Full text of DOI:10.1002/aic.690460607

Hindered Diffusion of Dextran and Polyethylene
Glycol in Porous Membranes
Jiahui Shao and Ruth E. Baltus
Dept. of Chemical Engineering, Clarkson University, Potsdam, NY 13699
The effecti®e diffusion coefficients of dextran and polyethylene glycol in track-etched
polycarbonate membranes were measured using membranes with nominal pore sizes of
0.03 ␮m, 0.05 ␮m, and 0.1 ␮m. Experiments were performed using narrow-size range
fractions of each polymer. When the Stokes-Einstein radius was used to describe solute
size, the obser®ed diffusi®ities for both polymers agreed closely and were larger than
®alues predicted for rigid spheres, as well as for linear polymers when only steric interac-
tions with the pore wall are assumed. These obser®ations cannot be explained by consid-
ering electrostatic interactions between the solute and the pore wall, or solute adsorption
on the pore wall. The experimentally measured diffusion coefficients agreed well with a
model that treats the polymeric solutes as rigid spheres and includes ®an der Waals
attracti®e interactions between the solute and the pore wall.
Introduction
2
Ž .
where Dϱ is the bulk solution diffusion coefficient cm rs ,
16
When the size of a solute is comparable to the pore size
through which it is diffusing, the effective diffusion coeffi-
cient of a solute within a pore is usually found to be less than
its value in bulk solution. This phenomenon is known as hin-
dered diffusion. An understanding of the equilibrium and
transport properties of macromolecules in porous media is
important in the design of controlled release devices for the
delivery of drugs and pesticides, in dialysis and other mem-
brane separations, in chromatographic separations, and in
heterogeneous catalysis.
y
Ž
.
kB is Boltzmann’s constant 1.38=10
Jrk , T is absolute
temperature K , fϱ is the molecular friction coefficient in
bulk solution kgrs , ␮ is the solvent viscosity N ؒsrm , and
Ž .
Ž
2
.
Ž
.
Ž
.
rs is the radius of the solute cm .
The reduced solute diffusivity that is typically observed with
macromolecular solutes in porous media results from two
phenomena. One is thermodynamic and leads to an intrapore
concentration driving force, which is less than the driving
force based on bulk solution concentrations. Steric, and, in
some cases, long-range, interactions between solute and the
pore wall exclude the solute from the region near the pore
wall. This is described with an equilibrium partition coeffi-
cient Keq, which is the ratio of the average solute concentra-
tion within the pore to that in bulk solution at equilibrium.
For a spherical solute in a cylindrical pore
Efforts to understand the phenomenon controlling hin-
dered diffusion have been ongoing for several decades. A re-
Ž
.
view article by Deen 1987 summarizes much of the theoreti-
cal and experimental research in this field. Useful discussions
of the hydrodynamic treatment of diffusion in porous media
Ž
.
Ž
.
are those of Bean 1972 , Anderson and Quinn 1974 , Bren-
Ž
.
Ž
.
ner and Gaydos 1977 , Malone and Anderson 1978 , and
1y ␭
Ž
.
Davidson and Deen 1988a . In general, the membrane is
treated as an array of cylindrical pores and the solute is as-
sumed to have both Brownian and hydrodynamic characteris-
tics. For spherical solutes in bulk solution, this leads to the
Stokes-Einstein equation
E Ž ␤ .rk
T
ey
B ␤d␤
H
0
Keq ␭ s
2
Ž .
Ž .
1␤d␤
H
0
Ž
.
where ␭ is the ratio of solute to pore size rsrrp , ␤ is the
kBT
fϱ
k␤T
Ž
.
,
dimensionless radial position of the solute in the pore rrrp
and E is the interaction energy between the solute and the
Dϱs
s
1
Ž .
6␲␮rs
Ž .
pore wall J .
The second effect is a transport effect. Because of the
proximity of the pore wall, the hydrodynamic drag experi-
Correspondence concerning this article should be addressed to R. E. Baltus.
AIChE Journal
June 2000 Vol. 46, No. 6
1149

enced by the solute in the pore is greater than the drag expe-
rienced in an unbounded fluid. This enhanced drag is charac-
terized by Ky1, which is the ratio of the friction coefficient of
the solute in the bulk solution to that within the pore
scribe the effects of long-range polymer-pore wall interac-
tions on the equilibrium partitioning coefficient. Results were
presented for square-well potentials, electrostatic double-
layer potentials, and van der Waals potentials.
