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Measurement of Diffusion Coefficients in Gels Using Holographic Laser Interferometry Nils Ove Gustafsson, Bengt Westrin, Anders Axelsson, and Guido Zacchi* Department of Chemical Engineering 1,University of Lund, P.O. Box 124, S-22100 Lund, Sweden
The accuracy and precision of holographic interferometry as a method to measure diffusion coefficients in gels are investigated both experimentally and theoretically. The standard deviations in the experimentally determined diffusion coefficient for ethanol in 4% (w/v) agarose gel were 3.3% for diffusion into the gel and 6.1% for diffusion out of the gel. These are in good agreement with the standard deviations obtained using Monte Carlo simulations. Systematic errors derived from an assumption of constant diffusion coefficients were also investigated.
Introduction Knowledge of the mass transfer and especially the diffusion in gels is crucial for the design and scaleup of bioreactors with immobilized enzymes and/or cells (Axelsson, 1990) and separation processes such as chromatography (Liapis, 1989) and gel extraction (Gehrke and Cussler, 1989). Diffusion coefficients in gels have in the past been measured by “indirect” methods such as the diaphragm cell, diffusion into and out of gel beads, etc. In these methods, the concentration of the diffusing solute is measured in the solution outside the gel. Some “direct” methods have alsobeen used, such as pulsed-gradientspinecho NMR (Stilbs,1987)and quasi-electriclight scattering (Sellen, 1986). These methods require advanced and expensive equipment as well as an advanced skill and may, in some cases, impose serious restrictions on the type of system that can be used. Recently, Ruiz-Bevih et al. (1989) showed that holographicinterferometry could be used for the measurement of diffusion coefficientsin gels. The technique has in the past been used to measure diffusion coefficientsin liquid systems (Becsey et al., 1971; Bochner and Pipman, 1976; Gabelman-Gray and Fenichel, 1979; Ruiz-Beviti et al., 1984). The method has several advantages. It is a direct method which avoids sampling and analysis of the liquid solution outside the gel. No mass balances are needed since the concentration in the gel is monitored directly, Le., one can actually “look inside the gel”. The fact that the diffusion process is visualized makes it possible to discover any disturbances during the experiments. It is also a rather simple technique which does not require expensive equipment. Furthermore, the technique can also be used to study the effect of external mass transfer, Le., the film resistance, and the existence or nonexistence of a convective flow in gels. The major aim of this work is to investigate the accuracy and precision of holographic interferometry as a method to measure diffusion coefficients in gels. A series of experiments was performed with ethanol diffusing into and out of an agarose gel. Monte Carlo simulationswere performed to study the effect of random errors in the experiment on the diffusion coefficients. The effect of systematic errors due to concentration-dependentdiffusion coefficients was also studied. Theory Holographic Interferometry. A hologram can be regarded as a photograph of an interference pattern 8756-7938/93/3009-0436$04.00/0
I
RECORDING
1aserbeam
transparent
holographic plate
object }. .............................................................................................................................................
BSERVING THE HOLOGRAM
Pobserver Figure 1. Recording and playback of a hologram.
obtainedwhen two light beams coincide. One of the beams passes through the object (the object beam), while the other comes directly from the light source (the reference beam). Laser light is used since monochromaticity and coherency are required. The holographic plate, on which the two beams coincide, is a glass plate coated with a lightsensitivefilm. After exposurethe plate is developed. When the holographicplate is reinserted into the referencebeam, the object can be seen through the plate (Figure l),just as if it was still present. In real-time holographicinterferometrythe holographic plate is, after development, placed in exactly the same position as at the time of exposure. If the object is not removed, the holographic picture is superimposed upon the real object. A change in the optical path of the object beam, caused by a change in the refractive index of the object, will result in an interference pattern. This can be seen as dark and bright areas when the object is observed through the holographic plate. A detailed description of holographicinterferometry is given by Srinivasan (1973). Diffusion into a Semi-infinite Slab. Instationary diffusion in one dimension with a constant diffusion coefficient is described by Fick’s second law:
-ac = D - a2c at ay2
(1)
For the case of diffusion into a semi-infinite slab with a constant concentration Co at the phase boundary and an initial concentration of zero inside the slab, the solution to Fick‘s law can be found in the literature (Crank, 1975)
0 1993 American Chemical Society and American Institute of Chemical Engineers
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fringe no.
