Diffusivity study of transparent liquid solutions by imaging beam deflection

Transcription

1 Diffuivity tudy of tranparent liquid olution by imaging beam deflection Arun Anand*, Vani K Chhaniwal and CS Narayanamurthy Photonic Laboratory, Applied Phyic Dept., Faculty of Tech. & Engg. MS Univerity of Baroda, Vadodra 39, INDIA *Preent addre: Intitute for Plama Reearch, Near Indira Bridge, Bhat, Gandhinagar 3848, INDIA arun_nair_in@yahoo.com ABSTRACT A non-interferometric technique to meaure the diffuion coefficient of tranparent liquid olution i propoed. The technique can be realized uing a white light ource. The bending of the light beam caued by the non-linear refractive index ditribution exiting within the experimental cell (gla cell) containing the diffuing olution, i utilized to find the diffuivity value. The bending of the light beam i imaged uing an optically active material or a birefringent material, placed between two croed polarizer and a CCD camera. A different portion of a plane wavefront paing through the diffuion cell travel different thickne inide the optically active/birefringent medium, the local refractive index variation i converted into a patial variation of output intenity. The diffuion coefficient could be calculated from a ingle image of beam bending or by uing image at two different intance of time. Key Word : Diffuion coefficient, Beam deflection. INTRODUCTION Diffuion i the movement of molecule due to their thermal energie under the influence of a concentration gradient. The tudy of diffuion i important in field a divere a chemical engineering, biology, pollution control, eparation of iotope etc. Knowledge of diffuivity i neceary for the deign of chemical equipment and for ma tranfer tudie. Diffuivity i therefore one of the mot important fundamental property of a chemical ytem,. There are many method to determine diffuion coefficient of tranparent liquid olution. Optical method are one of the mot accurate method ued to determine diffuivity value. The optical method include conventional interferometry, holographic interferometry and electronic peckle pattern interferometry 3-9. But the diadvantage of mot of optical method i that they impoe tringent optical requirement, which are difficult to employ in indutrial environment. So it become neceary to develop new method for diffuivity meaurement that are eay to implement and cheap. In thi paper we propoe a new method to meaure diffuion coefficient of tranparent binary liquid olution uing white light ource 7,8 and a birefringent/optically active medium. The method dicued i non -interferometric. Light ray paing through a nonuniform refractive index medium bend toward region of higher refractive index. The amount of the ray bending at variou intance of time will depend upon the refractive index gradient exiting inide the medium and hence on the diffuion coefficient. Therefore by meauring the bending of the ray the diffuion coefficient can be determined. Thi i achieved by uing an optically active/birefringent crytal to convert the beam bending into a patially varying intenity pattern. A wavefront ening device uing birefringent crytal ha already been developed. The output intenity caued by a wavefront paing through a birefringent plate placed between a pair of croed polarizer will depend upon the ditance each portion of the wavefront traveled through the birefringent medium. Therefore wavefront traveling more ditance will produce more output intenity and vice-vera. The ame principle applie for optically active material. The property of thee medium to rotate the plane of polarization of the incident plane polarized i utilized to determine the diffuion coefficient of tranparent liquid olution.

2 . EXPERIMENTAL SETUP AND THEORY The experimental etup ued i hown in Fig.. The light from a white ource i collimated and made monochromatic by uing a optical filter. The collimated light pae through a polarizer. The polarized light then pae through the cell containing the diffuing olution. A linear polarization rotator (LPR), which can be either an optically active crytal or a birefringent crytal and an analyzer, which i alo a polarizer with it polarization direction at 9 o to P, placed after the cell convert the beam deflection occurring inide the cell into a patially varying intenity ditribution. Thi intenity ditribution i imaged uing an imaging len and a CCD camera. Polarizer (P) Optically Active or birefringent medium (OAM) Diffuion Cell Analyzer (A) CCD chip Source To PC Collimating Len (L) Imaging Len (L) Fig. : Experimental etup Half of the experimental cell i filled with olution of lighter concentration. The heavier concentration olution i introduced below the lighter concentration olution, uing a capillary tube mechanim. The proce of diffuion tart after the experimental cell i filled. The olution inide the cell can be conidered a coniting of layer having patially varying refractive indice. Thi poition depended refractive index will give rie to a deflection of the incident beam (Fig. ). A diffuion progree the refractive index of thee layer will alo change, thereby changing the angle of deflection. If thi deflection can be recorded, it could be ued for diffuion coefficient meaurement. The theoretical explanation i a follow. Free diffuion proce Solution of the Fick econd law which govern free diffuion proce, in a binary liquid ytem coniting of liquid having concentration C and C (C >C ), eparated at x= when t=, conidering diffuion along only one direction (x-axi) i given by x Dt C + C C C C( xt, ) = + exp( η ) dη π () where C(x,t) i concentration at poition x and time t and D i the diffuion coefficient, which i a contant for the concentration range between C and C. The bracketed term i the error function 3 of x Dt. For the narrow range of concentration over which the experiment were conducted, the refractive index inide the experimental cell can be conidered a a linear function of concentration. Therefore the refractive index can be written a dn n( x, t) = C( x, t) + n dc ()

