Transport through polymeric membranes - ACS Publications

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The Journal of Physical Chemistry, Vol. 83, No. 77, 1979

M. C. Collins and W. F. Ramirez

(9) B. Valeur and J. Moirez, J. Chlm. Phys., 70, 500 (1973); J. Loeb and G. Cahen, Automatisme, 3, 479 (1963). (10) U. Wild, A. Holzwarth, and H. P. Good,Rev. Sci. Instrum., 48, 1621 11977). (11) D. G. brdner, J. C. Gardner, G. Laush, and W. W. Meinke, J. Chem. Phys., 31, 978 (1959). (12) M. La1 and E. Moore, int. J . Num. Methods Eng., 10, 979 (1976). (13) A. E. McKinnon, A. G. Szabo, and D. R. Miller, J: Phys. Cheh., 81, 1564 (1977). (14) (a) M. H. Hui and W. R. Ware, J. Am. Chem. Soc., 98, 4712 (1976); (b) J. C. Andre, D. O'Connor, and W. R. Ware, J . Phys. Chem., 83, 1333 (1979). (15) T. Nishikawa and K. Someno, Anal. Chem., 47, 1290 (1975). (16) K. W. K. Yim, T. C. Miller, and L. R. Faulkner, Anal. Chem., 49, 2069 (1977). (17) T. A. Maldacker, J. E. Davis, and L. B. Rodgers, Anal. Chem., 46, 637 (1974). (18) H. P. Larson and U. Fink, Appl. Spectrosc., 31, 386 (1977). (19) Y. G. Biraud, Astron. Astrophys., 1, 124 (1969). (20) S. W. Provencher, J. Chem. Phys., 64, 2772 (1976).

(21) S. W. Provencher, Biophys. J., 16, 27 (1976). (22) J. Schlesinger, Nucl. Instrum. Methods, 106, 503 (1973). (23) R. Bracewell, "The Fourier Transform and Its Applications", McGraw-Hill, New York, 1965. (24) J. W. Cooley and J. W. Tukey, Math. Comput., 19, 297 (1965). (25) W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, W. W. Lang, G. C. Maling, Jr., D. E. Nelson, C. M. Rader, and P. D. Welch, proc. rfEE, 55, 1664 (1967). (26) B. R. Hunt, I€€€ Trans. Audio Electroacoust., No. AU-20, 94 (1972). (27) H. F. Silverman and A. E. Pearson, If€€ Trans. Audio Electroacoust., NO. AU-21, 112 (1973). (28) F. J. Harris, Proc. I€€€, 66, 5 1 (1978). (29) D. V. O'Connor and W. R. Ware, J . Am. Chem. Soc., 98, 4706 (1976). (30) J. C. Andre, M. Niclause, and W. R. Ware, Chem. Phys., 28, 371 (1978). (31) J. C. Andre, M. Bouchy, and W. R. Ware, Chem. Phys., 37, 107, 118 (1979). (32) J. C. Andre, R. Lopez-Delgado, R. Lyke, and W. R. Ware, Appl. Opt., in press.

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Mass Transport through Polymeric Membranes Michael C. Collins and W. Fred Ramirez* Deparlment of Chemical Engineering, University of Colorado, Boulder, Colorado 80309 (Received November 3, 1977) Publication costs assisted by the University of Colorado

Mass transport through three polymeric membranes (Cuprophan, poly(acrylonitrile),and a poly(acry1onitrile) membrane with an adsorbed protein layer) is studied. A series of eight compounds ranging in molecular weight from 60 (urea) to 1355 (vitamin BIZ)were investigated. A complete set of transport properties are reported including the permeability coefficient, sieving coefficient, pressure-filtrationcoefficient, and frictionalcoefficients representing the interactions of solute-membrane, solute-solvent, and solvent-membrane. Solute permeability for diffusive transport correlates well for all membranes studied with solute molal volume except for sulfobromophthalein (BSP). Because of strong membrane-solute frictional interactions, BSP has a reduced permeability. The sieving coefficient, which characterizesconvective transport, also correlated with solute molal volume except for BSP and the lipophilic compound thiopental, The reduced convective transport of BSP is again due to its high frictional interactions with the membranes. The sieving coefficient for thiopental is markedly reduced although its permeability was normal. The membrane tortuosity factors for thiopental were also reduced. These facts indicate the existence of multiple diffusive pathways for thiopental and its exclusion from some solvent pathways in the membranes studied.

Introduction The purpose of this study is to evaluate the mechanisms of mass transport through polymeric membranes. Membranes are important separation devices extensively used in application such as renal dialysis and reverse osmosis. Although the theory of membrane transport is well established, there are few experimental studies of membrane transport mechanisms. The work of Kauffman and Leonard7y8and Ginzberg and Katchalsky6 are notable exceptions. In order to test theoretical concepts a wide range of solutes and two membranes often used in renal dialysis were chosen for study. Membranes are usually characterized by the following three parameters of the solute: permeability coefficient, w , the pressure filtration coefficient, L,, and the reflection coefficient, u. Often the sieving coefficient, S = 1 - u, is used instead of the reflection coefficient. The flux equations written in terms of these coefficients are (Keedem and K a t ~ h a l s k y ) ~ N , = wRTAC, + (1 - g)C8Jv (1) Jv = L,(AP - uRTAC,) (2)

flows. However, these three practical parameters provide little information about the detailed mechanisms of membrane transport. Spieglerlg proposed a frictional model to use with the principles of irreversible thermodynamics to give physical meaning to the phenomenological coefficients. He used the law of friction that the frictional force that hinders the motion of one object is proportional to the relative velocity between two objects (3) F , = -fij(ui - Uj) where f i j is the frictional coefficient between the ith and jth species. Kedem and Katchalsky'O related the frictional coefficients to the measurable macroscopic membrane transport parameters of permeability, filtration, and sieving by

