EXPERIMENTAL TECHNIQUES
Use of a Rayleigh Interferometer for Membrane Transport Studies Peter H. Bollenbeck and W. Fred Ramirez* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80302
Mass transport across a thin porous membrane is studied quantitatively with the use of a Rayleigh interferometer.Criteria are given for the use of a Rayleigh interferometer for membrane transport studies. The effect of optical wave front deflection is described and corrections are calculated. Concentration and flux data near the surface of a 0.4-pdiameter Nucleopore membrane are computed directly from the experimental data. The membrane is shown to have diffusive behavior for sucrose-water transport.
Introduction Mass transport through membranes has been traditionally studied by locating a membrane between two mechanically mixed fluid compartments and monitoring the composition of the solutions in each compartment. An overall mass flux is easily obtained in this manner. However, the stirring produces boundary layers a t each membrane surface so that the bulk concentration and the surface concentration are not the same. The overall resistance to mass transfer is due not only t o the membrane but also the diffusion through the boundary layers. Smith, e t al. (1968), have presented a good review and discussion for methods for finding the membrane resistance from stirred tank data. If a membrane is located in the test cell of an interferometer it is possible to observe directly the concentration a t each surface and to calculate the mass flux through the membrane from the concentration profiles on each side of the membrane. Boundary layers do not exist in this system because the solutions on each side of the membrane are not stirred. The freedom from stirring is particularly advantageous when physically weak membranes (e.g., some biological membranes) are studied. Ramirez and Bollenbeck (1970) have discussed the use of a modified Rayleigh interferometer for membrane studies. Spiegler, et al. (1965),have used a wedge-type interferometer to observe polarization effects on each side of a membrane in an electrodialysis cell. The Gouy (1880) interferometer and a modified version of the Rayleigh interferometer have been the most popular interferometers used for liquid diffusion studies. The modified Rayleigh interferometer is due to Philpot and Cook (1947), and Svensson (1944). The optical theory and use of the Gouy interferometer to measure liquid phase diffusivities has been presented by Gosting and Onsager (1952), and Kegeles and Gosting (1947). The data analysis was limited to binary systems with constant diffusivity and constant density. The optical theory of the modified Rayleigh interferometer has been reviewed by Bollenbeck and Ramirez (1970). A method for calculating diffusivities with the Rayleigh fringe pattern has been given by Svensson (1951) for binary systems with constant diffusivity and constant density. The modified Rayleigh interferometer was selected for use in this study of membrane transport for three primary
reasons: (1) the concentration profile on each side of a membrane is obtained directly from a fringe photograph for a binary mixture; (2) the light source slit is oriented parallel to the concentration gradient and orthogonal to the membrane so that diffraction due to the membrane is minimized; and (3) the potential accuracy of the interferometer has been demonstrated by Longsworth (1952) and Svensson (1951). When studying diffusion through membranes the desired information is the mass flux and concentration a t each surface. The classical data analysis techniques are limited to binary mixtures with constant diffusivity and constant density. These limitations present no problem when the only desired information is a diffusivity at a particular concentration. In membrane studies it is desirable to stimulate real systems in which there may be large concentration gradients and the diffusivity and density will not, in general, be constant. Hall (1953). Nishyima and Oster (1957), and Secor (1965) represent typical studies where variable diffusion coefficients have been considered. However, the constant density restriction still applies. Duda and Vrentas (1965a, b, 1966) have developed a numerical integration technique that handles diffusivity and density changes. They checked the method using a wedge interferometer (Duda, e t al , 1969) against the sucrose-in-water data of Gosting and Morris (1949) and obtained agreement to within 3%. The Duda and Vrentas method is built on a similarity transformation using both plus and minus infinity boundary conditions. A membrane represents a discontinuity between the boundaries, and the Duda and Vrentas method is not applicable. A data processing technique is developed and used in this work that is a numerical solution of the continuity of mass equation (Bollenbeck, 1972). The method is independent of diffusivity and density changes and needs a flux boundary condition on each side of the membrane in addition to the concentration profiles from the interference fringe patterns. Each side of the membrane is treated separately. The mass fluxes may be calculated at any point in the system and diffusivities may be calculated according to any desired constitutive relationships. The only serious limitation of the method is that it is currently limited to binary mixtures because the concentration profiles are derived from refractive index profiles. For the purpose of testing the modified Rayleigh interInd.
Eng. Chern., Fundarn., Vol. 13, No. 4 , 1974 385
t I
I
I
i I
t
5.
