenological theories are developedbased on the information that was obtained by the optical method[10–12]. The principle limitations of the optical methods are thatfirst the media should be transparent and have small dimensionsrelatively [1,10,13]. For instance, for nano-fluids with the initialmass fraction of nanoparticle is high, these methods cannot beused and various correlations along with numerical simulationmethods are alternative options [11]. The key principle of the newoptical method is based on the concentration profile in a diffusionphenomenon, which induces a gradient in the refractive indexinside the fluid domain [1,10,13].

Interferometry is conceivably the most broadly used opticalmeasurement method, and it includes various methods such asdigital interferometry [9,14], electronic speckle pattern interfero-metry [15–17], and holographic interferometry [18–20]. Eventhough all these methods rely on the change of the refractiveindex, each method includes the application of specific instru-mentation, with various experimental uncertainties [21]. A com-parison of these techniques has been published elsewhere [1,22].

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Optics and Lasers in Engineering

http://dx.doi.org/10.1016/j.optlaseng.2014.03.0090143-8166/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ1 416 979 5000x6418; fax: þ1 416 979 5265.E-mail address: zsaghir@ryerson.ca (M.Z. Saghir).

Optics and Lasers in Engineering 59 (2014) 72–81

It is claimed that the refractive indices of the components must beprecisely known precisely at the wavelength of light used [23]. Theholographic interferometry accuracy and precision for measuringthe diffusion coefficients in gels have been studied using theexperimental results and theoretical approach. Systematic errorsresulting from the constant diffusion coefficients assumption, havealso been discussed in detail [24]. It was found that the measure-ment uncertainties are on the order of 5�10�11 [m2 s�1] withinterferometry methods for binary mixtures. It is reported that thevalues of the diffusion coefficients obtained by interferometry aresensitive to the magnification of the optical system [1] because theslow process of diffusion reduces the sensitivity on the measure-ment of time [1].

The four diffusion coefficients of the ternary system NaCl–MgCl2–H2O at a simplified seawater composition were measuredat 25 1C [25–27]. The diffusion coefficients were determined usingboth Gouy and Rayleigh interferometry in these studies. A newmeasurement method for the diffusion coefficients using digitalholographic interferometry was developed that does not requireany contact with the domain in contrast to the conventionalholographic interferometric techniques. The diffusivity values alsocould be obtained in real time in this method [28].

In this study, a digital Mach–Zehnder interferometer (MZI) thatis equipped with two lasers of different wavelengths was used toconduct high resolution measurements of concentration profiles ofa ternary mixture inside a diffusion cell. An advanced imageprocessing method was employed to extract the phase image fromfringe images [8,29]. According to the measured refractive indexvariations inside the cell, the concentration profiles of the twocomponents were measured.

There exists a well-known analytical solution for the masstransfer of species in binary mixtures; it is also known that thecross diffusion coefficients of ternary mixtures are about one orderof the magnitude smaller than the pure diffusion coefficients inthese systems. Because of this magnitude difference, in most of thediffusion measurements of ternary mixtures the cross diffusioncoefficients were neglected, and the analytical solution for binarymixtures was used to determine the pure diffusion coefficients.Here, for the first time, the refractive indices inside a ternarysystem measured by two lasers were coupled with the analyticalsolution of mass transfer in the system for use in an evolutionaryalgorithm (EA) known as genetic algorithm (GA) based on Fick's

law were employed to simultaneously measure the pure and crossdiffusion coefficients of a transparent ternary mixture. In order tovalidate the proposed measurement method, the experimentalresults of the Selectable Optical Diagnostics Instrument-Diffusionand Soret Coefficient project (SODI-DSC project) on board theInternational Space Station (ISS) were analyzed using this techni-que, and the obtained results were compared with the currentlyused analysis techniques. Last, the proposed approach is explainedin detail in Section 2, which includes a mathematical descriptionof the problem for binary and ternary mixtures, the experimentalprocedures, and the optimization method. Section 4 provides theexperimental results gained by the proposed approach.

2. Proposed approach to measure diffusion coefficients

The proposed approach can be described by four sections (1)Mathematical differential equations that demonstrate the masstransfer in the ternary mixture, (2) analytical solutions for theternary system with specific initial conditions, (3) the concepts ofthe MZI experiment equipped with two lasers with differentwavelengths for monitoring the change of refractive index in thecross section of the cell, and (4) the optimization method, which isbased on an EA.

2.1. Mathematical description of the diffusion in a ternary mixture

2.1.1. Problem description using PDEs (partial differential equations)Diffusive mass transport in a ternary mixture in an isothermal

condition is described by two independent mass fluxes. There is noneed to define the third mass flux for the last component because thefinal closing relation describes the third mass flux completely. Given,i¼1, 2, and 3, the mass fluxes inside the domain are given by

J!

1 ¼ �ρðD11∇c1þD12∇c2Þ ð1Þ

J!