(4)

where N , is the solute flux and Jv the total volume flux. These three coefficients of permeability, pressure filtration, and sieving can all be determined experimentally and suffice to characterize membrane transport at low volume 0022-3654/79/2083-2284$0l .OO/O

0 1879 American Chemical Society

Mass Transport through Polymeric Membranes

The Journal of Physical Chemistry, Vob 83, No. 17, 7979 2295

TABLE I: Solutes Investigated solute

-

mol wt, daltons

V,a cm3/gmol

60.1 113.1 180.16 23 2.2 242.4 342.3 794.09 1355.42

58 117 195 238.2 286 364.5 535.1 1451

2) x 1 0 j a t d,& A 87 O C l C cm2/s

_ _ I _

urea (carbonyldiamide) creatinine ( 2-imino-l-rnethyI-4-imidezolidinone) glucose phenobarbital (5-ethyl-5-phenylbarbituricacid) thiopental (5-ethyl-5-(l-methylbutyl)-2-thiobarbituricacid) sucrose sulfobromophthalein (BSP) (phenoltetrabrornophthalein sulfonate) vitamin B,, (cyanocobalamine)


a

Method of LeBas.I3

Assuming spherical geometry, Farrell and Babb.4

The frictional interaction between 1 mol of solute and the solvent molecules surrounding it in the membrane is given by f,. The coefficient f, refers to the interaction of 1mol of solute and the membrane matrix in its vicinity. The interaction between 1 mol of solvent and the membrane in its vicinity is given by fm The magnitudes of the three frictional coefficients for various systems provides an understanding of the physical transport processes taking place in the membrane. Experimental Section Two membranes were investigated for their mass transfer properties. Cuprophan PM200, manufactured by Enka-Glanzstaff of West Germany, and RPAN 69 (PAN), manufactured by Rhone-Poulenc of France. The cellulosic cuprophane membrane is currently used in the majority of hemodialyzers. The RPAN-69 membrane is a copolymer of acrylonitrile and sodium methallyl sulfonate. It is currently being considered for use in renal dialysis. The membranes contain a number of impurities as a result of their manufacturing processes. Before any tests were performed, the membranes were washed with several changes of isotonic saline over a period of several days. Mass transfer evaluations were also performed on membranes which have been treated with protein in order to mimic the condition of a membrane in a clinical dialysis setting. The Cuprophan and RPAN membranes were treated with protein by soaking them in a 5 g YO bovine albumin-saline solution a t 37 "C for 24 h. Eight solutes were tested for their permeability characteristics. These solutes are listed in Table I, along with their molecular weight, molal volume, molecular diameter, and diffusivity at 37 "C. All the solutes used were of reagent grade. Solute molal volumes were calculated according to the method of LeBas13 as given in the Perry's handb00k.l~ Since all species were under 2000 molecular weight a spherical shape (Farrell)4 was used to compute the molecular diameters. The solute diffusivities were either obtained from Colton2 or calculated by use of the Wilke-Chang equation.20 Solute concentrations of 5 X M in isotonic saline solutions were used in all the studies except for thiopental M in which the concentration was approximately 1 X as a result of its low solubility. Concentration measurements were determined by liquid scintillation spectroscopy with isotope labeled compounds. Thiopental and sulfobromophthalein were labeled with sulfur-35, vitamin B12 with tritium, and the remainder of the solutes with carbon-14. All of the radioisotopes were obtained from the Amersham/Searle Corp. The use of radioisotopes enables the use of low solute concentrations and very high accuracy in measurements. When samples from an aqueous environment are counted the samples must be solubilized before they can be dispersed in the hydrocarbon solvent. Eastman Ready-to-Use11 scintil-

5.69 7.1.9 8.52 9.11 9.68 10.49 11.93 16.63

1.81 1.29 0.955 0.850 0.759 0.697 0.521 0.379

Data of Colton* or Willre-Chang correlation.20

TABLE 11: Membrane Distribution Coefficients at 37 "C with Cuprophan, PAN, and PAN-Protein Membranes solute Cuprophan PAN PAN-protein 0.821 i 0.131 urea 0.962 t 0.036 0.74 t 0.14 creatinine 0.883 t 0.016 0.991 f. 0.15 0.868 I0.043 0.989 t 0.222 glucose 0.969 f 0.025 0.997 z 0.1 0.661 i 0.175 3.192 i 0.12 4.341 i 0.394 Dhenobarbital thiopental 0.840 I 0.176 4.341 t 0.64 6.647 t 0.16 sucrose 0.911 i 0.035 0.786 i 0 - 1 1 0.901 4 0.097 2.5 5 0.2 7.442 t 0.545 1.07 ?r 0.2 BSP vitamin Bl2 0.738 i 0.021 0.828 t 0.11 0.873 i 0.07 TABLE 111: Ultrafiltration, Water Content, and Thickness Characteristics of the Cuprophan, PAN, and PAN-Protein Membranes membrane Cuprophan PAN PAN-protein

l o 5L,, cm min-l mmHg-' 0.428 5.621 2.365

I. t ?r

0.015 0.127 0.058

$ J ~vol , %

0.652 0.573 0.522

i

0.029

i

0.005

* 0.03

AX, cm

0.00299 0.00309 0.00309

lation cocktail was used. Each 500 & of sample was mixed with 10 mL of cocktail for counting. A Beckman LS-100C scintillation counter was used. Eppendorf pipets were employed for accurate sampling. The solute distribution coefficient is defined as the ratio of the equilibrium solute membrane concentration to the free solution concentration