The intrapore hydrodynamic resistance for flexible poly-
Ž
.
mers has been presented by Davidson and Deen 1988a . The
hydrodynamic resistance for a flexible polymer is predicted to
be less than that of a rigid sphere with the same ␭, when the
Stokes-Einstein radius is used to characterize the size of the
flexible polymer. However, the effective diffusion coefficient
is predicted to be smaller for a flexible macromolecule than
for a rigid sphere because the equilibrium partition coeffi-
cient is sufficiently smaller for the flexible molecule.
1y ␭
EŽ ␤ .rk
T
w f rf ␭,␤ xey
B ␤d␤
Ž
.
H
ϱ
0
1
Ky ␭ s
3
Ž .
Ž .
1y ␭
E Ž ␤ .rk
T
ey
B ␤d␤
H
0
Ž
.
where f ␭,␤ is the molecular friction coefficient of solute of
Ž
.
size ␭ at position ␤ in the pore kgrs .
These two effects are combined to define the effective dif-
Experimental investigations with flexible macromolecules
have yielded mixed results. Diffusion coefficient values of lin-
ear polystyrene measured in Nuclepore polycarbonate mem-
2
Ž
.
fusion coefficient D cm rs
Ž .
D ␭
branes by Cannell and Rondelez 1980 , and in porous mica
Ž .
1
s Keq Ky
4
Ž .
Ž
.
membranes by Kathawalla and Anderson 1988 , were in close
agreement with the theory of Davidson and Deen 1988a for
Dϱ
Ž
.
Ž
.
random coil polymers. Bohrer et al. 1984 measured the dif-
fusion rates of the polysaccharides dextran and ficoll as a
function of pore size using Nuclepore polycarbonate mem-
branes. Their results for dextran showed transport rates
greater than those predicted by either the Renkin equation
or the Davidson and Deen theory. Their results for ficoll were
in agreement with the Renkin equation, indicating that ficoll
behaves more as a rigid sphere than a flexible chain. A later
An important special case for a rigid spherical solute occurs
when only steric interactions are present between the solute
2
Ž
.
Ž
.
and the pore wall E s0 , leading to Keq s 1y␭ . Combin-
Ž
.
ing this with the assumption that f ␭,0 adequately approxi-
Ž
.
mates f ␭,␤ leads to the commonly used Renkin equation
Ž
.
Renkin, 1954
2
5
Ds D 1y␭ 1y2.104␭q2.089␭3y0.948␭
5
Ž .
Ž
.
study by Davidson and Deen 1988b also showed large diffu-
sivities for dextran and polyethylene glycol than those pre-
dicted for either spherical solutes or random coil polymers.
Diffusion coefficient values for polyisoprene measured by
ϱŽ
. Ž
.
A more exact expression was derived by Brenner and Gaydos
Ž
.
.
1977 using an asymptotic matching technique to obtain
Ž
Ž
.
.
Ž
.
Bohrer et al. 1987 and for poly ethylene oxide and
poly vinylpyrrolidone measured by Davidson and Deen
f ␭,␤
Ž
Ž
.
1988b were also larger than values predicted from the the-
9
Ž
.
ory of Davidson and Deen 1988a .
Ds Dϱ 1q
␭ ln ␭y1.54␭qO ␭
6
Ž .
Ž .
ž 8 /
In our laboratory, a study of the effect of solute concentra-
tion on hindered diffusion has been undertaken and the re-
sults from that study will be reported in a subsequent publi-
Values of DrDϱ predicted from Eq. 5 are in very good
agreement with values predicted using Eq. 6 when ␭-0.1. A
number of experimental investigations have confirmed the
validity of Eq. 5 for solutes that are generally rigid and spher-
Ž
.
cation Shao and Baltus, 2000 . In order to interpret trans-
port measurements at a high concentration, an accurate pre-
diction of solute diffusion at low concentration conditions is
needed. The objective of the study reported here was to mea-
sure the effective diffusion coefficient of narrow-size range
fractions of dextran and polyethylene glycol in track-etched
polycarbonate membranes under infinite dilution conditions.
The measured diffusivity values are compared to those pre-
dicted from established models, as well as to those from pre-
viously reported experimental measurements. A model is
proposed to explain the experimental results.