2 ¼ �ρðD21∇c1þD22∇c2Þ ð2Þ

∑3

i ¼ 1J!

i ¼ 0 ð3Þ

where ρ is the density of the mixture, ci is the mass fraction of theith component, Dii are the pure diffusion coefficients, and Dij when

Nomenclature

Δn Change in refractive indexΔC Maximum concentration differenceΔT Maximum temperature differencei,j Coordinate index of pixeli(x,y) Intensity of the fringe imageC0,i Initial concentration of ith componentT Temperature [K]L Optical path of the cell [mm]Δφ Phase distribution∂n∂C

� �Concentration contrast factor

t Time [s,min, h]τD Diffusion relaxation time[h]Dii Pure diffusion coefficient [m2 s�1]Dij, ia j Cross diffusion coefficient [m2 s�1 K�1]Cc Concentration at the cold wall [mass fraction]Ch Concentration at the hot wall [mass fraction]λ Laser wavelength [nm], Eight value of diffusion matrixx,y Cartesian co-ordinates in a 2D plane

Abbreviation and subscript

DSC Diffusion and Soret CoefficientEA Evolutionary algorithmexp ExperimentalGA Genetic algorithmh HourIm Imaginary part of complex numberISS International Space StationMN Movable Near infrared LaserMR Moveable Red Lasermin minuteRe Real part of complex numberref Referenceth ThermalSODI Selectable optical diagnostics instrumentSR Soret ratiost Steady

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–81 73

iaj are the cross diffusion coefficients in the ternary mixture. Massconservation equations, with previous phenomenological equationsand the assumption of a convectionless regime, allow writing thefollowing equations that govern the time evolution of the componentconcentration along with the general boundary and initial condi-tions:

∂c1∂t

¼ ρðD11∇2c1þD12∇2c2Þ;

c1ðz;0Þ ¼Δc1;0L

zþ2c1;0�Δc1;02

; c1ðz;1Þ¼ c1;0 ð4Þ

∂c2∂t

¼ ρðD21∇2c1þD22∇2c2Þ;

c2ðz;0Þ ¼Δc2;0L

zþ2c2;0�Δc2;02

; c2ðz;1Þ¼ c2;0 ð5Þ

∑3

i ¼ 1

∂ci∂t

¼ 0 ð6Þ

2.2. Analytical solutions for mass transfer for binary and ternarysystems

2.2.1. Analytical solution for binary mixturesWhen one assumes that the cross diffusion coefficients are

negligible in comparison with the pure diffusion coefficients, Eqs.(4) and (5) and can be simplified. The simplified forms of thoseequations are mentioned below

∂c1ðz; tÞ∂t

¼ ρD11∂2c1ðz; tÞ

∂z2ð7Þ

∂c2ðz; tÞ∂t

¼ ρD22∂2c2ðz; tÞ

∂z2ð8Þ

where ci,0 is the initial mass fraction of the ith component, Δci,0represents the initial concentration gradient of the ith component,and L is the distance between the points that define the bound-aries to Δci,0 at the beginning of diffusion process (or the length ofthe cell in this experimental method). The temporal concentrationvariation in the system can be expressed analytically as a functionof diffusion time

ΔC1ðz; tÞ ¼ C0;1�ΔC0;14π2

∑1

n;odd

1n2 cos

nπzL

� �exp �n2 t

τD11

� �" #ð9Þ

ΔC2ðz; tÞ ¼ C0;2�ΔC0;24π2

∑1

n;odd

1n2 cos

nπzL

� �exp �n2 t

τD22

� �" #ð10Þ

∑3

i ¼ 1Ci ¼ 1 ð11Þ

where τD11 ¼ L2=π2D11, τD22 ¼ L2=π2D22that demonstrate the diffu-sion times for components one and two, respectively, in theternary mixture.

In previous measurements of diffusion coefficients either inbinary or ternary mixtures, Eqs. (9) and (10) were used to estimatethe pure diffusion coefficients of the mixture. Consequently, thismethod was not able to measure the cross diffusion coefficients ofternary mixtures. However, in this study we did not ignore thecross diffusion coefficients, since it has been shown that in somecases the magnitudes of the cross diffusion coefficients arecomparable to those of the pure diffusion coefficients. In thesecases, neglecting the cross diffusion coefficients is not an accurateassumption. Consequently, in order to develop a general methodto measure diffusion coefficients of ternary mixtures, all diffusioncoefficients must be considered; therefore, Eqs. (5) and (6) shouldbe solved with no simplification.

2.2.2. Analytical solution for ternary mixturesEqs. (5) and (6) for the molecular diffusion of a ternary system

can be re-written in matrix form as

∂∂t

ρc1ρc2

" #¼

D11 D12

D21 D22

" #∂∂z

ρ∂∂z

c1c2

" # !ð12Þ

The only possible method to solve this system of differentialequations is by decoupling the two equations and then applyingthe results of binary mixture to the decoupled equations. A similarmethod has been used before for the same equations but withdifferent initial conditions [30–32]; however, for the specifiedinitial conditions mentioned in Eqs. (5) and (6), this is the firsttime that the analytical solution has been derived. The diffusionmatrix D can be diagonalized as

D¼ PEP�1 ð13Þwhere E is the eigenvalue matrix and the matrix P contains theeigenvectors