K = C,"/C,"

(7)

Experiments to determine this coefficient were carried out in 100-cm3polypropylene containers with tight fitting caps. A 25 cms solution was placed into the container and allowed to equilibrate a t 37 "C for 24 h. A swollen membrane having a known surface area of approximately 800 cm2and a known wet volume of approximately 2.5 cm3 was placed in the solution. Before placing the membrane in solution all the surface water was removed without allowing drainage from the interstitial spaces to occur, and the membrane-solution was equilibrated for 48 h at 37 "C. The container was shaken periodically. From a measurement of initial and final solute concentrations of the bathing solution, the distribution coefficient was determined from a mass balance. Three determinations were made for each solute-membrane combination. The distribution coefficient was also checked by a desorption experiment in which a solution containing no solute was allowed to equilibrate with the membrane of known solute concentration from a previous experiment. The results are presented in Table 11. Both average values and one standard deviation values are given. The dry and wet membrane thicknesses were determined with a Leitz microscope at 400X with a grid calibrated to 1.25 wm divisions. The water content of the membrane was determined by drying a water swollen

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The Journal of Physical Chemistry, Vol. 83, No. 17, 1979


membrane of known volume in an oven at 100 "C. Both wet membrane thickness, A X , and water content, &, results are given in Table 111. Ten determinations of the water content were done for each type of membrane. etermination of Membrane Permeability The dual closed loop technique was used to evaluate the membrane permeability to a given solute. The flow rates in both loops were adjusted to 1000 cm3/min at which point the fluid resistance to mass transfer was negligible (Collins).l A solution of known volume containing the solute for which the permeability was to be evaluated was recirculated through one side of the permeability test cell. A dilute solution was recirculated through the other compartment of the test cell. At the start of a run, 500-pL samples were taken from both compartments after allowing 10 min for equilibration. Samples were again taken from both solutions at the end of the run. This allows for a computation of a mass balance and a determination of the accuracy of the experiment. The membrane permeability coefficient was obtained from the equation for well mixed chambers:

where V is the volume of each loop, A is the membrane surface area, NT is the total solute mass of the system, CD is the dilute loop concentration, and t is the time of the run. The subscripts i and f refer to the initial and final conditions, respectively. The volume of each loop was 550 cm3which consisted of 500 cm3 in the reservoir and 50 cm3 in the tubing. The system was operated at zero mean transmembrane pressure. The volumes of both loops were checked a t the end of each run to determine if any significant ultrafiltration occurred. The permeability test cell and the fluid reservoirs were maintained at a constant temperature in a 45-L water bath controlled to h0.05 "C with a Brinkmann IC-2 heater-stirrer. A dual-head Cole-Parmer tubing exclusion pump with a variable speed drive was used to recirculate the solutions. The sample time for each run varied from 10 to 30 min. A permeability test cell designed by Laug and Stokesberryl* was used in the studies. This cell was developed by the National Bureau of Standards to test hemodialysis membranes. It consists of two identical plastic plates measuring 4 in. X 6 in. X 0.75 in. The two plates are aligned by guide pins and clamped together by ten stainless steel screws and wingnuts, holding a single sheet of membrane material. Each plate has a 2 in. X 4 in. X 0.050 in. channel surrounded by a small sealing lip. The membrane surface area is 51.6 cm2. The channel contains a plastic mesh, the main function of which is to promote fluid turbulence. Inlet and outlet ports provide access to the channel. With the definition of the dialysis mass transfer coefficient

N , = k&C,

(9)

the permeability coefficient is related to the mass transfer coefficient by w =

(k,/RT)

+ C,LP(l - a)a

(10)

At the concentration levels used in the present study, the second term on the righ-hand side of eq 10 is more than three orders of magnitude less than the mass transfer coefficient term and can be neglected. This also means the high flux correction of Mick1eyl5 to the Kedem-

M.

C. Collins and W. F. Rarnirez

TABLE IV: Membrane Resistances (min/cm) at 37 "C solute

Cuprophan

urea creatinine glucose phenobarbital thiopental sucrose BSP vitamin B,,

17.14 i 0.77 3 1 . 2 1 i 0.37 50.33 0.67 50.07 i 1.17 66.18 i 0.55 79.46 -t 3.13 123.3 i 0 . 9 1 197.5 ?. 9.1

*

PAN 13.5 20.7 26.0 28.2 35.2 38.6 77.2 68.9

_+

i i i f

0.4 1.1 0.9

0.4

0.7 1.0 i 1.6 i 2.3 ?:

PANprotein

____ 18.0 i 0.2 t 0.4 i 0.4 t 1.1 t 1.8

27.4 42.9 39.2 49.2 60.2 123.6 120.1

t 0.7 i

1.6

t 2.2

Katchalsky equations can also be neglected. The membrane resistances, which equals the reciprocal of the permeability coefficients, are given in Table IV. The membrane resistance for Cuprophan remained constant after soaking 24 h with bovine albumin. Therefore no evidence of protein adsorption was experimentally observed for the cuprophane membrane. The PAN membrane permeabilities were significantly reduced showing a definite adsorption effect on mass transport properties. P r e s s u r e Filtration At a zero concentration gradient across the membrane, the pressure filtration coefficient is related to the volumetric flux and hydrostatic pressure gradient by Lp