3 x Refractive index profile C Incident ray at z=, x =, x y= y θ Interface at x= Deflected ray to CCD camera z l C Fig. : Refractive index profile inide the diffuion cell and the reulting beam deflection where n i a contant and (dn/dc) i the mean value of the derivative for the applied concentration range. A the diffuion proce progree the concentration gradient and hence the refractive index change. The refractive index ditribution inide the diffuion cell i non-uniform (Eq. ) reulting in the bending of light ray entering the cell. The amount of bending will depend on the refractive index gradient and for a ray entering at z= and x, conidering diffuion along only the x direction i given by 9,, l nxt (, ) l nxt (, ) l dn C exp( x 4Dt ) C θ ( xt, ) = n = = x n x n dc (3) π Dt where l i the length of the diffuion cell and n i the refractive index at point x. 3. SIMULATIONS AND DISCUSSIONS The change in bending angle along the diffuion direction for different time i hown in Fig. 3 average concentration C avg =.9959 mol l - of ammonium dihydrogen phophate. It can be een that the ray are bending in the ame direction for the upper and lower part and the maximum bending occur at the interface. A the time progree the angle of bending alo decreae. A the bending depend upon the refractive index profile inide the cell, the diffuivity value could be determined from the amount of bending. The amount of bending i imaged uing the polarization rotator. y Bending angle θ (deg) t=4 t=36 t=6 t= x poition (mm) Fig. 3: Change in brending angle with diffuion direction at different intance of time

4 When a birefringent crytal i ued, the incident plane polarized light i plit into two eigen wave, which travel with different phae velocitie due to different effective refractive indice. Thee wave are combined at the exit face of the crytal and the polarization direction of the reulting wave depend upon the propagation length and hence on the beam bending. The rotation of plane of polarization of the plane polarized light i caued by the retardation, which i depend upon the difference of refractive indice along the two principal direction and hence on the angle of incidence. The analyzer will convert thi patially modulated polarization direction into a patially varying intenity pattern. In the cae of an optically active medium, the incident plane polarized light i plit into a right circularly polarized light and a left circularly polarized light, which travel with different velocitie inide the medium due to different refractive indice. They will combine at the exit face to form a plane polarized light with it plane of polarization rotated by an amount which depend upon the difference of refractive indice exiting for the let and right circular polarization. Here alo the amount of rotation at the exit face i patially varying due to varying propagation length of different point on the wavefront. The analyzer will convert thi polarization pattern into an intenity pattern. The diffuivity value could be determined from the image of the beam bending in two way ) from a ingle image and ) uing two image. Technique uing a ingle image The intenity at the CCD plane for ray entering the diffuion cell at two poition x and x at an intance of time t could be written uing Eq. (3) a. A A I ( x, t) = exp ( x 4Dt ) and I ( x, t) exp ( x 4Dt ) t = t (4) l dn C where C A = n dc πd of the local output intenitie 4 I I ( ) x, t ( ) x x = exp x, t 4Dt i a contant. Due to bending thee local output intenitie will be different. Taking the ratio (5) and diffuion coefficient can be written a x x D = (6) 4t ln[ I ( x, t) I ( x, t) ] Therefore by finding the ratio of the intenitie (gray level) at two poition in the diffuion cell will yield the diffuion coefficient. Technique uing two image From Eq. (3), the change in intenity due to bending for a ray entering the diffuion cell at poition x for two different intance of time t and t i given by exp ( x 4Dt ) ( ) exp x 4Dt δ I = I ( x, t ) I ( x, t ) = A (7) t t The graphical repreentation of Eq. 7 i hown in Fig. 4 (average concentration C avg =.9959 mol l - ). It can be een that thi curve ha three extreme. Thee extreme can be found by differentiating Eq. 7 with repect to x and equating to zero. Thi yield ( ) ( t exp x 4Dt = t exp x Dt ) (8) 4 One of the olution of Eq. (9) i x=, the interface. The other two olution can be found by taking it logarithm. Thi lead to the other two olution x and x. Uing x and x, the equation for diffuion coefficient can be written a,