(4
I
An X-
y-0
x b
3
2 1 % -
b
0
-1 %
2
b
Y-00
Figure 2. Schematicpicture of the interference pattern obtained from an experiment.
and is given by
expressed by
C = C, erfc(q)
(2)
where q
= L (3) 2(Dt)' I 2 If the diffusing molecule is homogeneous~y distributed in the slab at the beginning (concentration Co), and if a concentration of zero is maintained at the surface of the slab, the solution to Fick's law is given by
C = Coerf(q)
(4)
which thus describes the concentration profile for the solute diffusing out from a semi-infinite slab. Calculation of the Diffusion Coefficients. A schematic interference pattern resulting from a diffusion process is shown in Figure 2. This pattern can be translated into a concentration profile as described below. The diffusion coefficient can be determined by fitting this concentration-length profile to the theoreticalprofile given by eq 2 or eq 4. Each interference fringe represents a certain change in the optical path length compared with that at the time of exposure of the hologram. An interference fringe occurs when the change in optical path length equals an odd number of half-wavelengths (A/2). Since the thickness of the cell (b)is constant during the experiment, the change in the opticalpath length (Aa)can be expressed as a change in the refractive index with the following relation:
Aa = bAn, = (22 +
x 1)s
z = 0, 1,2,
...
(5)
where b is the gel slab thickness and Anz is the change in the refractive index of the gel at the position of fringe z, compared with that at the time of the exposure. For the case of diffusion into a slab, the total change in refractive index, Le., at the phase boundary, is given by (see Figure 2)
Anbt = (N + 1/2 + x)b x
y =0
where N is the total number of fringes and x is the fraction of a fringe spacing just below the phase boundary (see Figure 2) which can be obtained by extrapolation. If a linear dependence is assumed between the concentration and the refractive index, the concentration is
AC = K(nz- n,) = KAn, (7) whereK is an arbitrary constant. The assumptionof linear dependence is reasonably valid for small changes in concentrations. For the system investigated, i.e., aqueous solutions with 0-5 w t % ethanol, the relation between the concentration and the refractive index is completely linear (Handbook Of Chemistry and Physics, 1989)* For diffusion into a slab, the concentration at y = is zero, and consequently AC = C. The concentrations at an arbitrary fringe z and at the phase boundary can now be expressed as Q)
Cz = K ( Z+
x at fringe z 1/2k
and
C, = K ( N + '/2
+ x ) xg
at the phase boundary (9)
Combination of eqs 8 and 9 gives the following equation for a fringe:
Combination of eqs 2 and 10 gives the following working equation: erfc(7) =
z
+ l/2
N + 1/2 + x
(11)
The diffusioncoefficientis obtained by leastcsquaresfitting of the theoreticalleft-hand side of eq 11to the experimental values of the right-hand side. In a similar way, the diffusion coefficient for an experiment with diffusion out from a slab is obtained by fitting to the experimental values expressed by eq 12, obtained by combining eqs 4 and 10. erf(q) = 1-
z
+ '/2
N + 1/2 + x
This is the same as the relation we have in eq 11 since erf(q) is equal to 1- erfc(q).
Materials and Methods Preparationof Gel Slabs. A 4 % (w/v)agarosesolution was prepared by adding the agarose powder (kindly
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/,
Holes for pumping solution through the cell
,
Length scale
-Gel
Figure 3. Cell used in the holographic interferometry experiments.