= (Jv/Ap),c=o

(11)

Again the Laug and Stokesberry12test cell was used for the determination of the pressure filtration coefficient. Both compartments were carefully filled with fluid. A pressure of approximately 300 mmHg was applied to one side of the cell equipped with a calibrated pipet while the other side was open to the atmosphere. The volume flow was determined by measuring the rate of volume change in the pipet. Equilibration was allowed for 5 min before taking a measurement. The measurements were made at 37 "C. Kaufman*showed that the pressure filtration coefficient is a function of concentration for solutions greater than 0.01 M. However, his results indicate that for concentrations employed in this study, the coefficient should be independent of solute type and equal the solute-free value. At least ten measurements were made for each of the eight solutes. The values reported in Table I11 are the average of all data for each membrane. Also, one-standard-deviation confidence values are given. The pressure filtration coefficient was found to be independent of solution type. Sieving Coefficient The sieving coefficient, defined as one minus the reflection coefficient, is the coefficient for convective transport of the solute through the membrane. The sieving coefficient was measured by ultrafiltering a solution across the membrane a t a known volumetric flow rate, Jv, and measuring the resultant concentration gradient and solute flux, N,. Calculation of the sieving coefficient from the experimental data requires prior determination of the solute permeability coefficient, w. From eq 1 the sieving coefficient can therefore be directly calculated. An Amicon ultrafiltration test cell (Model No. 52) was used to determine the sieving coefficients. The membrane was placed on a porous support disk, and a brass ring and O-ring were placed over the membrane to give an accurate estimate of the membrane surface area and prevent leakage around the outside of the membrane. The effective membrane surface area was 9.51 cm2. The test cell holds 65 cm3of solution and is equipped with a magnetic stirrer

The Journal of Physical Chemistry, Vol. 83, No. 17, 1979 2297

Mass Transport through Polymeric Membranes

TABLE V: Sieving and Modified Sieving Coefficients a t 37 "C l____-l_l_-

Cuprophan solute

a

PAN-motein

PAN

Sa

S*b

Sa

S*b

urea

0.987 t 0.01

0.946

0.996

0.999

creatinine glucose phenobarbital thiopental sucrose BSP vitamin B ,2

0.955 c 0.03 0.891 i 0.031 0.896 -t 0.036 0.370 ?: 0.029 0.790 * 0.017 0.702 i 0.029 0.556 c 0.015

0.910 0.844 0.839 0.317 0.134 0.649 0.467

1.014 + 0.036 (1.000) 0.991 * 0.01 0.985 + 0.01 0.985 t 0.003 0.725 * 0.019 0.977 i 0.002 0.858 c 0.042 0.920 * 0.006

0.985 0.977 0.977 0.717 0.968 0.850 0.899

0.991 * 0,013 0.985 i 0,008 0.981 i 0.0045 0.154 i: 0.026 0.924 0.012 0.625 f 0.013 0.811 t 0.016

Sieving "coefficient.

i

0.992

0.004

0.982 0.975 0.968 0.141 0.910 0.615 0.785

Modified sieving coefficient, See eq 1 5 for definition.

TABLE VI: Cuprophan Frictional Parameters at 37 "C


10- "fsm,

solute

mmHg cm min/g mol

10-l 0 f S W , mmHg cm minlg mol

fsw/fswo


3.82 5.84 13.60 3.67 27.08 22.19 51.45 55.29

6.85 11.98 17.93 17.73 8.86 24.62 33.79 38.93

3.85 4.79 5.32 4.68 2.09 5.33 5.47 4.58

TABLE VII:

fswlfsm

1.79 2.05 1.32 4.83 0.33 1.11 0.657 0.704

10-7fwm, mmHg cm min/g mol 92.28 92.28 92.28 92.28 92.28 92.28 92.28 92.28

fsmlfwm 41.4 63.2 147.4 39.8 293.4 240.5 557.5 599.2

PAN Frictional Parameters at 37 "C l l _ _ _ l

solute urea creatinine glucose phenobarbital thiopental sucrose BSP vitamin B,,

lo-'ofSm, mmHg cm min/g mol 1.44 5.53 7.13 46.37 86.61 5.60 97.20 13.51

10- IOfSW, mmHg cm min/g mol 4.83 7.31 9.12 9.85 9.04 13.36 23.51 22.18

fsw/fswo

fSw/fsm

2.71 2.93 2.71 2.60 2.13 2.90 3.80 2.61

3.36 1.32 1.28 0.22 0.14 2.39 0.22 1.64

10-7fwm, mmHg cm m i n k mol 5.966 5.966 5.966 5.966 5.966 5.966 5.966 5.966

fsm/fwm

241.0 927.4 1194.3 7772.2 14517.4 938.0 16293.0 2263.7

TABLE VIII: PAN-Protein Frictional Parameters at 34 "C

solute

10-'Ofsm, mmHg cm min/g mol

10-'Ofsw, mmHg cm minlg mol

fsw/fswo


3.41 6.09 12.90 94.13 202.44 16.02 550.42 34.97

4.81 8.77 13.64 12.39 2.27 17.87 24.82 30.86

3.26 3.51 4.05 3.27 0.534 3.87 4.02 3.63

to prevent concentration polarization at the membrane surface. The cell was placed in a 1-L water bath maintained a t 37 f 0.05 O C . The test cell was pressurized to 10 psi and the volumetric flow rate was determined by a timed collection of 1-2 cm3 of filtrate in a calibrated 5-cm3 graduated cylinder. Equilibration for 30 min was allowed before a sample was taken. No more than 1 or 2 cm3 of filtrate was collected for each measurement so that concentration changes in the filtrand would be minimal. The solute flux was equal to the product of the filtrate concentration and the volumetric flow rate. The solute concentration in the filtrand was measured at the beginning and end of each run and the average value used in the calculations. Five determinations were made for each sieving coefficient. The mean values and onestandard-deviation values, obtained for the sieving coefficients, are listed in Table V.

