L. Palacio et al. Journal of Membrane Science 222 2003 41–51 43
Q = Q exp
− α
PC
b
µR
m
t +
t
α PC
b
µR
m
+ R
p
exp −α
PC
b
µR
m
t
p
dt
p
5 with R
p
evaluated from Eq. 4
. The integration over t
p
accounts for the time dependent blockage of the membrane surface.
Although Eq. 5
is relatively easy to evaluate nu- merically, Ho and Zydney
[11] have also developed a
much simpler analytical solution for the filtrate flow rate by assuming that the resistance of the protein layer
over the fouled region of the membrane is uniform at its maximum value given by
Eq. 4 with t
p
= 0.
Using this assumption, the coefficient multiplying the exponential in the convolution integral in
Eq. 5 be-
comes a constant and can be pulled outside of the in- tegral to give:
Q = Q exp
− α
PC
b
µR
m
t +
R
m
R
m
+ R
p
1 − exp −
α PC
b
µR
m
t 6
The first term in Eqs. 5 and 6
represents the flow rate through the open pores and is equivalent to clas-
sical pore blockage model. The second term describes the flow through the blocked pores. The flow rate is
thus described by three key parameters: a pore block- age parameter α, the ratio of the initial resistance
of the protein deposit to the membrane resistance β = R
p0
R
m
, and a parameter describing the cake growth γ = 2f
′
R
′
R
m
+ R
p0
. The best fit values of the parameters α, β, and γ
for the different protein mixtures were determined by minimizing the sum of the squared residuals between
the filtrate flow rate data and the model calculations Eq. 5
using the method of steepest descent.
Table 1 Physical characteristics of proteins
Protein Sigma catalog number
Source Molecular weight kDa ku
a
Isoelectric pH Albumin BSA
A7906 Bovine serum
67 4.7
Lysozyme A6876
Chicken egg 14
11.0 Pepsin
P6887 Pig stomach
36 1.0
a
1 kDa = 1 ku.
3. Materials and methods
3.1. Chemicals Phosphate buffered saline PBS solutions con-
sisting of 0.03 M KH
2
PO
4
, 0.03 M Na
2
HPO
4
· 7H
2
O, and 0.03 M NaOH were prepared by dissolving
pre-weighed quantities of the appropriate salts Sigma, St. Louis, MO in the desired volume of demonized
water obtained from a Barnstead water purification system BarnsteadThermodyne, Dubuque, IA with
resistivity greater than 18 M cm. All buffer solutions were pre-filtered through 0.2 m pore size Gelman
Supor-200 membranes Gelman Science, Ann Arber, MI to remove particulates and un-dissolved salts
prior to use.
Experiments were performed using bovine serum albumin, chicken egg white lysozyme, and porcine
pepsin, all obtained from Sigma St. Louis, MO. Pro- teins, either alone or in binary mixtures, were dis-
solved in PBS with the pH adjusted to 7 using NaOH as needed. All protein solutions were freshly prepared
before each experiment and used within 8 h of prepa- ration. Sigma catalog numbers and physical property
data for each protein are provided in
Table 1 .
3.2. Filtration experiments All filtration experiments were conducted using
a 25 mm diameter stirred ultrafiltration cell Model 8010, Amicon Corp. connected to an acrylic solution
reservoir that was air pressurized at 14 kPa. Data were obtained using 0.2 m polycarbonate track etched
PCTE membranes from Osmonics Livermore, CA. The stirred cell and solution reservoir were initially
filled with PBS, with the saline flux measured until steady state was attained usually within 30 min. The
stirred cell was then quickly emptied, refilled with a 2 g L
− 1
protein solution, and attached to a fresh
44 L. Palacio et al. Journal of Membrane Science 222 2003 41–51
reservoir containing additional protein solution. The system was re-pressurized within 1 min and the fil-
trate flow rate was measured by timed collection using a digital balance Sartorious Model 1580, Edgewood,
NY. At the end of the filtration, the stirred cell was rinsed with PBS, and the steady state PBS flux was
re-evaluated. All experiments were performed at room temperature 22 ± 2
◦
C and without stirring to avoid protein aggregation during the filtration run
[8,13] .
Additional details on the experimental methods are provided by Ho and Zydney
[11] .
3.3. Colloidal properties Zeta potentials for the proteins, both alone and in bi-
nary mixtures, were measured by a ZetaMaster
®
from Malvern Instruments Ltd. The measurement is based
on the dispersion of light caused by colloidal parti- cles or macromolecules moving under the action of
an electric field. The Doppler effect leads to a mea- surable frequency shift depending on the velocity of
the particle that can be related to the electrophoretic mobility and in turn the particle zeta potential.
Measurements were done with 5 g L
− 1
protein so- lutions at 0.01 M ionic strength. This higher protein
Fig. 1. Normalized permeate flux JJ vs. time for pure BSA and pepsin and some mixtures. The used mixtures are characterized by its
BSA MM.
concentration and lower solution ionic strength were necessary to give a detectable light dispersion. Zeta
potentials were measured over a range of pH, which was adjusted using phosphoric acid and NaOH.
Particle sizes and molecular weights were measured by laser light scattering using an Autosizer Lo-C
®
from Malvern Instruments Ltd. Data were obtained using 2 g L
− 1
protein solutions as in the filtration ex- periments.
4. Results and discussion