Ž
ical Beck and Schultz, 1972; Malone and Anderson, 1978;
Deen et al., 1981; Baltus and Anderson, 1983; Bohrer et al.,
1984; Davidson and Deen, 1988b .
In one of the earliest theoretical studies of the equilibrium
partitioning of flexible macromolecules, Casassa 1967 de-
rived expressions for random-flight polymer chains and cal-
culated partition coefficients using a mathematical analogy
.
Ž
.
Ž
.
to transient heat conduction. Casassa and Tagami 1969
extended the earlier work of Casassa to star-branched, ran-
Ž
.
dom-flight chains. Davidson et al. 1987 used Monte Carlo
simulations to calculate partition coefficients for flexible
polymers, including the effect of attractive interactions be-
tween chain segments and the pore wall in their model. Re-
sults showed that small, attractive interaction energies be-
tween chain segments and the pore wall produced dramatic
increases in the partition coefficient of chains with finite
Experimental
Membranes
Track-etched Nuclepore polycarbonate membranes with a
nominal pore radii of 0.1 ␮m, 0.05 ␮m, and 0.03 ␮m were
obtained from Corning Separation Division Corning Costar,
Ž
.
Acton, MA . These membranes contain straight, uniform,
Ž
.
length. Davidson and Deen 1990 performed the analogous
continuum ‘‘diffusion equation’’ calculation for the partition
coefficient adding an attractive square well interaction. Lin
cylindrical pores. The pore density and pore length were de-
Ž
.
termined by scanning electron microscopy SEM using a
JEOL JSM-6300 scanning electron microscope. Four mem-
branes were randomly chosen from a given lot, and six differ-
Ž
.
and Deen 1990 used a ‘‘diffusion-reaction’’ equation to de-
1150
June 2000 Vol. 46, No. 6
AIChE Journal

ent regions of each membrane were photographed in order
to calculate the average values. The pore density for each
membrane was determined by counting the number of pores
in surface micrographs of a known area at 16,000᎐30,000=
magnification. Each photograph contained 50᎐150 pores. The
pore length was determined by measuring the thickness of a
cross section. To do this, a piece of the membrane was em-
bedded in paraffin and then sectioned using a microtome
Ž
.
Reichert-Jung, 820 . Since pores in the membrane are essen-
tially uniform cylinders, pore length is very nearly equal to
membrane thickness, exceeding the membrane thickness only
to the extent that the pores are not aligned to the membrane
surface. The manufacturer states that the maximum devia-
tion from the normal is 29Њ. Assuming that all deviations from
0 to 29Њ are equally probable, the average pore length ex-
ceeds the membrane thickness by 6.8% and all calculations
employed this factor.
Figure 1. Experimental s ys tem us ed to perform s olute
diffus ion experiments .
Ž
.
Polyethylene glycol PEG with a weight-average molecular
The pore radius was measured using water flow measure-
ments. For a membrane containing cylindrical pores, the Ha-
Ž
weight of 10,890 was obtained from Pressure Chemical Pitts-
.
burgh, PA . The manufacturer reported polydispersity of this
gen-Poiseuille equation can be used to relate the flow rate
sample was 1.19. This polymer was used as received without
further fractionation.
The bulk phase diffusivity of dextran in water at 25ЊC
3
Ž
. Ž
.
Ž
.
Q
m rs through the pores and the pressure drop ⌬ P
Ž
.
across the membrane Pa
was determined using a dynamic light scattering system
4
Ž
.
²
:
Brookhaven Inc. . The incident light wavelength was 514.5
n␲ rp
Qs
⌬ P
7
Ž .
nm, and the scattering was observed at 90Њ. The photocount
autocorrelation function was obtained using a BI 2030 AT
correlator. For these measurements, dextran solutions were
prepared with concentration less than 5 mgrmL to avoid in-
teractions between dextran molecules.
8␮ l
where n is the number of pores and l is the pore length
cm . The notation rp indicates an average over all the
pores. By measuring the pressure drop across the membrane
at various flow rates, the average pore radius cm rp
4
Ž
.
² :
1r4
4
Ž
. ² :
was
Hindered diffusion measurements
determined from the slope of a plot of Q vs. ⌬ P. The experi-
ments were performed using a glass diffusion cell. Details of
the diffusion cell and the water flow measurement are de-
The diffusion experiments were performed using a glass
diaphragm diffusion cell that is described in detail elsewhere
Ž
.