L.
i
s;
t
I
Li
FP
Figure 1. Schematic representation of a modified Rayleigh interferometer for diffusion studies (top view).
ferometer it was desired to find a membrane and binary solution such that the membrane would represent only an area reduction to the diffusion process and where reliable diffusion coefficients were available for the solute in solution. Such a system would provide the simplest possible membrane transport process and simultaneously allow the instrument and data processing technique to be checked by comparing bulk diffusivities with accepted values. A General Electric Nucleopore membrane in combination with sucrose in water solutions meets the requirements. The Nucleopore membranes are made from a polycarbonate film, the pores are essentially cylindrical, pore sizes are available from 0.1 to 8 p , the pore size tolerance is +O to -10% and, as a n added feature, the membranes are about 10 p thick so that the applicability to thin membranes ( e . g , biological membranes) is also shown. There are no significant physical or chemical interactions between the polycarbonate film and sucrose-water solutions; sucrose in water has a molecular diameter of about 10-3 p (Guyton, 1966), and the binary diffusion coefficients for sucrose in water free diffusion have been measured to 0.1’70accuracy by Gosting and Morris (1949). In the absence of physical and chemical interactions between the membrane and solution, and with a concentration difference across the membrane as the only imposed driving force for mass transport through the membrane, any phenomena resulting in a departure from a simple area reduction model for the membrane would be due to geometric factors based on pore, solute, and solvent diameters. Theory and Mathematics of Rayleigh Interferometers A schematic drawing of a Rayleigh interferometer as seen from above is shown in Figure 1. An extended source of monochromatic radiation, SI,illuminates a long vertical slit, SI.The lens, L1, collimates the light from SI.The collimated light is limited by the two long vertical slits,&. This limiting by the slits S Z places the Rayleigh into a broad class of interferometers employing wavefront division. The two beams from Sz then pass through two cells, C1 and C Z . These cells contain fluids with refractive indices nl and n2, respectively. The lens. Lz, focuses the two beams in the horizontal plane to a region of the point, P, of the focal plane, FP. Lens LB is a cylinder with its power plane oriented vertically so that it does not affect the beams in the horizontal plane. The cylinder lens, Ls, combines with the lens Lz in the vertical plane to focus a plane near the back of the cells vertically onto the focal plane. This constitutes a bifocal system in that the focal length is not the same in the vertical and horizontal planes. It is this modification to a bifocal system that allows refractive index gradients to be observed. The interference patterns produced by a Rayleigh inter386
Ind.
Eng. Chern.,
Fundarn., Vol. 13, No. 4 , 1974
ferometer are shown in Figure 2. If the two cells in Figure 1 contain fluids with uniform refractive indices the interference pattern consists of straight, parallel, alternating bright and dark fringes (pattern “a” in Figure 2). If the fluid in one of the two cells contains a refractive index gradient in the vertical direction, and if the cylinder lens is suitably arranged, the straight fringes bend into a shape directly proportional to the refractive index profile (pattern “b” in Figure 2 ) . If the cylinder lens is removed from the system pattern “a” is all that will be observed even in the presence of a refractive index gradient. A complete development of the intensity relationship in the horizontaldirection between two beams of light producing interference patterns is presented by Bollenbeck (1972) and is
For laser light sources the complex degree of coherence, ylz ( e ) is greater than 0.95. The distribution of intensity in the vertical direction on the focal plane depends on the refractive index gradient in the test cell. When the refractive index gradient is zero the intensity is constant along any vertical line in the interference pattern. When there is a finite refractive index gradient in the test cell the intensity varies periodically along a given vertical line with a new maximum (one fringe shift) occurring whenever the difference between the test and reference optical path lengths changes by one wavelength. The variation of intensity along a vertical line in the presence of a refractive index gradient cannot be described by a simple expression such as eq 1, because of deflection of the incident wavefront by the refractive index gradient. Distortion of Wavefronts by Refractive Index Gradients If a collimated wavefront is transmitted through a medium in which there is a refractive index gradient with components normal to the direction of propagation, all points along the wavefront will not proceed a t the same velocity. The wavefront will be distorted and the outward normal vectors will have components in the direction of increasing refractive index. Wavefront deflection phenomena have been discussed and developed by Bollenbeck (1972) for visible light and moderate refractive index gradients. In this case the equation describing a ray (geometric optics) passing through a fractive index gradient is
where s is the arc of the ray, no is the initial refractive index, n is the refractive index, x is the vertical direction, and z is along the optical axis. Equation 2 shows that the ray bends in the direction of increasing refractive index and that the bending effect decreases as the overall refractive index increases. The cylinder lens causes some plane in the region of the cell to be focused vertically on the focal plane. The concept of focus says that all rays from a point on a n object plane are coincident a t a unique point on a corresponding focal or image plane. The paths of some rays passing through the cell are shown in Figure 3. The cell is viewed from the side. The membrane is represented by a horizon-
a
b
1Figure 2. Photographs of interferencepatterns obtained with the laboratory interferometer
still thicker shadow hut less upward displacement. Focusing on plane 3 gives a further increase in thickness but the displacement is now downward. Beach, .et ai. (1973), have described the effect of wavefront deflection in electrode boundary layers. When traversing the fringe pattern in a vertical direction. each. fringe shift. renreaenta nath ~ ~ ...~ ..D . ~ . ....., . ._._..._ ~ 1 - rhinos ~- in ...ontiral -I--.--I,--.leneth difference between the diffusion and reference sides of the cell of one wavelength. If bending is not considered, one fringe shift means that the refractive index of the solution in the diffusion cell changed by
ray 1
~
I IF1 I
Ax = ? L ! 1
(3)
Figure 3. Side view of test cell showing bending of light rays by the refractive index gradient. The dashed lines are my projections to vertical object planes.