supplied by Pharmacia-LKB AB) to hot, distilled water. The solution was heavily agitated for more than 15 min until it was homogeneous. The hot solution was then transferred into a cell using a preheated syringe. The cell, which can be seen in Figure 3, is made of Plexiglassexcept for the front and back windows, which are made of glass. When the solution cooled, a gel was formed. It was then carefully pushed out of the cell, and the upper part was sliced off with a razor blade in order to obtain a plane surface. The gel, with a length of 20 mm, was then reinserted into the cell, and distilled water was poured on top of it to prevent evaporation. Experimental Procedure. The optical setup for holographic interferometry is shown in Figure 4. A HeNe laser (Melles-Griot type 05 LHP 151) is used as the light source. The laser beam is split into an object beam and a referencebeam. Both beams pass through the spatial filters. The referencebeam then hits the holographicplate (AGFA 10E75) directly while the object beam passes through the cell before hitting the plate. The experiments were started by preparing the gel as described earlier. The cell was put into its place and the holographic plate was exposed to the laser light for l / s. ~ After exposure the plate was developed, dried, and then reinserted into the same position as during the exposure. After that, the diffusion process was started. The distilled water in the upper part of the cell was replaced by an ethanolsolutionof 5 wt % . The solutionwas pumped through the cell from a large vessel to obtain a constant concentration in the liquid phase. The pumping of liquid also ensured that no concentration gradient could evolve at the phase boundary. The experiments were performed at a constant room temperature. When the cell was observed through the holographic plate, a number of dark fringeswere observed moving slowly downward in the gel. This interference pattern was recorded with a camera attached to a stereo microscope. The diffusion process continued for about 5 h, and recordingsof the interference patterns were done each hour. For the case of diffusion out from the gel, the gel was first saturated with a 5 wt % ethanol solution. Instead of pumping the solution through the cell, the pure solvent was pumped. Simulation of Random and Systematic Errors. Random Errors in the Evaluation of the Location of the Fringes. With the evaluation method described earlier, the only measured variable necessary for determination of the diffusion coefficient is the location of the fringes (y) in the photographic picture of the interference pattern. Monte Carlo simulations were performed to
investigate the effects of random errors in the determination of the location of the fringes [for a more comprehensive description of the Monte Carlo simulation technique, see Press et al. (1986)l. The followingmethodology was adapted to this investigation: (1)All parameter values, including the number of fringes,were chosen to mimic the real experiments. A value of the diffusion coefficient (D) was assumed which was considered as the “real value”. (2) This value of the diffusioncoefficient was used to simulate an “ideal experiment”. The theoretical locations of the fringes at an arbitrary time were calculated using eq 11 for diffusion into a slab and eq 12 for diffusion out from a slab. A number of ideal data pairs (z,y*)were obtained. (3) Random errors were introduced in the locations of the fringes and in the location of the phase boundary, generating data for a “real experiment”. In this way a number of “real”data pairs (z,y)were obtained. (4) These new “experimental data” were used to calculate an apparent diffusion coefficient (Dapp) using the same fitting procedure as when real experiments were evaluated. (5) Points 3 and 4 were repeated 500 times. No parameters were varied, but a unique set of “real experimental data” was created each time. (6) The mean value and the standard deviationfor the diffusioncoefficients(Dapp) were then calculated. Systematic Errors due to a Nonconstant Diffusion Coefficient. The evaluationmethod described previously using either eq 2 or eq 4 assumes a constant diffusion coefficient. If the diffusion coefficient is dependent on the concentration, a systematic error is introduced. The procedure will still be employed (Ruiz-Bevi6et al., 1989) in spite of its invalidity, partly because of its simplicity, but mostly because a model considering concentrationdependent diffusion coefficients can be used only if the existence and the character of the nonideality are known in advance. The effect of this systematic error was investigated for a linear concentration dependence by adopting the following simulation procedure: (1) A linear concentration dependence was assumed:
D = D,(1+ kC*)
(13)
(2) An “experiment” is simulated taking this nonideality into consideration, i.e., the diffusion is described by eq 14 instead of eq 1. To solve this partial differential equation,
a numerical method has to be employed. The method of orthogonal collocation was chosen (Villadsen and Michelsen, 1978; Westrin and Zacchi, 1991). A number of “experimental” data pairs (z,y) were obtained. (3) The “experimental data” were used to calculate an apparent diffusioncoefficient(Dapp) using the same fitting procedure as described above when real experiments were evaluated. The three-step procedurewas repeated for different values of k, both for diffusion into and out from a slab. Errors due to the Assumption of a Semi-infinite Slab. The equations used to evaluate the experimental data, i.e., eqs 11 and 12, are valid for the case of a semiinfinite slab. As the experiments were performed with a finite slab, with a length of 20 mm, the validity of these equationswas checked. Data for the concentrationprofiles in the gel slab were generated with a model for a finite slab (using orthogonal collocation) with reflection at the end:
-d C - ~ aty=L
dY The data were generated using a diffusion coefficient of
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I-
\.“
5 “
A
2
A
v4
3
4
I
I
L
12
7
I
7
Figure 4. Layout of the equipment used in holographic interferometry: (1) laser, (2) laser beam, (3) camera body used as shutter, (4) polarization filters, (5)beam-splitter, (6) mirrors, (7) spatial filters, (8) diffusor, (9) the holographic cell, (10) holographic plate, (11) stereomicroscope with camera, and (12) screens.