E¼λ1 00 λ2

" #ð14Þ

P ¼1 D12

λ2 �D11

λ1 �D11D12

1

24

35¼

1 λ2 �D22D21

D21λ1 �D22

1

24

35 ð15Þ

P�1 ¼ λ2�D11

λ2�λ1

1 D12D11 � λ2

D11 � λ1D12

1

24

35¼ λ2�D11

λ2�λ1

1 D22 � λ2D21

D21D22 � λ1

1

24

35 ð16Þ

λ1 ¼D11þD22þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD11�D22Þ2þ4D12D21

q2

ð17Þ

λ2 ¼D11þD22�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD11�D22Þ2þ4D12D21

q2

ð18Þ

By substituting Eq. (13) into Eq. (12), the following equationwill be obtained:

∇t c!¼ PEP�1∇2

z c! ð19Þwhere c! is the column vector of mass fractions, ∇2

z is the secondderivative of location, and ∇t is the first derivative of time. If onemultiplies Eq. (19) by P�1, Eq. (20) will result.

∇t ϕ!¼ E∇2

z ϕ! ð20Þ

where ϕ!¼ P�1 c!which can be used as a new parameter. Eq. (20)

is a decoupled system of equations with a new parameter ϕ.Consequently, the analytical solution of each decoupled part of thesystem of equations has the same form as the general solution ofthe binary mixture, as

Δϕ1ðz; tÞ ¼ ϕ0;1�Δϕ0;14π2

∑1

n;odd

1n2 cos

nπzL

� �exp �n2π

2λ1t

L2

� �" #ð21Þ

Δϕ2ðz; tÞ ¼ ϕ0;2�Δϕ0;24π2

∑1

n;odd

1n2 cos

nπzL

� �exp �n2π

2λ2t

L2

� �" #ð22Þ

where ϕ!

0 ¼ P�1 c!0 and Δ ϕ!

0 ¼ P�1Δ c!0, and the rest of theparameters, such as λi and P, are as defined above [32]. Since thesolution for ternary mixtures is a general solution for mass transferdue to diffusion, mass transfer in a binary mixture can also beexpressed by these formulas when D11¼D22 and D12¼D21¼0. Itmust be noted that a numerical discretized form of Eqs. (4) and (5)can be solved numerically versus time for various estimatedparameters using an iterative method rather than an analytical

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–8174

solution. However, the accuracy of the numerical solution must bevalidated versus the analytical solution of the ternary mixture.

2.3. Application of Mach–Zehnder interferometry to visualize masstransfer

The refractive index is the factor by which the velocity and thewavelength of light are reduced with respect to their vacuumvalues. The speed of light varies in different media; consequently,the refractive index varies from a specific medium to another. Onecan use these refractive index differences between materials tomeasure the concentrations of specific materials or mixture. Forboth binary and ternary mixtures, the concentration dependenceof the refractive index is linear for small variations of concentra-tion [33]; therefore, Eq. (23) can be used to define the relationshipbetween the refractive index and concentration change for binarymixtures.

Δnðx; yÞ ¼ ∂n∂c1

� �T0 ;P;λ1

ΔC1ðx; yÞ ð23Þ

where ð∂n=∂cÞ is the variation of the refractive index due toconcentration. Thus, if the refractive index variation with respectto the homogeneous condition can be measured, the concentrationprofile of one of the components can be measured. It must benoted that Eq. (23) is only applicable when the concentrationchange is caused due to change of concentration alone. In otherwords, there should not be any local temperature gradient orchange of the temperature with respect to time. For the ternarysystem, there are two sources of change of refractive index causedby two components, as

Δn1ðx; yÞ ¼∂n1

∂c1

� �T0 ;C2 ;λ1

ΔC1ðx; yÞþ∂n1

∂c2

� �T0 ;C1 ;λ1

ΔC2ðx; yÞ ð24Þ

there are two contrast factors in Eq. (24) that relate to theconcentration change of components one and two. The refractiveindex (Δn) from one source (one laser) for a ternary system cannotprovide enough information to determine both Δc1 and Δc2. Onthe other hand, the refractive index of a material varies with thewavelength (and frequency) of light. As a result of this, if anotherlaser with a different wavelength is used simultaneously alongwith the first laser, two equations would result that makes themeasurement of Δc1 and Δc2 possible.

Δn2ðx; yÞ ¼∂n2

∂c1

� �T0 ;C2 ;λ2

ΔC1ðx; yÞþ∂n2

∂c2

� �T0 ;C1 ;λ2

ΔC2ðx; yÞ ð25Þ

Particularly, ð∂ni=∂cjÞp0 ;T0 ;Cm ;λ represents the rate of change of therefractive index with the change in the concentration of the jthcomponent, under the conditions of the initial pressure, tempera-ture, and constant mass fraction of the mth component, monitor-ing the ith laser at a constant wavelength. It must be noted thatthe contrast factors are functions of both composition and tem-perature [34–36]; however, they can be assumed constant whenthe change of the concentration or temperature is small.