where Xo is the wavelength of the light and 1 is the geometric path length through the solution. When bending is considered, the interpretation of the fringe pattern is no longer a simple matter. The optical path length through the solution in the diffusion cell is defined by the arc, s, of the ray and the refractive index along the ray so that
tal solid line and the refractive index is downward. The rays are initially orthogonal to the first glass surface and parallel to the membrane surface. As the rays pass through the solution they are bent downward by the refractive index gradient. The bending is exaggerated for the purpose of illustration. Ray 1 is the first ray above the top surface of the memhrane that passes completely through the cell. Planes 1, 2, and 3 represent three vertical object planes referred to in Rayleigh interferometer literature. If the system is focused on plane 1, the membrane shadow will he thicker than the membrane itself and will be displaced upward. Focusing on plane 2 gives a
Furthermore, the optical path length through the last optical flat, lenses, and air are not the same as for a horizontal ray and, depending on the plane of focus, the final position of the ray on the focal plane may not correspond to the position of initial entry into the cell. In addition since the refractive index gradient is the greatest near the center of the cell and zero at the extremes, the bending effect is a function of position in the cell. There is no loss of information along the bottom surface
P3 I
Ind. Eng. Chem., Fundam.. Vol. 13, NO. 4, 1974 387
of the membrane where the refractive index is increasing away from the membrane. There is a loss of information along the top surface of the membrane due to rays striking the membrane because the refractive index is increasing toward the membrane. If the membrane reflects or transmits any of these rays there may be secondary disturbances in the interference pattern. Correcting the Fringe P a t t e r n for Bending Effects The fringe pattern may be corrected for bending effects if the original entry positions and optical paths of the rays can be determined. In this way the refractive index of the solution a t the initial glass-solution interface can be found for each point along the interface. This would then define the concentration profile for the one-dimensional diffusion process. There are two ways of finding the original entry positions. The first way is to put a series of fine, opaque, horizontal lines along the first glass surface (plane 1 in Figure 3) on the diffusion side of the cell. If plane 1 is focused vertically on the focal plane, any diffraction due to the lines will be focused out of the system, so that the lines will be well-defined on the focal plane. These lines may then be used to reestablish the original vertical scale and thus locate the original entry points for the rays. The second way is to find a n object plane such that all rays are focused vertically to points corresponding to their original entry points. Deflection effects are greatest near the membrane where the refractive index gradient is the largest. A unique object plane does not exist, but since the deflection is the greatest near the membrane, the error due to deflection decreases for distances away from the membrane. Such a plane is defined by the intersection of the membrane and the dashed line for ray 3 in Figure 3. Locating the correct object plane for vertical focusing was the approach selected for this work since refocusing the system was the only required change.This was investigated by simulating a n ideal free diffusion process for a sucrose-water solution with an initial concentration difference of 1 wt % for the sucrose. The simulated cell had the same physical parameters as the cell used in the experimental work. Rays such as ray 3 in Figure 3 were traced through the cell by integrating eq 2 with a fourthorder correct Runge-Kutta technique and by using Snell’s law of refraction a t the optical flat interfaces. The results are summarized in Table I. The membrane was located a t x equal to zero. The values of z for x equal to zero represent the intersection of the dashed line through ray 3 in Figure 3 with the membrane. The results were obtained for both the real arc and for a straight line approximation. The results show t h a t a n acceptable location for the vertical focus is established in the first 3-4 min and that the straight line approximation is also acceptable. The location of the correct object plane is slightly to the left of the last solution-glass interface in Figure 3. It must be remembered that the location is dependent on the particular cell. If the last optical flat were thicker, the plane would move t o the right in Figure 3. If it were thinner the plane would move to the left. Iterative Technique for Removing Bending E r r o r s The information obtained by microscopically measuring a fringe pattern is a set of fringe numbers (referenced to the extreme boundary of the ringe pattern) and locations. That is