fswlfsm 1.703 1.440 1.058 0.132 0.00112 1.116 0.045 0.882

10-7twm, mmHg cm min/g mol 12.92 12.92 12.92 12.92 12.92 12.92 12.92 12.92

fsrlJfwrn 264.0 471.4 998.3 7285.8 15668.7 1239.7 4 2602.0 2707.7

Results The directly measured parameters included the solute-membrane resistance at zero hydrostatic pressure gradient, the sieving and distribution coefficients for the various solute-membrane combinations, the membrane pressure-filtration coefficients, and the membrane water content. From a knowledge of the membrane permeability, sieving, distribution, and pressure-filtration coefficients, the frictional coefficients of the Kedem-Katchalsky model can be calculated by use of eq 4-6. The coefficients f,,, f,,, and f,, are given in Table VI for cuprophane, in Table VI1 for the PAN membrane, and in Table VI11 for the PAN-protein membrane. Discussion Generally the solute-membrane interaction coefficient, f,,, tends to increase with increasing molecular volume as

2298

The Journal of Physical Chemktry, Vol. 83,No. 67, 1979

M. C. Collins and W. F. Ramirez

r tZ

e 8 I

I i 1 1 1 1

I

I io

I

CUPROPHAN PAN/PROTEIN PAN

I I I l l l l

I

I IOOOi

I 1 1 1 1 1 1 /

10,000

UREA GLUCOSE I ESP ViT. 812 CREATININE THIOPENTAL M O L A L VOLUME ( c c / g m o l e l

Flguro 1. Solute-membrane frictional coefficients as a function of solute molal volume for Cuprophan, PAN, and PAN-protein.

*


E

2

2 = w

m

I

CUPRODHAN PAN/PROTEIN

i

t

IO

I

M O L A L VOLUME ( c c / g r n o l e )

Figure 2. Solute-membrane resistance as a function of solute molal volume for Cuprophan, PAN, and PAN-protein.

shown in Figure 1. Urea, creatinine, glucose, sucrose, and vitamin B12 are compounds which do not exhibit any unusual membrane interactions and were used to determine the regression lines shown on the graph. For these regular or typical molecular species, the Cuprophan membrane shows the highest degree of interaction while the PAN membrane has much less interaction. Protein coating the PAN membrane brought its interaction levels up close to those of Cuprophan. Sulfobromophthalein (BSP) contains two large dipoles in the sulfonate groups and is a very polar molecular. This polar molecule showed larger than normal frictional interactions with the hydrophilic Cuprophan membrane. The membrane frictional interactions are even more pronounced with the PAN membrane which has a slight negative charge due to the sulfonate groups of the membrane. Phenobarbital is a moderately lipid soluble compound which has equally good solubility in aqueous environments. Phenobarbital showed a very low frictional interaction with the Cuprophan membrane. Thiopental on the other hand has a very high frictional interaction with the Cuprophan membrane. This is somewhat surprising in view of the fact that thiopental is nonpolar, has a much higher lipid solubility than phenobarbital, and an extremely low solubility in water. However, thiopental as a result of its low aqueous solubility, may be effectively rejected from some of the water pathways through the membrane. The diffusive solute membrane resislmce, R, = l/wRT, is plotted as a function of the solute molal volume for the Cuprophan, PAN, and PAN-protein membranes in Figure 2. The correlations are linear on log-log paper and the

CREATININE THIOPENTAL

M O L A L VOLUME (cc/gmole)

Flgure 3. Solute-membrane sieving coefficient as a function of solute molal volume for Cuprophan, PAN, and PAN-protein.

correlation coefficients are greater than 0.98 for all the membranes. BSP shows a relatively low diffusive permeability (high resistance) in the PAN and PANprotein membranes which is probably a result of its very high frictional interactions with the membrane and with the solvent. The permeabilities for both of the lipid soluble compounds phenobarbital and thiopental seem to be regular in the PAN and PAN-protein membranes. However, this is due to the averaging of two competing mechanisms. Both solutes are more soluble in the membrane phase as indicated by distribution coefficients of between 3 and 6.6 (Table 11), however, both solutes experienced greater than normal frictional interaction with the membrane (Figure 1). The net result is the regular permeability correlation shown in Figure 2. Also, both solutes appear to have normal permeabilities in the Cuprophan membrane. The sieving coefficient indicates the magnitude of the capacity for convective transport of the solute with the bulk flow of the solvent across the membrane. The sieving coefficients are plotted as a function of the logarithm of the molal volumes for the Cuprophan, PAN, and PANprotein membranes in Figure 3. The sieving coefficient for thiopental is markedly reduced in all membranes studied. However, as noted earlier, the diffusive permeability was normal. This suggests that the thiopental is excluded from a portion of the solvent pathways through the membrane, and the existence of an additional pathway for diffusive transport. Phenobarbital, the other lipophilic solute studied, did not have reduced sieving coefficients in any of the membranes. BSP, in addition to having a reduced diffusive permeability in the PAN and PAN-protein membranes, also had low sieving coefficients in these membranes. Its reduced transport coefficients are therefore concluded to be the result of high frictional interactions with the membrane. The modified sieving coefficient, S*, is defined relative to the solvent flow rather than volume flow. It is related to the standard sieving coefficient by 0

vs

s*=s--

LP

(12)