2.3.1. Experimental setup and conditionsThe diffusion coefficient is hardly difficult to determine at

ground conditions because of liquid convection and convectiveinstabilities caused by gravity [13,14]. On the other hand, micro-gravity provides conditions that eliminate convection that disturbsthe separation of components in a mixture [37]. Therefore, anoptical diagnostic instrument was implemented on board the ISSunder the overall supervision of NASA, ROSKOSMOS, and ESA, inorder to measure the various diffusion coefficients of selectedternary mixtures [38,39].

The optical setup for the Mach–Zehnder interferometer for obser-ving the spatial and transient variations of composition in the mixtureis shown in Fig. 1a. The actual experimental setup is illustrated inFig. 1b, which is installed inside Glovebox on board the ISS.

The sample was monitored with two lasers of constant frequencyat wavelengths λ¼670 nm and λ¼935 nm. The beamswere expandedby a spatial filter and then passed through a beam splitter. One of thebeams traveled through the cell holder and the experimental cellwhile the other bypassed the cell and was used as reference. Fig. 1ashows the two different paths inside the cell holder. One of the pathspassed through the front of the cell (in the direction perpendicular tothe thermal gradient) and the other one passed through the void areato be used as the reference beam. After passing mirrors, the twobeams interfered with each other at a second beam splitter. Theinterference fringes were then captured by the Pantera SA 2M30CCDcamera (Charge Coupled Device) manufactured by Dalsa [8,9,13].More details about the experimental cell, temperature controller andthe performance of the thermal control unit of SODI apparatus can befound in literature [9].

2.3.2. Experimental proceduresThe first step to taking a measurement was the thermalization of

the sample at the mean temperature to set uniform thermal andconcentration distributions inside the cell. The mean temperature thatwas first established at the top and bottom plates of the cell was 25 1C.Then, the temperature gradient was established over the next 6–8 h.Because of the Soret effect, the linear concentration variationsbetween the walls were also created by the presence of the thermalgradient [9,10,13,14]. After the completion of this step, the tempera-ture difference was removed, and the cell was brought back to themean temperature within three minutes. The data for the active cellwere then acquired for 4–6 h. After each run, the systemwas broughtto the initial isothermal and homogeneous concentration conditions.Data were received in the form of raw images, which needed to beprocessed and analyzed by a mean Fourier transform [13].

2.4. Advanced image processing aided with an evolutionaryalgorithm

2.4.1. Phase extraction from the fringe imageThe measured intensity distribution i(x,y) can be expressed in

the form

iðx; yÞ ¼ aðx; yÞþbðÞx; y cos ½ϕðx; yÞ� ð26Þ

Fig. 1. (a) Scheme of the interferometer moving bridge setup and (b) Photograph of the interferometer moving bridge setup.

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–81 75

where a(x,y) represents the background variation, b(x,y) describesthe local contrast of the pattern, and ϕ(x,y) is the phase of theintensity i(x,y). Since the cosine is an even, periodic function, everyphase calculated from one intensity distribution is infinite to anadditive integer multiple of 2π” [13]. The fringe pattern can bewritten as one zero peak term and two non-zero peak terms

iðx; yÞ ¼ aðx; yÞþcðx; yÞþcnðx; yÞ ð27Þwhere

cðx; yÞ ¼ 12bðx; yÞexp ½jϕðx; yÞ� ð28Þ

with j¼ffiffiffiffiffiffiffiffi�1

pand n denoting the complex conjugate. The

Fourier transform was the governing principle used to processthe images.

Iðu; vÞ ¼ Aðu; vÞþCðu; vÞþCnðu; vÞ ð29Þ

The peak terms (C and Cn) were placed symmetrically to theorigin.

The next step was to filter out either one of the two spectra onthe carrier. The unwanted background variation was filtered out atthis stage [8]. Then, applying an inverse fast Fourier transform(iFFT) algorithm to C(u,v) with respect to u and v resulted in c(x,y)

log ½cðx; yÞ� ¼ log12

� �bðx; yÞ

þ iϕðx; yÞ ð30Þ

From c(x,y), the interference phase was calculated point wiseby

ϕðx; yÞ ¼ tan �1Im½cðx; yÞ�Re½cðx; yÞ� ð31Þ

where Re and Im respectively represent the real and imaginaryparts of the interference phase [13]. As described earlier, therewere two lasers with different wavelengths (670 nm and 935 nm).In the case of ternary systems, in each loop of image processing,four images must be considered. Two reference images thatrequire knowledge of the initial conditions of the experimentand two other are current images that can be used to presenttemperature and concentration at each specific time. In otherwords, the reference images are the images taken in a condition ofuniform temperature and homogeneous concentration. These tworeferences remain without any change for the whole duration ofimage processing, and were employed to normalize the rest of theimages. Each image was then reconstructed by performing a two-dimensional Fourier transform of the fringe image. Then, thewindowed Fourier filter was used to select a special band of thespectrum that included one of the non-zero peaks [8,40,41], and,after that, the selected band was placed at the middle of thespectrum. The next step was to perform an inverse two-dimensional windowed Fourier transform of the filtered result.Consequently, an unwrapped phase of the intensity function wasextracted, which possessed continuous values from –π to þπ. It isworth mentioning that a similar procedure was applied to thereference images to evaluate ϕref ðx; yÞ ¼ ϕ ðx; y; tref Þ. Next, eachimage had the reference images subtracted from them, whichresulted in a wrapped phase-difference map. Thus, the requiredphase difference, Δϕ or phase distribution, was obtained by thedifference between the phase of the ith image and the referenceimage [8,13,40,42].