{Fi9 xi>
(5 1
where F1 is the number of the ith fringe and x l is the loca388
Ind. Eng. Chem., Fundam., Vol. 13, No. 4, 1 9 7 4
Table I. Results of Computer Simulation to Verify Existence of a Single Plane of Focus for Refocusing Deflected Rays Location of plane, cm
Deflection, cm Time, sec
Arc
St. line
Arc
St. line
60 120 180 240 300 600 900 1,200 1.500 1,800 2,100 2,400 2,700 3,000 3,300 3,600 7,200 14,400
-0.0256 -0.0182 -0.0149 -0.0129 -0.0116 -0.0082 -0.0067 -0.0058 -0.0052 -0.0047 -0.0044 -0.0041 -0.0039 -0.0037 -0.0035 -0.0033 -0.0024 -0.0017
-0.0258 -0.0183 -0.0149 -0.0129 -0.0116 -0.0082 -0.0067 -0.0058 -0.0052
0.963 1 0.9653 0.9656 0.9658 0.9658 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659
0.9661 0.9660 0.9660 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659 0.9659
-0.0047
-0.0044 -0,0041 -0.0039 -0.0037 -0.0035 -0.0033 -0.0024 -0.0017
tion in the fringe pattern. Since the optical system is focused so t h a t deflected rays are focused to their undeflected positions, the original entry points of the rays producing a given fringe are known so that the datum is
where xoL is the original entry point of the rays producing the ith fringe. A fringe shift in the interference pattern means that the optical path length difference between the reference and diffusion sides has changed by one wavelength. The optical path length difference relationship between two fringes along a line orthogonal to the membrane is (LDIFF,iii
- LREF,i+l)
-
(LDIFF,i
-
LREF,i)
= *IO
(7) where LDIFPand LREFare the diffusion and reference side optical path lengths, respectively. The optical path lengths are measured from the first solution-glass interface in the cell to the focal plane. The sign of XO depends on the refractive index change in the diffusion cell bel. The sign of A0 is negative tween fringe i and fringe i when the fringes are counted positive upward and the refractive index increases downward. Let F , be the fringe number for the ith fringe shift relative to the base (0th fringe shift). Equation 7 then gives the ith fringe path lengths in terms of the 0th fringe path lengths as
+
(LDIFF,i
- h E F , i ) - (LDIFF,o
-
LREF.0)
= *Fi,Xo
(8) The 0th fringe path lengths can be found by tracing without accounting for bending because the 0th fringe is a t the pattern boundary where there are no refractive index gradients (the boundary refractive index is a n experimental parameter).The diffusion path length for the ith fringe shift is found by rearranging eq 8 to give LDIFF,i
=
-t (LDIFF,O
-
LREF,O)
+ LREF,i
(9 )
Table 11. Computer Simulation Results Demonstrating Convergence of Iterative Scheme for Bending Corrections P e r cent e r r o r in derivative of r e f r a c t i v e index remaining after indicated iteration
Position in cell, cm
0
1
2
3
-0.0010 -0.0080 -0.0170 -0.0260 -0.0350 -0.0450 -0.0550 -0.0660 -0.0770 -0.0900 -0.1040 -0.1210 -0.1430 -0.1750
-3.512 -3.509 -3.496 -3.473 -3.441 -3.394 -3.335 -3.257 -3.165 -3.037 -2.877 -2.652 -2.308 - 1.704
0.103 0.104 0.105 0.106 0.108 0.112 0.116 0.121 0.128 0.137 0.148 0.164 0.188 0.230
-0.041 -0.041 - 0.042 - 0.043 -0.044 - 0.045 - 0.047 - 0.049 -0.052 - 0.056 -0.061 -0.068 -0.079 -0.098
0.000 0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.007 0.009 0.013
side
front
Figure 4. The glass diffusion cell. ‘2 ( a r d bottom)
-6
access ?art
cla-mping s c r e w s I
,
..
scale rm+
--....- .. .._ -...
y front
--...
-....
The data represented by eq 6 may be transformed using eq 9 to give
That is, the fringe pattern data is actually a set of optical path length-origin point pairs. A given fringe pattern is corrected for bending when the refractive index profile is found that produced the data pairs in expression 10. An iterative algorithm for correction is presented by Bollenbeck (1972). The results of a computer simulation to test the iterative bending correction technique are presented in Table 11. The results show that the error due to bending was effectively removed from the refractive index gradient in two iterations.