9 X 10-lo m2/s. The concentration profiles for diffusion out from a gel, obtained for times between 1 and 5 h, are shown in Figure 5. The concentration at the end of the slab is not changed even after 5 h, which is the maximum time used in any of the experiments. Most of the experiments were evaluated from the concentration profiles obtained at 2-3 h. Data generated using the assumption of semi-infinite slabs, i.e., using eqs 10 and 12, are shown in the same figure for comparison. The conclusion is that, for the conditions used in this investigation, the system is very well approximated by a model for semi-infinite slabs.
Results ExperimentalResults. Two series of experiments to determine the diffusion coefficients for ethanol in a 4% (w/v) agarose gel were performed; six experiments with method “in” and eight with method “out”. In Figure 6 a photographicpicture of an interference pattern, from one of the experiments,is shown. For each experiment,several interference patterns were recorded at different times. Although the diffusion coefficient values could differ slightly,no time dependency was observed. A mean value was calculatedwhich was consideredto represent the entire experiment. The values were recalculated to 20 “C by assuming that D is proportional to T/paccording to the Stokes-Einstein law. The results are tabulated in Tables I and 11. The arithmetic mean value obtained from all of the experiments with diffusion into the gel is 7.81 X 10-lO m2/s with a standard deviation of 3.3 % . For the experiments with diffusionout from the gel, the values are 10.98 X m2/s and 6.1 % , respectively. Simulation Results. The effect of random errors in the locations of the fringes (y) and of the phase boundary ( x ) is shown in Figures 7 and 8,respectively. The standard deviation of Dapp is plotted against the length of the 95 % confidence interval of the variable, i.e., y in Figure 7 and x in Figure 8. A value of 0.2 mm in Figure 7 indicates that the observed y is within fO.l mm of the ideal, simulated value. Figures 7 and 8 are valid both for diffusion “in” and diffusion “out”, since the working equations for the two methods, i.e., eqs 11 and 12, are identical. A reasonable, but rather subjective, estimation of the maximum errors in the two variables for a real experiment is y = *0.06 mm and x = f0.25 mm. When both of these errorsare introduced at the sametime, a standard deviation of 3.5% is obtained. This should be compared with
0
Y u
0
4
8
12
16
20
Length (mm)
Figure 5. Concentration profiles in the gel after 1,2,3,4, and 5 h calculated using the model for finite slab (-) and the model for semi-infinite slab (eq 12) (0).
Figure 6. Interferencepatternrecorded after 3 h. The millimeter scale is seen to the left in the picture.
