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Measurement of Soret and Fickian diffusion coefficients by orthogonal phase-shifting interferometry and its application to protein aqueous solutions
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Image of FIG. 1.
FIG. 1.

Sketch of the stability diagram of a binary solution inside a Soret cell expressed in terms of the critical Rayleigh number as a function of the Soret coefficient of the denser component. The configuration heated from above ( < 0) corresponds to the one used in most conventional convectionless methods to measure the Soret coefficient.

Image of FIG. 2.
FIG. 2.

The Soret cell where the binary solution is injected. (a) Three-dimensional model of the cell, (b) its lateral cross-section, and (c) a horizontal cross-section view of the cavity where the sample is placed. The lateral walls of the cell are transparent to allow the passage of two orthogonal test beams. The temperature control of the upper and lower copper boundaries is conducted by Peltier modules integrated to a PID system. Two holes were drilled in a quartz wall to fill the cavity with the solution while avoiding bubbles, and to introduce the thermocouples in order to measure the temperature just at the copper/liquid boundary.

Image of FIG. 3.
FIG. 3.

PID temperature control at the copper/liquid boundaries. The temperature is measured just at the liquid interface by extra thin T-type thermocouples. The temperature control for the upper and lower boundaries is simultaneously performed to set a constant temperature difference Δ and a mean temperature . The overshoot and settling time depend on PID gains chosen in the control routine.

Image of FIG. 4.
FIG. 4.

Two orthogonally aligned phase-shifting interferometers. The polarization state of light is shown for each segment. The Soret cell is placed inside the interferometer to visualize thermodiffusion and isothermal diffusion from two orthogonal directions. The lower right inset shows the rotating polarizer, which is synchronized with the CCD camera in order to take interferograms at angle and time intervals of Δ = 2π/3 and Δ = 1/30 s, respectively.

Image of FIG. 5.
FIG. 5.

Image processing of the interferograms in (a) quasi-real-time, and (b) post-experimental treatment. In the quasi-real-time image processing, three consecutive interferograms , , , are treated with the phase-shifting equation, i.e., Eq. (7) , in order to obtain the phase-shifted data (, ), which is also called wrapped phase map. An infinite fringe condition is achieved by adjusting the tilt of mirrors and (see Fig. 4 ). However, the infinite fringe condition may not be achievable when placing the Soret cell in the test section due to possible optical defects of the quartz walls. These optical defects are translated into the unwrapped phase map that is a measurement of the deformed wavefront, as shown in (b). The unwrapped phase difference may be later used as background to obtain the real thermodiffusion field.

Image of FIG. 6.
FIG. 6.

Determination of Δ for a dilute NaCl solution. (a) The unwrapped phase diagram obtained from the phase-shifted data shown in the insets. The phase diagram profile is the unwrapped data averaged in the horizontal direction perpendicular to . (b) The phase difference Δ between the upper and lower boundaries. The phase difference Δ averaged in time is then divided by the concentration difference to obtain the ratio Δ. This value is used in the thermodiffusion experiments to determine the concentration profile within the Soret cell.

Image of FIG. 7.
FIG. 7.

Isothermal diffusion of ethanol-water solution at ethanol mean concentration = 38.52 wt.% and Δ = 1.94 wt.%. (a) The upper view of the diffusion cell with its dimensions, the direction of each test beam, and the coordinate system. (b) The diffusion process observed with the test beam (OP 2 mm). (c) The diffusion process observed with the test beam (OP 10 mm). The phase difference between the upper and lower boundaries is taken after = 45 s. The vertical positioning of the diffusion cell is adjusted until = 85 s.

Image of FIG. 8.
FIG. 8.

Optical measurements from an isothermal diffusion experiment in an ethanol-water binary solution at ethanol mean concentration = 38.52 wt.% and Δ = 1.94 wt.%. (a) The portion surrounded by the vertical rectangle of the phase-shifted data shown in Fig. 7(c) is unwrapped and then averaged in the direction; the resulting profiles are translated (adjusted) in the phase difference domain assuming that the mass flux through both boundaries is equal. (b) The temporal variation of Δ, which is the difference between the unwrapped phase at the lower and upper boundaries. Δ remains constant as long as there is no mass flux through the immersed boundaries, as indicated by the phase-shifted data at = 80 s.

Image of FIG. 9.
FIG. 9.

Transient concentration profiles within the diffusion cell for the isobutylbenzene-dodecane binary solution. The experimental and numerical profiles are indicated by gray circles and solid lines, respectively. The visualization range is from = 0 to = 2.83 mm. Each numerical profile, which is based on Fick's law assuming a constant diffusion coefficient, is fitted to the experimental results by inverse analysis. In the numerical calculations, the first experimental profile is used as initial condition. The phase-shifted data for = 0, 100, and 500 s is shown in the insets. The diffusion coefficient at = 500 s is = 9.56 × 10 m/s. The average diffusion coefficient for all fits and its corresponding standard deviation is shown in Table II .

Image of FIG. 10.
FIG. 10.

Experimental transient concentration profiles of the protein aqueous solutions of lysozyme-water at = 1 mg/ml. The OP is 20 mm. The phase-shifted data (modulo 2π) corresponding to the initial and final concentration profiles are shown in the insets. The diffusion process starts at = 0 s.

Image of FIG. 11.
FIG. 11.