Diffusion Cell A completely new type of interferometer diffusion cell was designed so that diffusion through membranes could be observed. The diffusion cell had to hold the membrane perfectly flat with its surfaces parallel to the collimated light rays and normal to the optical flats comprising the cell windows behind the masking slits. There had to be a way of sealing the membrane into the cell to prevent leaks. The desire to observe interference fringes right up to the surfaces of the membrane eliminated conventional gasketing techniques. Furthermore, minimizing the number of separate pieces would minimize the amount of alignment necessary whenever a new membrane was mounted. Figure 4 is a drawing of the interferometer membrane diffusion cell designed and built for this study. The cell consists of a sandwich of three fused quartz optical flats (flat to 5 2 t h wavelength). The center flat had slots cut in it and then the sandwich was joined with a special lens bounding cement that maintained the lhzth wavelength tolerance. The slots in the center flat form the reference and diffusion compartments of the cell. The sandwich was then cut in half (normal to the glass surfaces) to form an interface for a membrane. The two cut surfaces were then ground ( 5 p compound) to one another so that they mated perfectly and were perfectly orthogonal to the surfaces of
\ shimmirg screws
Figure 5. Diffusion cell cage and masking slit assembly.
the optical flats. Ports were drilled in the ends of the fluid compartments for taking fluids in and out of the cell. The excellent glass fabrication work was done by the Rocky Mountain Instrument Co. in Longmont, Colo. A membrane is placed on the ground-glass interface of one half of the cell with a very thin layer of grease for sealing purposes and the other half of the cell is put in position, sandwiching the membrane in bewteen. Since all other surfaces are bonded, no further alignment or sealing is required. The design parameters for the cell were based on both physical and optical considerations. The optical flats needed to be about 1 cm thick to provide rigidity and avoid distortion due to physical stresses. The 1-cm light path through the fluid compartments was arrived a t after considering the desired sensitivity to sucrose concentration changes. The change in refractive index equivalent to one fringe shift without wavefront deflection is given by eq 4. Figure 5 is a drawing of a cage that holds the two halves of the cell together and provides a means of attaching the cell to the optical bench mount. Adjustable masking slits are on the front surface of the cage and the small screws are used to align the cell within the cage so that the membrane interface is orthogonal to the masking slits. The cage is made of several detachable pieces so that it can be literally built-up around the glass cell. The cage attaches to the bench mount with 1/8-in. machine screws and alignment pins so that the cage may be removed and replaced without destroying the basic cell alignment within the optical system. Ind. Eng. Chem., Fundam., Vol. 13, No. 4 , 1974
389
Experimental D a t a Three runs were made using sucrose-water solutions of 0-1, 2-3, and 4-5% by weight of sucrose. This range of solutions covers the range of the Gosting and Morris data. The same membrane was used in all three runs so that variation between membranes would not be a factor.The refractive index as a function of concentration was found by measuring refractive indices of solutions with a Pulfrich refractometer a t a wavelength of 0.633 p. The results correlated well to a linear dependency on molar sucrose concentration. Values of density as a function of composition were taken from the work of Browne (1941). Again, a good linear correlation was obtained (correlation coefficient = 1.00000). The composition and refractive indices of the sucrosewater solutions used in three experiments performed with the 0.4-p pore size membrane are given in Table 111. The experimental runs were made a t 25 f 0.1"C. The optical system was focused in the vertical plane to compensate for wavefront deflection by the refractive index gradient. As a result, no vertical movement of the membrane shadow on the diffusion side of the cell was observed during the experimental runs. This means that the iterative scheme developed by Bollenbeck (1972) for removing the effect of wavefront deflection may be used with this data. Typical fringe position data taken from the fringe pattern photographs is given in Table IV. Note that the distances are measured a t the focal plane.That is, the vertical magnification of the optical system is not yet accounted for. Dividing each distance by 0.4631 gives the fringe location relative to the cell. Figure 6 shows fringe numbers when plotted as a function of position. The fringe profiles must be checked and corrected for deflection errors before molar fluxes and diffusivities are calculated. The iterative scheme for removing deflection errors requires that the apparent refractive index and gradient be calculated a t each iteration. This may be done individually for each fringe or may be found from a suitable correlation (or set of correlations) giving the apparent refractive index as a function of position in the fringe pattern. A correlation is preferred because small errors in measuring the fringe positions are magnified when derivatives are taken. A correlation would smooth out these irregularities. A sigmoid correlation is desirable in the extremes of the profiles because the sigmoid function possesses the necessary asymptotic properties. The error function solution to the ideal diffusion equation (Crank, 1957) is cb,
t)
=
1
Co erfc2 2
X
m
where c is the concentration scaled from zero to one, x is the diffusion coordinate, t is the time, D is the binary diffusion coefficient, and erfc is the complemented error function. Equation 11 implies the existence of a sigmoid function of refractive index in terms of the distance parameter x of the form
n = m erf(Ax)
+
b
(12)
That is, the refractive index is linear in terms of erf(Ax). By fitting the data to the error function form we are smoothing the data over a limited range of distance and/ or time. Mass Flux and Diffusivity Calculations The usual computational techniques for finding diffusion coefficients from concentration profile data involve 390
membrane
24
Ind. Eng. Chem., Fundam., Vol. 13, No. 4,
1974
900 sec 02460 sec 5160 sec o
u
si
:I,
c 3
,
,
,
,
, &I .