Δϕðx; y; tiÞ ¼ ϕðx; y; tiÞ�ϕref ðx; yÞ ð32Þ

2.4.2. Refractive index extraction and unwrapping procedureNumerous fringe analysis techniques such as Fourier transform

fringe analysis, phase stepping, and wavelet transform fringeanalysis are conducted using the arctangent function[43,44].

In the case where there are no disturbances in the phase data, itis straightforward to calculate the unwrapped phase by integratingthe phase gradients over the whole data set [45–47]. However,actual experiments depend on the quality of the apparatus and theenvironment, in which there are several sources of disturbances.This simple unwrapping approach fails in the vast majority ofcases [45–47].

A complete review of the 2D phase unwrapping techniques areprovided by Ghiglia and Pritt [48]. All the algorithms that areknown to be robust are also computationally intensive; the propermethod must be chosen to get accurate results that remove noisein the domain. A sample of the fringe pattern obtained bythe interferometer on board the ISS, its phase map, and theunwrapped images are shown in Fig. 2. The arrows inside thefringe pattern (Fig. 2a) point out various sources of noises in theimage processing procedure that were caused by the opticalsystem. By using a proper image processing, phase extraction,and an advanced unwrapping method, the negative impacts of thisnoise can be removed from the profile. As is shown in Fig. 2b and c,the noise effects are completely removed from the phase and theunwrapped image.

ϕcontinous ðxj; yiÞ ¼ ϕðxj; yiÞþ2πnðxj; yiÞ; i; j¼ 1;2;… ð33Þ

In general, a smooth two-dimensional profile of phase changecan be obtained over the entire domain using Eq. (34) in which ncan be obtained from Eqs. (35) and (36). The first equation givesthe phase unwrapping in one dimension, and the second oneexpands it along the second dimension. “Basically, during thisprocedure, the wrapped phase of a pixel was compared with aneighboring pixel that was previously validated. If the differencebetween two neighbor pixels was less than π, n remainedunchanged. The phase was changed to 2π plus its wrapped phaseif the difference was less than �π. Finally, if the differencebetween the two pixels was more than π, the phase was equal toits wrapped phase minus 2π” [9]. It must be noted that unlike mostother unwrapping techniques, the unwrapping process startedfrom the pixel that was located at the center of the image, movingto both edges to avoid expanding any effects of noise close to theboundaries [13].

nðxm=2; yn=2Þ ¼ 0; ð34Þ

nðxm=2; yiÞ ¼nðxm=2; yi�1Þ if jϕðxm=2; yiÞ�ϕðxm=2; yi�1Þjoπ

nðxm=2; yi�1Þþ1 if ϕðxm=2; yiÞ�ϕðxm=2; yi�1Þr�π

nðxm=2; yi�1Þ�1 if ϕðxm=2; yiÞ�ϕðxm=2; yi�1ÞZπ

8><>:

ð35Þwhere

j¼m2;mþ22

;mþ42

; :::;m

and

j¼m2;m�22

;m�42

; :::;1:

nðxj; yiÞ ¼nðxj�1; yiÞ if jϕðxj; yiÞ�ϕðxj�1; yiÞj oπ

nðxj�1; yiÞþ1 if ϕðxj; yiÞ�ϕðxj�1; yiÞ r�π

nðxj�1; yiÞ�1 if ϕðxj; yiÞ�ϕðxj�1; yiÞZπ

8><>: ð36Þ

where j¼ n2;

nþ22 ; nþ4

2 ; :::;n and j¼ n2;

n�22 ; n�4

2 ; :::;1 and also where iand j represent different coordinate indices of pixels in the y and xdirections, respectively, and m and n are the number of pixels inthe x and y directions, respectively [8,13,47].

The phase differences obtained by the described unwrappingprocess for both of the lasers were used to measure refractiveindex profiles, Δn(x,y). The profiles then allowed the measurement

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–8176

of the concentration profiles in the cell, and as a result themaximum ΔC in the cavity. The relationship between phasedifference Δϕ and Δn is described below

Δni x; yð Þ ¼ niðx; yÞ�ni;ref ðx; yÞ ¼λi2πL

Δϕiðx; yÞ ð37Þ

where L is the optical path, i.e., the thickness of the liquid layerthat the laser light passed through, and i is a laser with a specifiedwavelength. From the above equation, the variation of refractiveindex was equivalent to the change of the optical phase. A generalsample of the change of refractive index between the walls duringthe entire experiment is plotted in Fig. 3. At first, there is nochange of refractive index. Then, there is a sudden jump in therefractive index caused by the application of the temperaturegradient. Gradually, the concentration gradient built up betweenthe walls. Subsequently, the temperature gradient was removedand in the last step the pure diffusion process occurs.