Shao, 2000 . It consists of two half-cells, one with a volume
Ž
.
scribed elsewhere Shao, 2000 .
of 12.64 cm3, and the other with a volume of 15.87 cm3. The
membrane, supported by a stainless steel screen support, was
sandwiched between the two half cells. The cell was stirred
internally by Teflon coated bar magnets, which were acted
upon by a magnet external to the cell. The stirring speed of
the external magnet was controlled by a DC motor speed
Polymer solutes
Ž
.
Dextran T40 was obtained from Pharmacia Piscataway, NJ
with weight-average molecular weight of approximately 40,000
grmol and MwrMn f1.5. Because hindered diffusion rates are
sensitive to solute size, the polydispersity of the polymer sam-
ple can have a significant effect on observed diffusivities.
Dextran T40 was fractionated on a preparative Superdex 200
Ž
.
controller Bodine Electric Company . The diffusion cell was
jacketed for water circulation to maintain a constant temper-
Ž
.
ature 25.0"0.1ЊC . The procedure used for the diffusion ex-
periments is shown in Figure 1. Initially, one-half cell was
Ž
.
pg gel column 16 mm I.D.=600 mm, Pharmacia to obtain
narrow molecular weight distribution samples. For each run,
2 mL of a 100 mgrmL T40 dextran sample in 0.02 M ammo-
nium acetate was injected into the column. The same buffer
was used as the eluent, which was pumped at a flow rate of
Ž
.
filled with water low concentration side , while the opposite
cell was filled with polymer solution of known concentration
Ž
.
0.2᎐0.4 mgrmL . Water was pumped at a constant flow rate
through the low concentration cell using a syringe pump. Dis-
crete samples, each with a volume of about 3 mL, were col-
lected from the effluent and were analyzed for solute concen-
tration using a Waters 410 differential refractometer. The
time required for each diffusion experiment varied between 7
and 50 h depending on the relative size of the solute and
pores. Assuming perfect mixing in each half cell, the mea-
sured sample concentration was related to the mass-transfer
Ž
.
0.8 mLrmin. Four dextran fractions each about 10 mL were
automatically collected. Each fraction was then desalted and
Ž
.
concentrated using Centriprep concentrators Amicon . Fi-
nally, the desalted and concentrated dextran samples were
extensively freeze-dried to remove residual water. Each dex-
tran fraction was analyzed using analytical gel permeation
Ž
.
chromatography GPC with a Shodex OHpak KB-80M gel
Ž . Ž .
sampled reservoir
coefficient
k
mLrh using a mass balance written on the
Ž
. Ž
.
column 8 mm I.D.=300 mm, Waters Shao, 2000 . The
number and weight average molecular weight of each frac-
tion was determined from the resulting chromatogram using
ASTM Standard Method D 3536-91. The ratio MwrMn for
each fraction was less than 1.2.
C
CHo
kq
VLVm r yr
Vtrq
m s
er tyer t dt
8
Ž .
1
2
Ž
.
. H
Ž
Vty1rq
1
2
AIChE Journal
June 2000 Vol. 46, No. 6
1151

Table 2. Properties of Polymer SolutesU
r1,r2
2
2
†
‡
Mw
kqq
VL
k
kqq
VL
k
2k kyq
Ž
.
Dϱ
rs
y
q
"
q
y
2
7
(
Ž
.
Ž
.
Solute
Mw
Mn
Mn
cm rs=10
nm
ž
/ ž / ž /
VH
VH
VLVH
s
Dextran
2
T40
37,100
62,600
39,600
26,300
16,600
10,890
26,000
52,700
37,000
23,400
13,600
9,150
1.43
1.19
1.07
1.12
1.22
1.19
4.6
3.6
4.5
5.4
6.8
8.2
5.3
6.8
5.4
4.5
3.6
3.0
Ž .
F 1
9
Ž .
Ž .
F 2
Ž .
F 3
where Cm is the solute concentration of the ith sample, CHo
is the initial solute concentration in the high concentration
cell, VL is the volume of the low concentration cell, VH is the
volume of the high concentration cell, Vm is the volume of
the ith sample, V is the total volume collected up to and
including the ith sample, V y1 is the volume collected up to
Ž .