-24 -20
-T
l
h
-12 -8 -4 0 4 8 12 16 bottorr * top distance frorr membrane, cm X102 (at focal plare)
20
24
Figure ti. Run 1 fringe profiles.
Table 111. Initial Sucrose in Water Concentrations and Refractive Indices of the Solutions Used in the Three Experiments with the 0.4-11 Pore Size Membrane Sucrose concn, mol/cc (refractive index) Run no.
Top of cell
Bottom of cell
1
0 (1.331313) 6.433 x (1.334464) 1.244 x (1.337407)
2.988 x (1.332775) 9.457 x 10-5 (1.335946) 1.539 x (1.338852)
2 3
the use of some similarity variable whether or not the diffusion coefficient and/or density are constant over the profiles. Further, these methods require continuous profiles over the entire distance between boundaries. The presence of a membrane gives rise to a discontinuity in the profile so that the methods based on similarity solutions to diffusion equations no longer apply. Moreover, the information needed in membrane studies is the flux and concentration a t a membrane surface. A method was developed in this work that solves the continuity of mass equation numerically to obtain mass fluxes. This method has the added advantage that no constitutive relationships need be assumed. The method is quite simple. Consider the mass flux of component A, Nq, in the x direciion, and along a uniform cross sectional area. The unsteady-state mass balance between any xg and x positions is for a continuous concentration profile, and when there are no sources or sinks of component A between xg and x
Once the concentration profiles are known for several different times and a reference flux is known at x g , the flux a t any other x is found with eq 13. Note that the data are first integrated (a smoothing process) and then differentiated ( a noise producing process). If diffusivities are to be determined the concentration gradient must be found by yet another differentiation of the data. The reference flux, N N I ~is~only , known a t the bottom limit of the cell where it is zero as long as the concentration gradient is zero. There will in general be a nonzero flux a t the top of the cell even in the absence of a concentration gradient because of density changes caused by the diffusion process in the middle portion of the cell. This means that the flux at the bottom surface of the mem-
Table IV. Run 1Fringe Positions Measured a t the Focal Plane for Six Times F r i n g e location relative t o the membrane, c m ~~
Fringe no. 1 2 3 4 5 6 7 8 9 10
900
1620
2460
3660
sec
sec
sec
sec
5 160 sec
0.0742 0.05765 0.0465 0.0384 0.03 13 0.02475 0.01905 0.01335 0.0083
0.0972 0.0758 0.0612 0.05035 0.04085 0.0325 0.0249 0.01745 0.0107
0.1245 0.09655 0.07815 0.0643 0.05255 0.04245 0.03275 0.0235 0.01475 0.00665
0.1516 0.1168 0.096 15 0.0788 0.0651 0.052 0.0401 0.0287 0.0185 0.0081
0.192 0.1473 0.11865 0.0986 0.0816 0.06645 0.0516 0.03 86 0.0254 0.0135
-0.00515 -0.0102 -0.01545 -0.02075 -0.02695 -0.033 1 -0.0402 -0.04 845 -0.0587 -0.0748
-0.0075 -0.0145 -0.02 12 5 - 0.02885 - 0.03695 - 0.0448 -0.0546 - 0.066 - 0.0803 -0.1000
-0.00995 -0.0186 -0.02715 -0.0362 - 0.04 58 5 -0.05695 -0.06775 - 0.0823 5 -0.0997 -0.1279
- 0.01 145 -0.0217 -0.03255 -0.04405 -0.05565 -0.0685 -0.08295 -0.10035 -0.12105 -0.1536
-0.00575 -0.016 -0.02865 - 0.04075 -0.0544 -0.06885 -0.0838 -0.1017 -0.1214 -0.14815 -0.18695
7560
sec 0.2223 0.17395 0.1415 0.11645 0.0959 0.07825 0.06125 0.0453 0.03045 0.015665
Membrane 13 14 15 16 17 18 19 20 21 22 23
brane is easily found but the flux a t the top surface may only be found by integrating eq 13 across the membrane or by independently measuring the mass velocity a t the upper limit of the cell. The membrane used in this work is only 14 p thick so that the flux is assumed the same a t both surfaces. The integral of eq 13 is found from the fitting eq 12 as
CA dx =
cy
A
[Ax erf(Ax)
+
1
ze"Ax'
2
SCdx =
G x - Co2 2
x erf-
X
m
-0.00745 -0.02235 -0.03685 -0.05155 -0.06785 -0.08595 -0.10425 -0.12495 -0.1503 -0.1828 -0.23275
-
Equation 17 suggests the functional form
]
Since the molar concentrations of water and sucrose are linearly related over the range of compositions used in this work, the molar water concentration, CH, may be expressed in terms of the molar sucrose concentration by
Then, the water flux is
The time derivative in eq 13 may be evaluated from a suitable correlation,by finite differences, or graphically. The correlation approach is preferred because it has the advantage of smoothing out small irregularities in the data before the derivative is evaluated. The error function relationship given in eq 11 was successful in representing the concentration profiles individually a t particular times and it seemed that a useful functional form might be derived from it for correlating the profile integrals as a function of time. If eq 11 is integrated with respect to x the result is