The concentration profiles can then be calculated combiningEqs. (24) and (25), when the refractive index profiles and theconcentration contrast factors of the mixture are known. Themaximum refractive index variations measured by both lasers

during the diffusion phase for the test mixture in this study isillustrated in Fig. 4 (more detail on the experimental results areprovided in the next section).

So, the mass fraction of the first two components can becalculated as follows:

ΔC1ðx; yÞΔC2ðx; yÞ

" #¼

∂n1∂c1

∂n1∂c2

∂n2∂c1

∂n2∂c2

24

35

�1

T0 ;P0

Δn1ðx; yÞΔn2ðx; yÞ

" #ð38Þ

When the separation profiles of two of the components aremeasured, the third profile can be found based on the conserva-tion of mass in the system. Thus, the temporal concentrationprofile of each mixture was obtained. These profiles were used toextract the diffusion matrix. The aided genetic algorithm (GA) withfast robust discretized smoothing was implemented to find thecurve that provided the best fit to the experimental concentrationdifference.

0 100 200 300 400 500 600

0

1

2

3

4

5

6x 10-3

Time [min]

Ref

ract

ive

Inde

x D

iffer

ence

| Δ

n|

λ= 670 nmλ= 935 nm

End of Thermal Time

Diffusion Phase

Thermal Time, Maximum Seperation

Reference Image

Fig. 3. Maximum refractive index difference between the hot and cold walls duringthe entire duration of the MZI experiment.

10 60 110 160 2100

0.5

1

1.5

x 10-3

Time [min]

λ= 670 nmλ= 935 nm

Δ

duri

ng d

iffus

ion

phas

en

Fig. 4. Maximum refractive index variations of both laser with different wave-lengths during the diffusion phase.

Fig. 2. (a) A sample of fringe pattern, (b) its phase image, and (c) unwrapped phase image.

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–81 77

2.4.3. Genetic algorithm and implementation of robust smoothingIn any experiment, it can be shown that interpolation is

unsatisfactory because it fails to account for the errors or uncer-tainty in the data, and the uncertainty, error, and rough data mustbe removed from the temporal results. We have instead opted touse data smoothing techniques, which eliminate noise and extractreal trends and patterns [1,13,15,37,49–51]. Basically, signals areassumed to vary smoothly, perhaps with a few abrupt shifts. Theroughness of the data can be considered as the sum of additivenoise from a symmetric distribution with mean zero and a certainvariance or spiky noise [13,50].

In signal processing, image processing, or in general statistics,smoothing means performing a procedure that results in anapproximating trend that endeavors to obtain the real behaviorof the patterns. In smoothing, the data points of a signal aremodified so that individual points are raised or lowered to becloser in value to their adjacent points, leading to a smoothersignal [13,50]. In the proposed method here, a fast robust versionof a discretized smoothing spline was used [50,52] based on adiscrete cosine transform (DCT) that allows robust smoothing ofequally spaced data in one and higher dimensions. The accuracy ofthis model has been illustrated in a few cases using MATLAB[13,50]. The applications of the robust smoothing DCT on realexperimental results are shown with the solid and the dashedlines in Fig. 5.

After applying the robust smoothing method to the experi-mental data points, the analytical solution for the mass diffusion internary mixtures (Eqs. (21) and (22)) must be used to find the bestfit to experimental results (or the best smooth curve of experi-mental results). The best fit lines can estimate the diffusion matrix,which includes four unknowns: D11, D22, D12 and D21. In the casethat the number of free parameters is more than the numberequations, those parameters can be evaluated and correctedduring each iteration using the curve-fitting problem with an EAapproach. In this case, for any pair of images taken by the lasersduring the diffusion experiment, there were two equations withfour free parameters that needed to be evaluated. Moreover, therewere at least four constraints to be considered in order to find anoptimal solution that satisfies the physics of the problem. Theseconstraints were introduced in the literature [32] as

D1140 ð39Þ

D2240 ð40Þ

D11D22�D12D2140 ð41Þ

ðD11�D22Þ2þ4D12D21Z0 ð42Þ

In addition to the constraints mentioned above, finding theoptimal solution in the 4-dimensional domain using a gradientbase method has been shown to be impossible [53]. Evolutionarymethods (such as GA) are faster and more accurate than gradient-based methods for such problems as this, because these methodsdo not become trapped in local optimal points, and they canconsistently report the global solution. Therefore, the applicationof a GA for such nonlinear problems is highly recommended. Theobjective function must be defined in a way to minimize thedifference between the predicted analytical curve and the experi-mental results (or smoothed experimental curve). Moreover, astime passed during the experiment, the concentration differencebetween the walls decreased, because of the diffusion process.When the concentration variation was small the influence of thenoise and errors were more pronounced. Since at the end of thediffusion process (last hour of the diffusion process) the ΔCbetween the walls was very small (less than 10�4), the influenceof noise caused fluctuations in the curve of separation versus time.These fluctuations can cause noticeable variation between theexperimental and fitted curves; variation that increases theobjective function and has a negative impact on the final optimumsolution. Thus, the objective function must be a function of time inthe form of a least square error function, and it must be minimizedduring the optimization procedure. The suggested form of theobjective function is introduced here as [13]

Err¼ ∑n

i ¼ 1

1nðτD�tÞ Cexp� smooth�Cf it� curve

� �2 ð43Þ

where n is the number of images obtained by each of the lasers,and t is the time used to intensify the influence of each data point.Use of the the diffusion timeτD in an objective function caused alinear variation between the first and the last data points in a waythat the effect on the objective function of the first image was twotimes larger than the effect of the last one [13].