F 4
PEG
U
Ž . Ž . Ž .
Ž .
T40 is the original sample from Pharmacia; F 1 , F 2 , F 3 and F 4 are
fractions collected using gel permeation chromatography.
†Obtained from light scattering.
t
‡Stokes-Einstein radius determined from Dϱ and Eq. 1.
t
the ith sample, and q is the flow rate through the low con-
centration cell. A best fit value for k was determined using a
fitting procedure that involved minimization of the function
In addition to water flow measurements, the pore radius of
several membranes was also determined by measuring the rate
of glucose diffusion. The experimental procedure for these
measurements was the same as that described for the poly-
mer diffusion experiments. Because glucose is small, bulk
phase diffusivity can be assumed in the pores. The pore ra-
dius was determined from the measured value of k using Eq.
Ý Cm,pred yCm,meas 2, where Cm,pred is the sample concentra-
Ž
.
tion predicted with a given value for k, Cm
is the mea-
,meas
sured sample concentration, and the summation was per-
formed over all collected samples. In Eqs. 8 and 9, all vol-
umes are in mL and all concentrations are in mgrmL. By
assuming negligible boundary layer resistances, the effective
diffusion coefficient of the polymer in the pore was deter-
mined from the measured value of k with the following rela-
tionship
y
5
10 and a literature value for Dϱ 0.67=10 cm2rs at 25ЊC,
w
Ž
.x
David, 1994 . The pore radius values determined from wa-
ter flow and glucose diffusion were in good agreement, indi-
cating that the pore-size distribution in these membranes is
narrow.
Characteristics of the original T40 dextran sample and the
four fractions obtained from this sample are presented in
Table 2. The following relationship between Dϱ and Mw was
obtained for dextran in water at 25ЊC
n␲ rp2D
ks
10
Ž
.
l
The validity of the assumptions of perfect mixing and negligi-
ble boundary layer resistances were verifed experimentally
Ž
.
Shao, 2000 . For each membrane, water flow measurements
Dϱs7.58=10y5Myw 0.485
cm2rs
11
Ž
.
were performed before and after each diffusion experiment
to monitor any change in the pore radius.
The bulk diffusivity of polyethylene glycol in water at 25ЊC
Ž
.
was predicted using the following equation Singh et al., 1998
Res ults
Dϱs1.465=10y4Myw 0.557
cm2rs
12
Ž
.
The pore length and pore density obtained from SEM im-
ages of the three different pore size membranes are summa-
rized in Table 1. The values are compared to the nominal
values provided by the manufacturer. The values determined
from our measurements were used for all subsequent calcula-
tions. For each pore size, all membranes came from the same
lot.
Equations 11 and 12 were used with the Stokes-Einstein
equation Eq. 1 to determine an effective radius rs for each
solute.
The hindered diffusion data for the four dextran fractions
and PEG are shown in Table 3. The dimensionless diffusivity
values DrDϱ are plotted as a function of dimensionless solute
size ␭ in Figure 2. The experimental values are also com-
pared to values predicted for a solid, spherical solute using
the Renkin equation, as well as to values predicted by the
Ž
.
Table 1. Comparison of Pore Length and Pore Density of
Nuclepore Membranes Determined from SEM to Nominal
Values Provided by CorningU
Ž
.
theory developed by Davidson and Deen 1988a for a flexi-
ble, linear polymer. Only steric interactions between the sol-
ute and pore wall were considered when making these pre-
dictions. In the Davidson and Deen theory, ␣ is a permeabil-
ity parameter which characterizes the resistance to solvent
flow through the macromolecule. The prediction presented
in Figure 2 was obtained using ␣ s34. It has been shown
that the predicted diffusion coefficient is relatively insensitive
to ␣ when 34F␣ F60 and that ␣ s40 is an appropriate
Nominal
Pore Length
Pore Density
Poresrcm =10
2
y8
Ž
.
Ž
.
Ž
.
Size ␮m
Method
␮m
SEM
Corning
5.80"0.44
3.28"0.25
0.1
6.41UU
3
SEM
Corning
5.81"0.43
5.66"0.54
0.05
0.03
6.41UU
6
SEM
Corning
5.95"0.44
5.35"0.43
6.41UU
6
Ž
.
value for dextran Davidson and Deen, 1998b . It is apparent
from Figure 2 that the experimentally determined hindered
diffusion coefficients for both solutes are in good agreement
and are larger than those predicted by either the Renkin
UReported errors are the 95% confidence limits obtained from images
of 4 different membranes.