The modified sieving coefficients are given in Table V along with the standard sieving coefficients. Pappenheimer16 and Renkinls define the area available for the diffusive transport of a given solute by LlX

The Journal of Physical Chemktty, Vol. 83, No. 17, 1979 2299

Mass Transport through Polymeric Membranes 0.9999r-

0 5

CUPROPHAN

PAN

2.

S O 4

w

3

a 0.3

w

a u. 0.2

*

0 31

a

-

W

0 PAN/PROTEIN 0.11 m PAN

0 .I

-0

LL

0

0

I

,

I

, , , I l l

I


Figure 4. Solute-membrane modified sieving coefficient as a function of solute molal volume for Cuprophan, PAN, and PAN-protein.

TABLE IX: Effecbive Membrane “Pore” Dimensions dpPa ‘4

Cuprophan 15.4 PAN 28.8 PAN-protein 20.2 Diameter. Standard deviation.

o‘,b A

6.4 9.75 6.2

From the basic relations of irreversible thermodynamics and the frictional model the solute area is (Co1lins)l

A, =

Kf,W0 ~

fsw

-t

fm

and the area available for solvent transport is

A W = -4-w= T

50

)

TABLE X : Membrane Tortuosity Factors tortuosity computed from

, , # < / I

M O L A L V O L U M E (cc/gmole)

membrane

30 40 DIMENSIONS ( 4

Figure 5. Effective pore size distribution for Cuprophan and PAN.

PHENOB A R B I T A L SUCROSE I

IO 20 E F F E C T I V E “PORE”

4Wf,W0

fsw

Therefore the ratio of A, to A, is the modified sieving coefficient A8/Aw = S+ (16) If solute and solvent transport occurs through a normally distributed porelike structure, then a plot of S* vs. the logarithm of solute molal volume would be linear on probability paper. These plots are given in Figure 4 for the Cuprophan, PAN, and PAN-protein membranes. With the exception of the solutes having unusual membrane interactions, the plots are linear. The effective mean pore diameters and standard deviations for the normal distribution are given in Table IX. The calculated membrane pore-size frequency distributions are plotted in Figure 5. Based on this tranport data, the Cuprophan membrane shows a smaller mean pore diameter than the PAN membrane and a tighter distribution or increased selectivity. The ratio of the solute-solvent to solute-membrane frictional coefficients, f,,/f,,, are given in Tables VI and VI1 and also indicate that Cuprophan is the more selective membrane since this ratio decreases as selectivity increases. The adsorbed protein layer on the PAN membrane resulted in a decrease in both the effective mean pore size and pore-size distribution of the PAN membrane, increasing its selectivity. The mean pore size of 15.4 A estimated for the Cuprophan membrane is lower than previous estimates for cellulose membranes which have ranged from 19 to 41 A (Renkin,18 D ~ r b i nGinzburg,6 ,~ Fritzinger6). A pore-size

membrane type

transport data

random walk theory

Cuprophan PAN PAN-protein

4.86 i 0.57 2.74 * 0.139 3.60 f 0.32

4.26 6.21 8.03

distribution was not considered in arriving at previous estimates and it was assumed that a molecule larger than the effective pore radius could not penetrate the membrane. From Figure 5 the maximum effective pore diameter for Cuprophan is approximately 35 A which is in agreement with other maximum estimates. The solutesolvent frictional coefficient, f,, is a measure of the interaction between 1 mol of solute and the solvent in its vicinity inside the membrane. This coefficient is also a measure of the tortuosity of the membrane, since the interaction of the solute and solvent in the membrane will increase as the solvent pathway increases in length for a given membrane thickness. The free solution frictional coefficient between solute and solvent for a binary liquid system is given by (Laity? fSWO

= RT( 1

%i ):

+ Cs-

For ideal solutions, eq 17 reduces to the Einstein equation. The tortuous path of the solute molecules due to the presence of the membrane can be characterized by a tortuosity factor T , such that TAX represents the overall solute path in the membrane. If no unusual solutemembrane interactions occur, the tortuosity factor can be computed by the solute-solvent frictional interaction and is given by 7

= f*W/fSW0

(18)

Their values are reported in Tables VI-VIII. Thiopental has a low tortuosity factor (fsw/fswo) in the Cuprophan membrane which is consistent with its exclusion from part of the solvent pathway and the existance of an alternate pathway for diffusion. Its tortuosity is not quite so low relat,ive to the other solutes in the PAN membrane. Thiopental has a very low tortuosity factor in the PAN-protein membrane and is virtually excluded from the solvent pathway. BSP has a very high tortuosity factor in the PAN membrane as a result of its interactions with the membrane sulfonate groups. The mean membrane tortuosity factors are given in Table X. These calculated values did not include thiopental for the Cuprophan membrane nor thiopental and BSP for the PAN and PAN-protein membranes due to their unusual transport mechanisms. These calculated membrane tortuosity factors are relatively constant for a given