The results of the optimization procedure satisfied the con-straints of the problem, which are mentioned in Eqs. (39)–(42).The first two constraints were automatically be satisfied when thelower search band of the pure diffusion coefficients were assignedzero. However, the last two constraints were applied to theoptimization procedure using a penalty method. In the case thatall constraints in Eqs. (39)–(42) were not satisfied, the objectivefunction in Eq. (43) was increased by a factor of three plus aconstant positive value. Doing this assured that the optimizedsolution satisfied the constraints. The specific parameters of thegenetic algorithm that were recommended for the estimation ofthe pure and the cross diffusion coefficients are provided inTable 1. These bands, limitations, and populations may changefrom case to case depending on the properties of the mixture andthe experimental conditions.

3. Application of the proposed method for space experiments

The SODI-DCMIX1 (Selectable Optical Diagnostics Instrument-Diffusion Coefficients in Mixtures 1) project included a series of

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

Δc 1=

c h-cc[-

]

0 1 2 3 4 5 6-0.2

-0.15

-0.1

-0.05

0

Time[hr]

Δc 2=c

h-cc[-

]

Experimental ResultSmooth Curve

Experimental ResultSmooth Curve

Fig. 5. Application of robust smoothing discrete cosine transform on experimentalresults of diffusion experiment.

Table 1Genetic algorithm specification.

Iteration Population Estimation of purediffusion coefficient

Estimation of crossdiffusion coefficient

Lower band Upper band Lower band Upper band

300 500 0 1�10-9 �5�10-10 5�10-10

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–8178

diffusion and thermodiffusion experiments that were implemen-ted under the overall supervision of NASA, ROSKOSMOS and ESA.Measurement of the diffusion and thermodiffusion coefficients ofselected hydrocarbon ternary mixtures was the main goal of thisproject, which took advantage of the microgravity environment onboard the ISS [54]. A mixture of tetrahydronaphthalene–isobutyl-benzene-dodecane (THN 45 [wt%], IBB 10 [wt%], C12 45 [wt%]) washosted on board the ISS. The liquid mixture was contained in a10 mm�10 mm�5 mm (w, l, h) cell and was monitored by a MZIsetup. The experiment was performed at a mean temperature of298 K. The optical properties of this ternary mixture at this specificcomposition can be found in literature [55].

The maximum concentration difference, ΔC, for the test mix-ture was extracted from the chart shown in Fig. 4. The calculatedconcentration variations between walls are illustrated in Fig. 6 forC12 (at the top) and IBB (at the bottom). The smooth patterns ofconcentration differences of the components (especially for IBB)can be observed in Fig. 6. The separation magnitudes of both C12

and IBB were similar, which means that for this case the measure-ment of the diffusion coefficient of THN is not easily made,because the maximum concentration gradient of THN at thebeginning of the diffusion experiment is not noticeable.

The GAwas applied to the obtained concentration variations forthe diffusion phase of the experiment, which lasted 4 h. Inaddition to this procedure, the analytical solution for the binarymixture was used to find the best fit to the experimental results.Interestingly, both fitted curves (one obtained by the analyticalsolution of the ternary mixture and the other one from the binarymixtures) overlapped each other completely (see Fig. 6). Thisshows that both procedures provide the same accuracy to estimatethe best curve to fit the experimental results. However, the crossdiffusion coefficients calculated in the binary mixture case wasignored. If one compares the values of D11 obtained by both (seeFig. 6), then the small difference between these two pure diffusioncoefficients is observed. It must be added that the pure diffusioncoefficient estimated by the proposed method in this study has asmaller value than that estimated by the other method. Thisdifference can be explained by considering the cross diffusioncoefficient that is also estimated by this technique. When theanalytical solution of the binary mixture is being used instead ofthe ternary one, pure diffusion represents the summation of theeffects of both the pure and cross diffusion coefficients. Using theproposed technique, on the other hand, distinguishes the impactof these two diffusion coefficients and reports them as indepen-dent values.

4. Advantages and limitations of the proposed method

The proposed method is applicable to most of the ternarymixtures and has an important advantage that other methods donot have. To measure the diffusion coefficients, there must be alocal concentration gradient for at least one of the components ofthe mixture at the beginning of the procedure (t¼0). Mostexperimental methods propose two mixtures containing similarcomponents but different initial concentrations of the componentsin contact of each other at the beginning of the experiment.Proposing these two mixtures at t¼0 introduces many technicaldifficulties to the experimentalists, and it also requires directcontact between the fluid media. This connection can disturb thepure diffusion procedure and may have an influence on themeasurement of the diffusion coefficient. On the other hand, inthe method proposed here, the Soret effect is used as the drivingforce to induce a linear variation of the concentration gradientinside the domain. This method does not need any contact withthe medium and also provides a linear variation of the concentra-tion as the initial condition of the problem.