The profile integrals were correlated to eq 18 by using a single parameter search to find the value of B that gave the best overall fit in terms of a, b, and e. Five profile integrals representing five different times were used for each correlation in order to have more points than parameters. The time derivative could then be evaluated for the central time. The correlations were limited to spacial positions between the membrane and the last fringe position for a given set of profiles. The molar flux of sucrose a t the bottom surface of the membrane is then found by taking the time derivative of the profile integral correlating function, eq 18, and substituting the result into eq 11 to give
The molar flux of water is found from eq 16. The calculated molar fluxes for run 1 are given in Table V. Validity of Results in Terms of Diffusivities A value for the molar flux through the membrane along with the concentrations a t the membrane surfaces is the desired membrane transport data. Before proceeding with more membrane calculations the reliability of the calculated fluxes must be checked. This is best accomplished by using the fluxes and profile gradients to calculate binaInd. Eng. Chern., Fundarn., Vol. 13,N o . 4 , 1974
391
Table VI. Run 1 Binary Diffusion Coefficients Computed
Table V. Run 1 Molar Fluxes at 2460 Sec for the Bottom Half of the Cell. A Positive Flux Is Upward
Both with and without the Bulk Flow Contribution to Flux Binary diffusivity @ 25"C, c m 2 / s e c
FIW, mol/cm2-sec x loio Cell position, c m
0 (Membrane)
-0.0202 -0.0404 -0.0606 -0.0807 -0.1009 -0.1211 -0.1413 -0.1615
Sucrose
3.828 3.789 3.692 3.540 3.342 3.105 2.839 2.563 2.292
-44.74 -44.28 -43.14 -41.38 -39.06 -36.29 -33.18 -29.95 -26.78
ry diffusion coefficients and then checking the coefficients against accepted values. The diffusion coefficients obtained from the flux and profile data for run 1 are given in Table VI. The diffusion coefficient was calculated both with and without the bulk motion contribution as a means of checking the errors involved when bulk motion is neglected. The diffusion coefficient for the sucrose-water system should decrease with increasing sucrose concentration (Gosting and Morris, 1949). This trend is present in all three runs with the exception of the last three values in Table VI. These three values are in the tails of the concentration profiles where the fringe shifts are widely spaced. This means that the profile correlations would most likely be favorable to the region near the membrane where there are more data points. The fluxes are calculated using profile integrals so that errors in the flux should be reasonably uniform over the entire profile. However, taking the derivative of a single profile to find the concentration gradient is probably not as accurate away from the central region as it is in the central region. A slightly inaccurate value for the concentration gradient probably accounts for the error in the last three diffusivities in Table VI. Typical diffusion coefficients from the three runs are compared with the Gosting and Morris data in Table VII. The diffusivities from the three runs lie on either side of the Gosting and Morris data indicating that there are no consistent differences. The magnitude of the differences increases with increasing sucrose concentration. The bulk flow contribution also increases with increasing sucrose concentration due to the increasing mole fraction of sucrose. If bulk flow were not considered, the diffusivities from runs 1 and 3 would compare more favorably with Gosting and Morris but run 2 would compare less favorably. Duda and Vrentas (1966) developed a numerical ,technique for nonideal free diffusion and obtained agreement with Gosting and Morris to within 3%. It seems then that the differences in Table VI1 must be attributed to the more complex data analysis plus some possible unknown experimental errors. The method developed here for use with membranes is within the 3% accuracy of the Duda and Vrentas method for free diffusion. Membrane Calculations As pointed out in the Introduction, the Nucleopore membrane should represent only an area reduction to diffusion provided that there is no observable membrane warpage due to osmotic effects. The flux through the membrane has already been found and no warpage has been observed. The concentration a t the surfaces is available from the profile correlations. The flux through the 392
Ind. Eng. Chern., Fundarn., Vol. 13, No. 4,1974
x 106
Water
%
Sucrose concn, mol/cc x lo5
No bulk
Bulk
Difference
1.256 1.402 1.546 1.685 1.818 1.942 2.056 2.160 2.253
5.260 5.247 5.237 5.230 5.225 5.223 5.223 5.239 5.293
5.274 5.262 5.254 5.249 5.245 5.244 5.245 5.263 5.318
- 0.27
-0.28 -0.32 -0.36 -0.38 -0.40 -0.42 -0.45 -0.47
Table VII. Comparison of Diffusion Coefficients from These Data with the Data of Gosting and Morris (1949) Diffusion coefficient, c m 2 / s e c x lo6
Run 1 Run 2 Run 3
%
Sucrose concn, mol/cc x lo5
Data
Costing
Difference
1.402 8.136 14.12
5.262 4.971 4.991
5.215 5.035 4.880
0.90 -1.27 2.27
C
.