Phase-shifting interferometric measurements of the thermodiffusion and isothermal diffusion fields of the ethanol-water binary solution (ethanol 39.12 wt.%) inside the Soret cell. The views from the orthogonally aligned interferometers with light sources (OP 20 mm) and (OP 10 mm) are shown at different experimental times. The entire height of the cell, i.e., 2 mm, is recorded by the phase-shifted data. The induction of the thermodiffusion field is conducted by applying a temperature difference of Δ = 5 °C ( = 25 °C) and adjusting the tilt of the mirror . The temperature reaches a linear profile between the copper blocks within 75 s. The mirror adjustment is finalized at 80 s, which is the time at which the measurement of the thermodiffusion phase is initialized. At = 7000 s, the ethanol concentration profile has reached a quasi-steady state. At = 7001 s, the binary solution is set to an isothermal state ( = 25 °C) and the Fickian diffusion process is then visualized with the test beam until the binary solution becomes homogeneous.

Image of FIG. 12.
FIG. 12.

Temporal variation of the concentration difference Δ between the upper and lower walls for (a) ethanol in the ethanol-water solution, and (b) dodecane in the isobutylbenzene-dodecane (IBB-C12) solution. The lightest component in each liquid pair, i.e., (a) ethanol and (b) dodecane, is taken as reference for the concentration difference. The total experimental time is 12000 s for the former case and 7000 s for the latter case. Some snapshots of the phase-shifted data for the ethanol-water experiment are shown in Fig. 11 . The temperature difference between the upper and lower boundaries in the thermodiffusion phase of ethanol-water and isobutylbenzene-dodecane is Δ = 5 °C and Δ = 6 °C, respectively. The OP in the measurement shown in (a) is 20 mm for the thermodiffusion phase and 10 mm for the isothermal diffusion phase. The OP in the measurement shown in (b) is 10 mm for the thermodiffusion phase and 20 mm for the isothermal diffusion phase. The scattering for the Δ points is larger for the short OP measurements.

Image of FIG. 13.
FIG. 13.

Transient concentration profiles for the ethanol component in the ethanol-water binary solution during its thermodiffusion phase. Thermodiffusion drives the ethanol molecules to the upper region, i.e., the high temperature region, and the water molecules to the lower region, i.e., the low temperature region. Therefore, the ethanol and water molecules have thermophilic and thermophobic behavior, respectively.

Image of FIG. 14.
FIG. 14.

Beam deflection problem through the Soret cell with focus on the lower cold boundary (for < 0). The cold region inside the solution has a higher refractive index than the upper hot region. Therefore, the beam is curved towards the bottom of the cell.

Image of FIG. 15.
FIG. 15.

Transient concentration profiles during the isothermal diffusion phase of the isobutylbenzene- dodecane binary solution (50 wt.%). The profiles are shown at times separated by Δ = 100 s starting at = 3700 s. After the thermodiffusion phase, the isobutylbenzene molecules have accumulated in the lower cold part of the cell, i.e., the isobutylbenzene molecules are thermophobic. The phase-shifted data at two different times is shown in the insets.

Image of FIG. 16.
FIG. 16.

Phase-shifted data for two thermodiffusion experiments of a dilute aprotinin-water (3 mg/ml) binary solution inside a Soret cell of = 1.5 mm. In the first experimental run (a) Δ = 5 °C, = 25 °C, in the second experimental run (b) Δ = 10 °C, = 30 °C. The fringes observed at = 0 s are due to optical defects of the quartz walls. The fringes due to thermophoresis are clearly observed in both the thermodiffusion and isothermal diffusion phases. The beam deflection problem at Δ = 10 °C and OP = 20 mm becomes more prominent.

Image of FIG. 17.
FIG. 17.

Transient concentration profile during the isothermal diffusion phase of the second experimental run of the aprotinin-water binary solution. The height of the cell is = 1.5 mm, and the temperature difference applied during the thermodiffusion phase is Δ = 10 °C. Three phase-shifted data at = 3690 s, 3740 s, and 6000 s are shown in the insets. The aprotinin molecules exhibit a thermophilic behavior ( < 0) within the aqueous surrounding.

Image of FIG. 18.
FIG. 18.

Concentration profiles for the thermodiffusion phase of a lysozyme-water binary solution (3 mg/ml). The temperature difference is set to Δ = 5 °C between the liquid/copper boundaries separated by = 2 mm. The concentration difference Δ is too small to obtain a reliable measurement of the Soret coefficient. Nevertheless, the thermophobic nature ( > 0) of the lysozyme molecules is evident from this experimental result.


Generic image for table
Table I.

Components used in the four liquid pairs and the corresponding molecular mass and density (at 25 °C) of each pure liquid. Distilled water, dodecane, and isobutylbenzene are abbreviated as HO, C12, and IBB, respectively. The lot number and the suppliers, Wako Pure Chemical Industries (Wako), Ltd. and Sigma-Aldrich Co. (Sigma), are indicated. The proteins were supplied in crystalized form and, therefore, their density is omitted.

Generic image for table
Table II.

Measurement of the contrast factor and diffusion coefficient for the four binary solutions at mean concentration . The concentration in the contrast factors is given for the denser components, which are underlined for each solution. The reference values for ethanol-water (water 60.13 wt.%) and isobutylbenzene-dodecane (IBB-C12 50 wt.%) are included. The factor is not included for the protein aqueous solutions to avoid confusion due to the different units used in the concentration term.

Generic image for table
Table III.

The Soret coefficient , the isothermal diffusion coefficient , and the thermodiffusion coefficient for the two benchmark binary solutions measured in the present study at a mean concentration . The reference component is underlined. Literature values are written in parentheses for comparison.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Measurement of Soret and Fickian diffusion coefficients by orthogonal phase-shifting interferometry and its application to protein aqueous solutions