UUThe thickness reported by the manufacturer is 6 ␮m; the pore length
was assumed to be 1.068=thickness.
1152
June 2000 Vol. 46, No. 6
AIChE Journal

Table 3. Membrane Pore Sizes and Measured Diffusion
Coefficients of Dextrans and PEG
U
avg
UU
1r4
4
²
:
rp
⌬rp
␭
D
1r4
4
2
7
Ž
.
Ž
.
Ž
²
:
.
Ž
.
Solute
␮m
␮m
rsr rp
cm rs=10 DrDϱ
avg
Dextran
Ž .
F 1
0.0885
0.0535
0.0324
y0.002
0.08
0.13
0.21
3.3"0.2
2.7"0.2
1.9"0.1
0.91
0.75
0.53
y0.0004
y0.0001
Dextran
Ž .
0.0887
0.0516
0.0326
y0.002
0.06
0.11
0.17
4.3"0.2
3.3"0.2
2.7"0.2
0.95
0.73
0.60
F 2
y0.0013
y0.0002
Dextran
Ž .
0.0885
0.0522
0.0300
y0.002
0.05
0.09
0.15
4.9"0.3
4.2"0.3
3.3"0.2
0.91
0.78
0.61
F 3
y0.0014
y0.0012
Dextran
Ž .
0.0842
0.0521
0.0310
y0.0037
y0.0025
y0.001
0.04
0.07
0.12
6.1"0.4
5.9"0.4
4.9"0.3
0.90
0.87
0.72
F 4
PEG
0.0840
0.0530
0.0320
q0.0001
y0.0002
y0.0008
0.04
0.06
0.09
7.4"0.4
7.0"0.4
6.7"0.3
0.90
0.85
0.82
Figure 3. Meas ured diffus ion coefficient in different
( )
ionic s trength s olutions for fraction F 2 dex-
tran.
UAverage of pore size determined before and after each diffusion mea-
surement using water flow measurements.
UUThe pore size measured after solute diffusion measurement minus the
pore size measured before the solute diffusion measurement.
The same 0.1 ␮m pore size membrane was used for each
experiment.
The bulk phase diffusivity of dextran in different ionic
y
2
M, 10y3 M and 10y4 M KCl was
Ž
.
strength solutions 10
determined by dynamic light scattering. The measured values
of Dϱ were essentially the same, indicating that the Stokes-
Einstein radius of dextran does not change with solution ionic
strength. Hindered diffusion coefficients of dextran were also
measured in different ionic strength solutions. The results are
presented in Figure 3 and show that the ionic strength of the
solution has no effect on the hindered diffusion coefficients.
If there were electrostatic interactions between dextran and
the pore wall, one would expect the diffusion coefficient to
decrease as the ionic strength of the solution decreases.
Therefore, it is reasonable to assume that electrostatic inter-
actions between dextran and pore wall are negligible. It was
also assumed that electrostatic interactions between PEG
Ž
.
also an uncharged polymer and the pore wall were negligi-
ble.
Dis cus s ion
Figure 2. Dimens ionles s hindered diffus ion coefficient
DrDϱ as a function of the s olute to pore-s ize
ratio ␭.
The hindered diffusion coefficients of dextran measured in
this study are larger than those predicted using existing theo-
retical predictions and are in good agreement with values re-
Ž
.
ported by Davidson and Deen 1988b for dextran in polycar-
bonate membranes. The measured hindered diffusion coeffi-
cients of PEG are in good agreement with the values deter-
mined for dextran. Davidson and Deen also measured hin-
dered diffusion coefficients of PEG, but used material with a
weight average molecular weight of 21,000, which is about
twice as large as the PEG used for our measurements. Be-
cause of this difference in the size of the PEG used in the
equation or the Davidson and Deen theory. Our measured
effective diffusion coefficient values are also in good agree-
Ž
.
ment with those reported by Davidson and Deen 1988b for
dextran, but are larger than those reported by Davidson and
Ž
.
Deen 1988b for PEG.