2
3
4
5
6
7
8
time, seconds X c 3
Figure 7. Run 1concentration differences across the membrane.
membrane may then be regarded as a simple diffusion phenomenon and written as
where Af is the effective free area for diffusion. The concentration differences across the membrane are quite small (10-5in refractive index) so that the diffusivity and total concentration may be considered constant through the membrane. Small errors in fringe pattern reading show a t this point and it is necessary to time-smooth the concentration differences to obtain the best value to use in eq 20. A relationship for free diffusion with an interfacial resistance presented by Crank (1957)indicates that the concentration difference across an interface separating two ideal diffusion systems should decay exponentially with time. That is In AC = at
+
b
(21)
A typical plot of the log of the concentration differences as a function of time is shown in Figure 7 (the straight
Table VIII. Free Diffusion Area Results for the 0.4-p Pore Diameter Nuclepore Membrane (Membrane Thickness = 0.0014 cm) Run no.
Sucrose flux, mol/cm2 -sec
1 2 3
3.83 x lo-'' 3.36 x lo-'' 3.76 x 10-l'
Diffusivity, cm2/sec 5.27 x 4.98 x 4.98 x
Ac9
mol/cc 1.84 x 1.71 x 1.94 x
A f,
cm2/cm2 0.0557
0.0559 0.0560
line is the regression line for eq 21). The free areas calculated with eq 20 are in Table VIII. The three free area values agree to within 0.5%. A series of water flow measurements made by Mickley (1972) gave a free area of 0.085 cm2/cm2. This value is based on assuming laminar flow through smooth 0.4-p diameter tubes. Since there are more restrictive geometry assumptions in :he water flow technique and the diffusion results are extremely consistent, we prefer the interferometric technique for membrane transport evaluation.
Conclusions This paper has presented information necessary for the design of a Rayleigh interferometer which can be used in the analysis of liquid phase diffusional studies and mass transport studies across thin porous media such as membranes. A technique is developed for correcting the interference fringe pattern for light bending effects due to concentration gradients in the system. A new diffusion cell is presented for use in membrane transport studies. A method is presented for directly calculating concentration and flux data from interference fringes when a discontinuity such as caused by a membrane is present in the system. The validity of this method was checked by comparing bulk phase diffusivities obtained in this work to that of accepted values. The diffusivities agreed to within 3%. By using flux and concentration data obtained a t the membrane surface, a 0.4-p diameter Nucleopore membrane was shown to exhibit diffusional behavior for sucrose-water transport over a sucrose concentration range of 1 x to 14 x mol/cm3. The free area for transport was 5.6%. Nomenclature a = slit width C = concentration, M D = diffusivity, cm2/sec
D , = membrane diffusion coefficient f = focal length of lens L, Fi = number of ith fringe I = intensity 1 = geometric path length through diffusion cell solution L = pathlength n = refractiveindex N A = molar flux of species A, mol/cm2 sec R = gasconstant s = arc of a ray of light t = time, sec T = absolute temperature x = vertical direction, cm XA = mole fraction of A y = displacement from optical axis z = distance along optical axis Greek Letters a = phase difference between two light beams y = complex degree of coherence X = wavelength of light A = osmotic pressure u = reflection coefficient
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Receiced f o r reuzeu October 23,1973 Accepted June 28, 1974
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