standard deviationsof 3.3 % and 6.1 % ,which are the values for the experiments given previously. The systematicerror due to a concentration dependence of the diffusion coefficientshowed a surprising regularity. For the diffusion into a slab, the measured value of Dapp was always numerically equal to the value obtained when C* = 0.725 is inserted into eq 13, regardless of the value of k. For the diffusion out from a gel slab, the measured value of Dapp was always equal to the value obtained when C*= 0.275 is inserted into eq 13. The value of Do did not
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Table I. Diffusion Coefficients Obtained from Exwriments with Diffurion into the Gel. expt D, (XlO'O m2/s)
Discussion and Conclusions The experimentalr d t s show no significant differences in the standard deviations between diffusion out of and into the gel, although it is somewhat higher for method "out", 6.1%, compared with method "in", 3.3%. The former method could still be advantageous since the pure solvent is pumped through the cell. This means that a smaller amount of the diffusing molecule is needed. If the molecules are sensitive to high shear forces (e.g., large proteins), it could also be a disadvantage to pump them. The difference in the values for the diffusion coefficients obtained from diffusion into and out from a gel cannot be accounted for by randomerrors. One plausible explanation is that the diffusion coefficient is dependent on the concentration. If the dependence is linear, the diffusion coefficient obtained from the experiments with diffusion into the gel would correspond to an ethanol concentration of 3.6 wt % ,while the value obtained from the diffusion out from the gel would be for a concentration of 1.4 wt '% (according to the simulation section above). The liquid diffusion coefficient for ethanol in water shows a linear concentration dependence (Landolt-Biknstein, 1969) in the ethanol concentration range 0-5 wt %
8.10 7.99 7.87 7.99 7.56 7.37 7.81
1 2 3 4 5 6
arithmetic mean a Values were recalculatedto 20 "C.
Table 11. Diffusion Coefficients Obtained from Exwdments with Diffusion out from the Gel. expt D, (X1010 m2/s) 11.81 10.95 11.75 10.73 11.35 10.87 9.74 10.67 10.98
1 2 3 4 5 6 7
8 arithmeticmean 0 Values were recalculated to 20 "C.
.
DJD,= (1 + k,C)
e
h
6*3
0
.1
.2
.3
Length of 95%confidence interval for fringes (mm)
Figure 7. Effect of random errors in the determination of the position of the interference fringes.
E !
U-
0
.1
.2
.3
Length of 95%confdence interval for the phase boundary (mm)
Figure 8. Effect of random errors in the determination of the
position of the phase boundary.
affect the results. These results are similar to those obtained in a study of diffusion into and out from gel beads (Westrin and Zacchi, 19911,although with different values for C*.
(15)
If the dependence is assumed to be the same for diffusion in gels, Le., that the value of kz is the same as for diffusion in liquid, which is a very reasonable assumption, the diffusion coefficients could be recalculated to the same concentration. The data from Landolt-B6rnstein (1969) give k2 = -0.026. Using this value for k2,eq 15 can be used to calculate the value of DOfor diffusion "in" and "out", respectively. Equation 15 can then be used to calculate the diffusion coefficients for an arbitrary concentration in the range 0-5 wt %. The diffusion coefficients were recalculated to 2.5 wt % A value of 8.06 X m2/swas obtained for method "in" and 10.65 X 10-lom2/eformethod "out". The remaining large difference between method "in" and method "out" can hardly originate from differences in the physical process. Therefore, an artifact associated with the observation of the process must be suspected. One suchpossible artifact is the effect of refraction. When a light beam passes through a region with a refractive index gradient, i.e., the cuvette, it will be refracted in the direction of increasing refractive index. The light beam leaves the cuvette at a different y-coordinate than that at which it entered, and the direction will be changed. The refraction phenomenon would thus give too low a value of the diffusion coefficient for method "in" and too high a value for method "out". A method to reduce the effects of refraction is to use a symmetric system and to evaluate the difference between the fringes in the two layers (Fenichel and GabelmannGray, 1981),one with a positive and the other with a negative refraction. This can be arranged by joining two gel slabs, one containing the diffusant and the other containing the solvent. A preliminary investigation (Dahlqvist,1992)showed very promising results, although some practical problems with getting a good contact between the slabs still remain to be solved. The value of the diffusion coefficientfor 2.5 w t % ethanol in 4 w t % ' agarose gel has been determined to be 8.7 X 10-'0m2/s by Westrin (1991). In that study, the diffusion was measured with several different methods such as the diaphragm cell and NMR. A curiosity is that this value is between the values of the two diffusion coefficients
.
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obtained with methods "in" and "out" in this study (the mean value is 9.35 x W O m2/s). The results obtained, from the experimental investigation as well as the Monte Carlo simulations, show that holographic interferometry is an accurate method of measuring diffusion gels. It is also a reasonable, inexpensive method and can be used to visualize the diffusion processes of which no other method is capable. Further experiments are required to investigate the effect of the refraction.