The pore size of the membrane was measured before and
after each polymer diffusion measurement and the change in
observed pore size for each experiment is listed in Table 3.
Pore sizes were found to decrease, but only slightly. The pore
radius changes during each diffusion experiment were less
than 3.7 nm. This small change in membrane pore size is an
indication that dextran adsorption is insignificant. Similarly
small pore-size changes were also observed for the experi-
ments with PEG.
Ž .
two studies, the range of dimensionless solute size ␭ from
the two studies only overlaps slightly. The results do seem to
indicate that the values reported from our measurements are
larger than the values reported by them. It is possible that
this difference may be attributed to different polymer confor-
mations experienced because of differences in the molecular
weight of the two polymers. In their study, Davidson and
AIChE Journal
June 2000 Vol. 46, No. 6
1153

Ž .
Deen found significant adsorption of PEG on the pore wall,
as indicated by changes in pore size. Our results showed min-
imal change in pore size during a PEG diffusion experiment,
indicating insignificant adsorption of PEG. Therefore, it is
also possible that the difference in diffusion coefficients mea-
sured in the two studies might be due to different extents of
adsorption. The PEG used by Davidson and Deen was ob-
where A is the Hamaker constant J between the solute and
polycarbonate membrane in water and h is the distance be-
Ž
.
tween the particle surface and the pore wall cm . Substitut-
ing Eq. 14 into Eq. 13 yields the following expression for the
interaction energy between the solute at position ␤ and the
pore wall
Ž
.
tained from Toyo Soda Varian Associates, Sunnyvale, CA ,
A
␭
␭
1y␤ y␭
1y␤ q␭
E
␤ s
q
qln
15
Ž
.
Ž .
while the PEG used for our study was obtained from Pres-
½
5
6
1y␤ y␭ 1y␤ q␭
Ž
.
sure Chemical Pittsburgh, PA . A possible explanation for
the difference in apparent adsorption of PEG from the two
studies is some difference in chemistry arising from differ-
ences in preparation procedure or purification of the poly-
mers.
Attractive polymer-pore interactions could be responsible
for the observed increase in effective diffusion coefficients
relative to those predicted from models that do not include
these long-range interactions. Davidson et al. 1987 consid-
ered attractive interactions between the chain segments of
the polymer and the pore wall by means of a square-well po-
tential in a region of thickness dE adjacent to the pore wall.
The chain was regarded as having nq1 mass points con-
nected by its n segments. Each mass point within the square
well was assumed to have an energy ⑀kBT. They applied a
partitioning model using Monte Carlo simulations which in-
corporated this attractive potential in combination with the
hydrodynamic calculations for a noninteracting random coil.
Their experimental results fit the model when a moderate
As h™0, Eq. 15 predicts that the attractive interaction en-
ergy E ™ϱ. Since dextran and PEG did not adsorb and im-
mobilize on the pore wall, it is assumed that there is a mini-
mum value for h such that the attractive interaction energy
does not go beyond some reasonable value. Therefore, the
pore is separated into two regions: one away from the pore
wall where van der Waals attractive interactions are present
and one near the pore wall where only steric interactions are
present. A similar approach was taken by Malone and An-
Ž
.
Ž
.
derson 1978 in modeling latex diffusion through mica mem-
branes. They chose hs0.05rs as the point of separation be-
tween these two regions. In our calculations, hs0.05rs was
also chosen as the demarcation point between steric and van
der Waals interactions. An expression for the equilibrium
partition coefficient Keq is then obtained
1y1.05␭
1y ␭
Keq s2
exp y E ␤ rk T ␤d␤ q2
␤d␤
Ž
.
Ž
.
H
H
B
Ž .
interaction energy ⑀ of 0.26 was used for dextran and 0.20
0
1y1.05␭
was used for PEG. In their model calculations, the interac-
tion distance dE was set equal to the chain segment length.
They found that the partition coefficient values predicted us-
ing this approach were extremely sensitive to the thickness of
the interaction region and suggested that any conclusions
based on these calculations should be made with caution.
In this study, we have used an alternative approach to in-
corporate attractive interactions between the solute and the
pore wall on predictions of the partition coefficient. Here, we
assume that there are van der Waals interactions between
solutes and the pore wall which result in an increase in the
partition coefficient and, hence, a corresponding increase in
the effective diffusion coefficients in the pore relative to those
predicted without long-range interactions. The dextran and
the PEG were modeled as solid spheres. A semianalytical re-
sult for the van der Waals interactions between a spherical
particle and a cylindrical pore surface has been reported
16
Ž
.