2300

The Journal of Physical ChemWy, Vol. 83, No. 17, 1979

membrane with deviations approximately 10% from the mean value. The tortuosity factor for flow through water-filled capillaries as derived by MacKay14from the random walk treatment of diffusion is given by = ( 2 - d)w)2/d)wz

(19)

Values for the membranes studied are also given in Table


X. The random walk theory predicts that the tortuosity should be independent of solute and only a function of the water content of the membrane. This is in agreement with the results presented in this study if those solutes with unusual membrane interactions are excluded. The theory also predicts that the tortuosity will increase as the water content decreases. This was not found to be the case with the Cuprophan and PAN membranes. The agreement of the random walk tortuosity to the actual value is fairly good for the Cuprophan membrane. Values are totally out of agreement for the PAN membrane. This can be explained by the differences in structure of the two membranes. Cuprophan is a highly swollen amorphous membrane wit,h little structure. Solvent pathways are not well defined and are constantly changing. PAN, on the other hand, is only slightly swollen by water and is primarily crystalline in nature with a great deal of structure. Its water pathways are well defined which results in a reduction of the tortuosity factor. The adsorbed protein layer on the PAN-protein membrane is highly swollen with water and this has the effect of increasing the tortuosity of this membrane from that of the normal PAN membrane. The frictional coefficient ,f quantifies the interaction between solvent and membrane. As expected for dilute solutions, the coefficients are all equal for a given membrane as shown in Tables VI-VIII. The interaction of water with the polymer matrix of the Cuprophan membrane was more than 15 times as great as that with the PAN membrane. This is due to the hydrophilic, highly swollen, and amorphous nature of the Cuprophan membrane compared to the structured, hydrophobic nature of the PAN membrane. Furthermore, in laminar flow, the viscous friction factor is inversely proportional to the pore diameter to the fourth power. The PAN membrane with its larger pore size would therefore be expected to show a lower solvent frictional coefficient. The parameter f,,/f,, can give a relative indication of the magnitude of membrane filtration to solute permeability since

Values of fsm/fwm are given in Tables VI-VI11 for the membranes studied. This ratio is much greater for the PAN membrane than that for Cuprophan. The adsorbed protein layer did not significantly alter the PAN values. This is a detrimental characteristic for the PAN membrane in hemodialysis applications since dialyzer surface areas are typically limited by maximum allowable filtration capacity. Conclusions A complete set of membrane transport data is presented for a wide range of solutes for two important dialysis membranes. The following are the major conclusions drawn from this study: Cuprophan is a highly swollen, hydrophilic, amorphous membrane. Its measured tortuosity factor agreed with that

M. C.Collins and W. F. Ramirez

computed from random walk theory indicating that the membrane has little internal structure. The mass transport occurs randomly through the water swollen matrix. Pore-size distribution measurements from mass transfer data shows a narrow distribution giving the membrane a high degree of selectivity. The RPAN 69 (PAN) membrane is hydrophobic and swells little in an aqueous environment. The low value for the tortuosity factor indicates a high degree of internal structure for the PAN membrane with water transport pathways separate and independent of the membrane matrix. Pore-size distributions showed a broad distribution which gives a low degree of selectivity. The PAN membrane irreversibly adsorbs protein from aqueous solutions and this adsorption changes the transport characteristics of the membrane. Pore-size distributions, calculated from transport data, were narrower than for PAN giving increased membrane selectivity. The tortuosity increased from that of the PAN membrane, indicating that the protein layer is highly swollen and amorphous. The solute permeability or membrane resistance for diffusive transport correlates well for all membranes studied with solute molal volume except for sulfobromophthalein (BSP). Strong membrane-solute frictional interactions for BSP reduced this permeability. A significant degree of positive interaction was observed between the lipophilic solutes and the hydrophobic PAN membrane, as indicated by high values of distribution coefficients. However, the concomitant increase in membrane-solute frictional coefficients cancelled out the enhancing effect of increased solubility in the membrane phase on solute permeability. The sieving coefficient, which is the convective transport coefficient, also correlated with solute molal volume except for BSP and thiopental. The reduced convective transport of BSP seems due to its high frictional interactions with the membranes studied. The sieving coefficient for thiopental is markedly reduced in all membranes studied. However, its diffusive permeability was normal. This suggests that thiopental is excluded from some of the solvent pathways through the membrane, and that additional pathways exist for diffusive transport. The tortuosity factor for thiopental was also reduced in the membranes studied, again indicating its exclusion from some of the aqueous pathways. membrane surface area (cm2) chemical activity area for solute transport (cm2) area for solvent transport (cm2) low side solute concentration (g mol/cm3) mean concentration across membrane (g mol/cm3) diffusivity (cm2/s) molecular diameter (A) pore diameter (A) frictional force between i and j (rnmHg cm2/g mol) frictional coefficient of ith and j t h species (mmHg cm min/g mol) total volume flux (cm/min) distribution coefficient pressure filtration coefficient (cm/min mmHg) solute flux (g mol/cm2 min) total solute mass of the system (g) gas constant (cm3 mmHg/g mol K) membrane resistance (s/cmz) modified sieving coefficient sieving coefficient temperature (K)

Isotropy Failure Theory Application to Effusion Data

t

V

P ui AC,

AP Ax cr a'

&, w T

TO

time (min) volume of each side of permeability experiment

graciously provided laboratory space, equipment, and supplies needed for the experimental phase of this study.