5 d x x δθ (deg) x poition (mm) Fig. 4: Change in angle of deflection in the diffuion direction (t =4, t =9) [( t ) ( t )] d D = (9) 6 ln / ( t 3/ t 3 ) where d=x -x i the eparation between the two extreme. By ubtracting, intenity pattern at time t from time t, the eparation between the extreme point can be obtained. Subtituting thi in Eq. (9) yield the diffuion coefficient. Simulation Figure 5a to 5c how the imulated image for Ammonium dihydrogen olution of concentration.9959 mol l -. Fig. 5a and 5b are the imulated image at time 4 econd and 6 econd repectively. Fig. 5c i the difference image of Fig. 5a and 5b. Uing a ingle image or the difference image one could calculate the diffuion coefficient. diffuion direction (cm) - (a) (b) (c) Fig. 5: Image of beam bending; (a) at 4 ; (b) at 6 ; (c) difference image of (a) and (b)

6 4. CONCLUSIONS A method for diffuion coefficient from the beam bending due to the non-linear refractive index uing a birefringent/optically active crytal i propoed. The diffuion coefficient could be calculated from a ingle image or uing two image. The main advantage of the method i that it i non interferometric and could be performed with white light. From the imulated image, it can be een that, the method yield accurate reult. ACKNOWLEDGEMENTS One of the author (AA) thank Prof. Karten Bue for helpful dicuion regarding thi work. VKC like to thank Center for Scientific and Indutrial Reearch (CSIR), Govt. of India for providing her financial aitance through Senior Reearch Fellowhip. REFERENCES. E. L. Culer, Diffuion-ma Tranfer in Fluid ytem, Cambridge Univerity Pre, Cambridge (997).. W. Jot, Diffuion in Solid, Liquid, Gae, Academic, New York (969). 3. N. Bochner and J. Pipman, A imple method of determining diffuion coefficient by holographic interferometry, J. Phy. D: Appl. Phy. 9, (976). 4. L. Gabelmann-Gary and Henry Fenichel, Holographic Interferometric tudy of liquid diffuion, Appl. Opt. 8, (979) 5. J. Szydlowka and B. Janowka, Holographic meaurement of diffuion coefficient, J. Phy. D: Appl. Phy. 5, (98). 6. W. D. Seufert and R. N. O Brien, Determination of diffuion coefficient from progreion of interference fringe, J. Phy. Chem. 88, (984). 7. F. Ruiz-Bevia, A. Celdran-Mallol, C. Santo-Garcia and J. Fernandez-Sempere, Holographic Interferometric tudy of diffuion: anew mathematical treatment, Appl. Opt. 4, (985) 8. J. A Rard and D. G. Miller, Mutual diffuion coefficient of BaCl -H O and KCl -H O at 5 o C from Rayleigh interferometry, J. Chem. Eng. Data 4, 4-6 (99). 9. D. Paoletti, G. Schirripa Spagnolo, V. Baigini and M. Santariero, A new method for meauring diffuivity of liquid binary mixture uing DSPI, Pure. Appl. Opt., (993).. G. Schirripa Spagnolo, D. Ambroini, A. Ponticiello and D. Paoletti, Evaluation of diffuion in liquid by digital peckle pattern interferometry: computer imulation and experiment, Eur. J. Phy. 7, 5-59 (996).. G. Schirripa Spagnolo, D. Ambroini, A. Ponticiello and D. Paoletti, A imple method of determining diffuion coefficient by digital laer peckle correlation, J. Phy. III France 6, 7-5 (996). D. Paoletti and G. Schirripa Spagnolo, Speckle decorrelation tudy of liquid diffuion, Opt. La er Eng. 6, 3-3 (997). 3. E. E. Alani, G. C. Romero and C. C. Martinez, Interferometric meaurement of diffuion coefficient through a canning laer beam, Opt. Eng. 39, () 4. N. Rahidnia and R. Balaubramaniam, Development of an interferometer for meaurement of the diffuion coefficient of micible liquid, Appl. Opt. 4, (). 5. D. Ambroini, D. Paoletti, A. Ponticeillo and G. Schirripa Spagnolo, Speckle decorrilation tudy of liquid diffuion, Opt. Laer Eng. 37, (). 6. A. Anand, V. K. Chhaniwal, S. Mukherjee and C. S. Narayanamurthy, Diffuion tudie in liquid by multiple beam interferometer, Opt. Laer. Technol. 34, () 7. V. K. Chhaniwal, A. Anand, S. Girhe, D. Patil, N. Subhramanyam and C. S. Narayanamurthy, New optical technique for diffuion tudie in tranparent liquid olution, J. Opt. A: Pure Appl. Opt. 5, S39-S337 (3).

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