The second integral represents the region over which only
steric interactions E s0 are considered. The effective dif-
fusion coefficient was determined by combining the expres-
sion for the intrapore diffusivity the second term in paren-
theses on the righthand side of Eq. 5 with the value for Keq
Ž
.
Ž
.
from Eq. 16.
Ž
.
In the calculation of E ␤ , a value of the Hamaker con-
stant A for solute and polycarbonate interaction in water is
needed. The Hamaker constant A312 for interactions be-
tween material 2 and material 3 in medium 1 can be ex-
Ž
.
pressed as Hiemenz and Rajagopalan, 1997
As A312 s A13r32 y A1r2 A12r22 y A1r2
17
Ž
.
Ž
11 .Ž
.
11
Ž
.
Bhattacharjee and Sharma, 1995 . These results show that,
where A11 is the Hamaker constant for interaction between
material 1 and material 1 separated by a vacuum and A22
and A33 are defined in a similar manner. In this study, mate-
rial 1 is water, material 2 represents dextran or PEG, and
material 3 is polycarbonate. The Hamaker constant for water
Ž
.
for relatively small particles ␭-0.2 , a simpler model for a
sphere interacting with a flat plate provides a reasonable esti-
mate of the interaction force between a sphere and a cylin-
drical surface. Because ␭-0.2 for all of our experiments, we
have used the simpler flat plate model to describe the attrac-
tive interaction between solute and pore wall. The attractive
interaction energy for this geometry was given by Hamaker
y
20
Ž
.
Ž .
A11 is 3.51=10
J Visser, 1972 , and the Hamaker con-
20
stant for polycarbonate A33 is 7.0=10y
Ž
.
Ž
J Mitchell and
.
Deen, 1984 . Hamaker constants for dextran and PEG are
Ž
.
1937
not available. The Hamaker constant for polymers is pre-
20
dicted to range from 6.15=10y to 6.6=10y20
J Visser,
Ž
A
1
x
h
1
x
y
20
.
1972 . Assuming that A for dextran and PEG is 6.38=10
E x s
Ž .
q
q2ln
13
14
Ž
Ž
.
.
½
xq1 5
Ž
.
12
xq1
1y␤ y␭
2␭
J the middle value in this range , Eq. 17 predicts As5.0=
21
10y J for interactions between dextran or PEG and poly-
carbonate in water. The prediction for D obtained using this
value for A is shown in Figure 2. A comparison of this pre-
xs
s
2rs
1154
June 2000 Vol. 46, No. 6
AIChE Journal

diction to the experimental values shows excellent agree-
ment.
In the interpretation of our data, we have chosen an alter-
native approach to explaining our results rather than using
the modified random coil model used by Davidson and Deen
to describe the solute-pore wall interactions in the model cal-
culations is in agreement with a value derived from literature
values for polymer-polymer and water-water interactions in a
vacuum.
Ž
.
1988b to describe their hindered diffusivity values for dex-
Acknowledgment
This study was supported by the National Institutes of Health un-
der project No. 1 R15 GM51012-01.
Ž
.
tran. As described earlier, the Davidson et al. 1987 parti-
tioning model includes a square well potential between the
polymer and the pore wall and requires a number of arbitrary
parametersᎏthe number of segments in the polymer chain,
the thickness of the square well, and the interaction energy
between each segment and the pore wall. By treating the dif-
fusing solutes as rigid spheres, we are able to rely on previ-
ously developed theories describing van der Waals attractions
between a sphere and a wall, and to utilize literature values
for the Hamaker constant for the materials in our systems.
While dextrans may not be spheres, results from light scat-
tering measurements indicate that they are also more com-
pact than ideal random coils Suzuki et al., 1982 . Light scat-
tering measurements with poly ethylene oxide have shown
that this polymer behaves as a linear random coil for molecu-
lar weights as low as 80,000 Devanand and Selser, 1991 .
However, the poly ethylene glycol used for our measure-
ments had a molecular weight considerably smaller than this
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Ž
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Manuscript recei®ed May 21, 1999, and re®ision recei®ed Jan. 12, 2000.
1156
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AIChE Journal