Notation a optical path length (m) b thickness of the cell (m) concentration of the diffusing molecule (mol/m3 C gel) initial concentration at the phase boundary of the CO diffusing molecule (diffusion into gel) or initial concentration of diffusing molecule inside the gel (diffusion out of gel) (mol/mg gel) concentration at fringe number z of diffusing C* molecule (m0l/m3 gel) C* normalized C, defined as C/Co diffusion coefficient in gel phase (m2/s) D reference value of D at C 0 (m2/s) Do apparent measured value of D (m2/s) DaPP value of D at concentration c (m2/s) Dc k proportionality constant in eq 13 proportionality constant in eq 15 ((wt %)-l) k2 K constant defined in eq 7 (mYmol) an integer greater or equal to zero m refraction index n refraction index at the position of fringe z nz initial refraction index in gel n, N total number of interference fringes t time (8) fraction defined in Figure 2 X distance from phase boundary (m) Y fringe number (see Figure 2) z Greek Letters Aa change in optical path length (m) change in refraction index An total change of refraction index in the gel Anw x wavelength in air for the laser light (m) parameter defined by eq 3 v
-.
OD
standard deviation of D,, (m2/s)
Literature Cited Axelsson,A. Maw transfereffectsin bioreactorswith immobilized enzymesand cells. Ph.D. Thesis,Report LUTKDH/(TKKA1001)/1-56/Lund University, Lund, Sweden, 1990. Becsey,J. C.; Jacson, N. R.; Bierlein, J. A. J.Phys. Chem. 1971, 75, 3374.
Bochner, N.; Pipman, J. J. Phys. D: Appl. Phys. 1976,9,1825. Crank, J. The Mathematics of Diffusion; Oxford Univ. Press: Oxford, 1975; p 32. Dahlqvist,K. A study on diffusion between two gel-slabs using holographic interferometry. Internal Report (in Swedish) LUTKDH/(TKKA-5008),Lund University, 1992. Fenichel, H.; Gabelmann-Gray,L. Holographic interferometry applied to measurementa of diffusion in liquids. In Optics in four dimensions. Am. Inst. Physics. Conf. Proc. (Machado, M. A., Narducci, L. M., Eds.) 1981,65, 586593. Gabelmann-Gray,L.; Fenichel, H. AppZ. Opt. 1979,18,343. Gehrke, S. M.; Cussler, E. L. Chem. Eng. Sci. 1989,44 (3), 559. Handbook of Chemistry and Physics (CRC) 70th ed.; CRC Press: Boca Raton, FL, 1989-1990, D-230. Landolt-B&nstein Zahlenwerte und Funktionen; Springer Verlag: Berlin, 1969; 6 Adage, I1 Band, 5 Teil, p 640. Liapis, A. I. J. Biotechnol. 1989,11,143. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. The Art of Scientific Computing; Cambridge Univ. Press: Cambridge, 1986; pp 529-531. Ruiz-BeviB,F.; Celdran-Mallol,A.; Santoa-Garcia,C.; FernhdezSempere, J. Can. J. Chem. Eng. 1984,63,765. Ruiz-BeviB,F.; Fernbdez-Sempere,J.;Colom-Valiente,J. AIChE J. 1989,35,1895. Sellen, D. B. Br. Polym. J. 1986, 18, 28. Srinivasan, V. S. Advances in Electrochemistry and Electrochemical Engineering; Muller, R. H., Ed.; Wiley: New York, 1973; Vol. 9, pp 369-422. Stilbs, P. Progr. NMR Spectrosc. 1987,19, 1. Villadsen,J.; Michelsen, M. L. Solution of differential equation models by polynomial approximation; PrenticeHalk Englewood Cliffs, NJ, 1978. Westrin, B. Diffusion measurement in gels: a methodological study. Ph.D. Thesis, Report LUTKDH/(TKKA-1003)/1-149/ Lund University, Lund, Sweden, 1991. Westrin, B. A.; Zacchi, G. Chem. Eng. Sci. 1991,46,1911. Accepted March 19, 1993.