partial molar volume (cm3/g mol) velocity of species i (cm/min) solute concentration difference across membrane (g mol/cm3) pressure difference across membrane (mmHg) membrane thickness (cm) reflection coefficient standard deviation volume fraction of water in the membrane (vol %) solute permeability coefficient (cm2mmHg/g mol) tortuosity factor random walk tortuosity factor

References and Notes

Subscripts


The Journal of Physical Chemistry, Vol. 83, No. 17, 1979 2301

Superscripts

m = membrane f = final s = free solution i = initial m = membrane p = pressure s = solute w = solvent Acknowledgment. The authors acknowledge the support of Cobe Laboratories of Denver, Colorado. They

Collins, M. C. Ph.D. Thesis, University of Colorado, Boulder, Colo., 1977. Colton, C. K.; Smith, K. A.; Merrlll, E. W.; Farrell, P. C. J . Biomed. Mater. Res. 1971, 5 , 459. Durbin, R. P. J . Gen. Pbysiol. 1960, 4 4 , 315. Farrell, P. C.; Babb, A. L. J. Biomed. Mater. Res. 1972, 6. Fritzinger, B. K.; Brauman, S. K.; Lyman, D. J. J . Biomed. Mater. Res. 1971, 5 , 3. Ginzburg, B. Z.;Katchalsky, A. J . Gen. Pbysiol. 1963, 47, 403. Kaufman. T. 0.:Leonard, E. F. AIChE J . 1966, 74, 110. Kaufman, T. G.; Leonard, E. F. AICbEJ. 1968, 74, 421. Kedem, 0.; Katchalsky, A. Biocbim. Biopbys. Acta 1958, 27, 229. Katchalsky, A. J . Gen. Physioi. 1961, 45, 143. Kedem, 0.; Laky, R. W. J. Pbys. Cbem. 1959, 63, 80. Laug, 0.B.; Stokesberty, D. P. National Bureau of Standards Report 9872, July 1968. Le Bas "The Molecular Volume of Llquid Chemical Compounds"; Longman's: London, 1915. MacKay, D.; Meares, P. Trans. Faraday Soc. 1959, 55, 1221. Mlckley, M. C. Ph.D. Thesis, University of Colorado, Boulder, Colo., 1976. Pappenhelmer, J. R.; Renkin. E. M.; Borrero, L. M. Am. J. fbysiol. 1951, 767, 13. Perry, J. H., Ed. "Chemical Engineering Handbook", 4th ed.; McGraw-Hill: New York, 1963. Renkin, E. M. J . Gen. Pbysiol. 1954, 38, 225. Spiegler, K. S. Trans. Faraday Soc., 1958, 5 4 , 1405. Wilke, C. R.; Chang, P. AICb€J. 1955, 7 , 265.

Application of the Isotropy Failure Theory to Effusion Data of Ewing and Stern P. G. Wahlbeck Jnstitutt for Siiikat og h&temperaturk]emi,Norges tekniske h&skoie, 7034 Trondheim, Norway and Department of Chemistry, Wichita State University, Wichita, Kansas 67208 (Received February 26, 1979)

Ewing and Stern have reported the experimental measurements of total mass flux of several alkali halides from Knudsen cells with near-ideal orifices as a function of varying gas phase conditions. Molecular flow and the transition region were studied with care. These experimental measurements are compared with predicted values from the isotropy-failure theory. The agreement is excellent and is considered to be a confirmation of the reliability of the isotropy-failure theory.

Introduction The effusion of gases through orifices has been used extensively for the determination of vapor pressures of condensed phases.' Because of this use and because of interest in the flow of gases, experimenters have studied the total flow of gases through both ideal orifices (hole in a plane) and nonideal orifices. These experimenters include Knudsen2with his classic work; Liepmann? Carlson, Gilles, and Thorn;4 Wey and W a h l b e ~ k and ; ~ recently Ewing and Stern6 In these cases the total number of molecules effusing from an orifice has been measured as a function of pressure of the vapor. The results have indicated pressure conditions for both molecular flow and the transition region between molecular and hydrodynamic flows. Angular distribution studies for molecules effusing from orifices have been measured by Ward;7 Wang and Wahlbeck;8Adams, Phipps, and W a h l b e ~ kand ; ~ Grimley et al.IO Wahlbeckll proposed the isotropy-failure theory as an explanation of the changes in the orifice transmission probabilities and angular distributions as pressure increases so that the flow changes from molecular flow to the transition region to hydrodynamic flow. The central 0022-365417912083-2301$01.OO/O

idea is that the assumption of isotropy of the gas in a Knudsen cell fails for molecules near the orifice as the mean free path decreases to be about the orifice dimensions. The magnitude of the isotropy failure is proportional to the solid angle subtended by the orifice at the last collision site for a molecule in the gas container. The probability for escape of the molecule at that last collision site is increased by the solid angle subtended by the orifice since the area of the orifice is not a source for molecules traveling toward the last collision site. The isotropy-failure theory is developed with a treatment typical of the kinetic theory of gases. For the ideal orifice, Wahlbeckll found that the orifice transmission probability, W , the ratio of the number of molecules which effuse to the kinetic theory number (I14NC;N is the number of molecules cm-3 and is the average molecular velocity), is = 1 + 2/3[&3(0)/27r] (1)

w

where 6a(O) is derived from the solid angle subtended by the orifice at the last collision site, and 7rr 6i1(0)/27~= 1 - -[H,(r/X) - Yl(r/A)] + r/X (2) 2x

In the above equation Hl(x) is a Struve function and Yl(x) @ 1979 American Chemical Society

Transport through polymeric membranes - ACS Publications

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