Despite this benefit provided by the Soret effect, using thisphenomenon to create the concentration gradient has its limita-tions. In order to have a large enough concentration gradient(�8�10�3 [wt]) to enable clear measurements and detection ofthe diffusion process, the liquid mixture should have a noticeabletendency to separate in the presence of a thermal gradient. Inother words, this technique does not work for mixtures with smallSoret coefficients (o10�3). This also depends on the othercharacteristics of the experiment, such as ΔT, the initial concen-tration gradient, and the cell size.

One may suggest applying a higher temperature gradient forthe cases with low values of Soret coefficient in order to compen-sate for this limitation. In the presence of higher temperaturedifferences, higher concentration variations will be caused for anybinary or ternary mixture. However, there are two points thatmust be considered. First, there is a constraint to increases in ΔT.The maximum temperature difference applied to the system mustbe in a region that the contrast factors can be assumed constant.Since the contrast factors are a weak function of temperature, themaximum value for ΔT has an approximate upper band at 20 K.The second limitation is only true for ternary mixtures and can beexplained as follows. If the separation of one of the components incomparison with the two others is small in the ternary mixture, itis not possible to measure the diffusion coefficients for thatcomponent. However, it must be noted that even in this case thediffusion coefficients of the two other components can be easilymeasured using this method.

The quality of the images, the number of fringes, the angle ofmirror, and the length of the path must all be optimized in the MZIsetup to obtain an accurate and precise result. For instance, if thereare fewer fringes, this can in turn decrease the resolution of theconcentration profiles, but can be controlled by the angle ofmirrors and the optical path length.

The last limitation of this technique is the high sensitivity of theresult to the concentration contrast factors. The concentrationdependence of the diffusion coefficient provides insight into themolecular behavior inside the binary or ternary solution[1,4,27,33]. The contrast factor matrix in Eq. (38) is the key pointto relate the refractive index variation to the concentration. Sincethe concentration contrast factors of a specific mixture at variouswavelengths are remarkably close, and also Δn measured from theexperiment on board the ISS by both lasers are quite similar (seeFig. 4 that shows the similar behavior of both lasers for the testmixture), a small change in the contrast factor will change the finalresult. It should be emphasized that this is a general limitationof MZI to measure the concentration profiles for ternary mixtures.

0

5

10

15

20x 10

-3

ΔC

C12

, Ch-C

c [-]

0 0.5 1 1.5 2 2.5 3 3.5

-20

-15

-10

-5

0

Time [hour]

ΔC

IBB, C

h-Cc [-

]

Experimental Data,IBBFitted Curve to Diffusion Phase (Ternary), D =9.60e-11,D =4.92e-10[m s ]

Fitted Curve to Diffusion Phase (Binary), D =5.22e-10[m s ]

Experimental Data,THNFitted Curve to Diffusion Phase (Ternary), D =5.09e-10, D =1.34e-10[m s ]

Fitted Curve to Diffusion Phase (Binary), D =5.47e-10[m s ]

Fig. 6. Temporal concentration variation between the walls during the spaceexperiment and the fitted curves obtained based on the analytical solution of thebinary and ternary mixtures.

A. Ahadi, M.Z. Saghir / Optics and Lasers in Engineering 59 (2014) 72–81 79

In order to minimize this effect, two lasers that are used in theexperiment must have the maximum possible wavelength differ-ence. For instance instead of using lasers with λ¼670 and 935[nm], it is recommended to use two lasers that have λ¼450 and935 [nm].

5. Conclusions

The goal of this research was achieved by proposing a newexperimental method to measure the pure and cross diffusioncoefficients of any transparent liquid mixture using Mach–Zehn-der interferometry. The essential post-processing procedure thatwas required to extract the diffusion matrix from the fringe imagesis provided in detail. The Soret phenomenon was used asthe driving force to initiate the concentration variation at thebeginning of the diffusion process. The analytical solution of thediffusion mass transfer in a ternary system along with an optimi-zation method using an evolutionary algorithm was employed toestimate the values of the diffusion matrix. The purpose of thistechnique was examined against the experimental resultsobtained on board the International Space Station. The comparisonbetween the measured diffusion matrix found by the proposedmethod and the previous technique (using the analytical solutionfor a binary mixture) proves the accuracy of the proposedtechnique. Furthermore, this technique can in fact provide moredetails. Eventually, the advantages and limitations of this techni-que were discussed and the essential recommendations andsuggestions were made to minimize the influences of the men-tioned limitations.

Acknowledgments

The authors acknowledge the financial support of the CanadianSpace Agency (CSA), the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for this project, and alsoespecially thanks to the European Space Agency (ESA) andROSCOSMOS for allowing us to use the hardware on board theInternational Space Station.

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