Determination of a Setup Correction Function to Obtain Adsorption Kinetic Data at Stagnation Point Flow Conditions

Abstract

This paper is the first report on the characterization of the hydrodynamic conditions in a flow cell designed to study adsorption processes by spectroscopic ellipsometry. The resulting cell enables combining the advantages of in situ spectroscopic ellipsometry with stagnation point flow conditions. An additional advantage is that the proposed cell features a fixed position of the "inlet tube" with respect to the substrate, thus facilitating the alignment of multiple substrates. Theoretical calculations were performed by computational fluid dynamics and compared with experimental data (adsorption kinetics) obtained for the adsorption of polyethylene glycol to silica under a variety of experimental conditions. Additionally, a simple methodology to correct experimental data for errors associated with the size of the measured spot and for variations of mass transfer in the vicinity of the stagnation point is herein introduced. The proposed correction method would allow researchers to reasonably estimate the adsorption kinetics at the stagnation point and quantitatively compare their results, even when using different experimental setups. The applicability of the proposed correction function was verified by evaluating the kinetics of protein adsorption under different experimental conditions.

Full text

Determination of a setup correction function to obtain adsorption kinetic data
at stagnation point flow conditions
Maria F. Mora
a,1
, M. Reza Nejadnik
a,1
, Javier L. Baylon-Cardiel
b
, Carla E. Giacomelli
c
, Carlos D. Garcia
a,*
a
Department of Chemistry, The University of Texas at San Antonio, United States
b
Departamento de Ingeniería Eléctrica y Computacional, Tecnológico de Monterrey, Mexico
c
INFIQC-Departamento de Fisicoquímica, Fac. de Ciencias Químicas, Universidad Nacional de Córdoba, Argentina
article info
Article history:
Received 16 December 2009
Accepted 11 February 2010
Available online 13 February 2010
Keywords:
Spectroscopic ellipsometry
Stagnation point
Adsorption kinetics
Polyethylene glycol
Protein adsorption
abstract
This paper is the first report on the characterization of the hydrodynamic conditions in a flow cell
designed to study adsorption processes by spectroscopic ellipsometry. The resulting cell enables combin-
ing the advantages of in situ spectroscopic ellipsometry with stagnation point flow conditions. An addi-
tional advantage is that the proposed cell features a fixed position of the ‘‘inlet tube” with respect to the
substrate, thus facilitating the alignment of multiple substrates. Theoretical calculations were performed
by computational fluid dynamics and compared with experimental data (adsorption kinetics) obtained
for the adsorption of polyethylene glycol to silica under a variety of experimental conditions. Addition-
ally, a simple methodology to correct experimental data for errors associated with the size of the mea-
sured spot and for variations of mass transfer in the vicinity of the stagnation point is herein
introduced. The proposed correction method would allow researchers to reasonably estimate the adsorp-
tion kinetics at the stagnation point and quantitatively compare their results, even when using different
experimental setups. The applicability of the proposed correction function was verified by evaluating the
kinetics of protein adsorption under different experimental conditions.
Published by Elsevier Inc.
1. Introduction
The adsorption of macromolecules, colloids, and bioparticles to
solid surfaces has been widely reported in the literature [1–5] with
studies that include different methodologies, surfaces, and applica-
tions [6–8]. The time-dependent nature of the adsorption pro-
cesses as well as the significance of their initial steps for various
biomedical and industrial applications, highlight the importance
of kinetic studies. Regardless of the selected analytical technique
to follow the process, there are two main experimental approaches
to study adsorption kinetics of particles: batch and flow experi-
ments. Batch experiments are generally performed by monitoring
the depletion of adsorbate in a dispersion of sorbent particles [9].
Although this approach is attractive due to its minimal instrumen-
tal requirements, it involves non-uniform hydrodynamic condi-
tions, and is limited to slow adsorption processes. A more
efficient way to study adsorption processes, particularly those
involving shorter time-scales, is by flowing a solution of adsorbate
over the sorbent surface. Several authors have pointed out the
advantages of performing such studies including the well-con-
trolled hydrodynamic conditions that allow accurate measure-
ments, particularly regarding the initial stages of the adsorption/
desorption phenomena [10,11]. Various flow displacement geome-
tries have been used for such adsorption studies [12–15]. Among
them, setups yielding stagnation point flow conditions are fre-
quently used for measurement of adsorption kinetics [1,16–20].
Stagnation point flow conditions are obtained by perpendicularly
impinging a jet of solution to the sorbent surface through a cylin-
drical channel. The stagnation point is defined as the intersection
of the symmetry axis of the cylinder with the surface [1,21]. The
main advantage of this arrangement is that the hydrodynamics
of the mass flux at the stagnation point can be accurately described
[1,22,23].
Stagnation point flow conditions have been used for adsorption
studies in conjunction with several techniques such as microscopy
[24], quartz crystal microgravimetry [20], evanescent wave light
scattering [25], and reflectometry [8,20,22,26,27]. Our group is par-
ticularly interested in the application of spectroscopic ellipsometry
(SE) for adsorption studies because it can provide real-time infor-
mation regarding the kinetics of the adsorption process as well
as the structure of the adsorbed layer for a broad range of materials
and substrates. Ellipsometry is an optical technique that measures
changes in the reflectance and phase difference between the paral-
lel (R
P
) and perpendicular (R
S
) components of a polarized light
beam upon reflection from a surface [28]. The intensity ratio of
0021-9797/$ - see front matter Published by Elsevier Inc.
doi:10.1016/j.jcis.2010.02.019
*Corresponding author. Address: One UTSA Circle, San Antonio, TX 78249, USA.
Fax: +1 210 458 7428.
E-mail address: carlos.garcia@utsa.edu (C.D. Garcia).
1
Both authors contributed equally to this work.
Journal of Colloid and Interface Science 346 (2010) 208–215
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science
www.elsevier.com/locate/jcis

R
P
and R
S
can be related to the ellipsometric angles (
W
, amplitude
and
D
, phase difference as functions of wavelength or time) using
Eq. (1):
tanðWÞe
i
D
¼R
P
R
S
ð1Þ
Spectroscopic ellipsometry allows the measurement of the elli-
psometric angles as a function of the wavelength of the incident
light beam. Because ellipsometry measures the ratio of two values
originated by the same signal, the data collected are highly accu-
rate and reproducible. The output values of ellipsometry are extre-
mely sensitive to the thickness (down to the monolayer level),
optical constants, and microstructure (such as surface roughness,
index grading, and intermixing) of the films. This monolayer sensi-
tivity is useful for real-time studies of film deposition, including
the formation of layers of biological molecules on a wide variety
of substrates [29–31]. It is worth noting that although dynamic
adsorption studies performed by ellipsometry have been widely
reported in the literature, the flow conditions in these reports have
not been properly characterized [32–35]. In this regard, this paper
provides the first fully characterized flow cell to perform ellipso-
metric adsorption studies under stagnation point flow conditions.
One common issue associated with the use of optical methods,
such as reflectometry and ellipsometry, to perform adsorption ki-
netic studies is that these techniques typically underestimate the
initial adsorption rate at the stagnation point [1,19,36]. The differ-
ence between the experimental and predicted values have been
attributed to the fact that these instruments collect signals from
a spot (few square millimeters) rather than an infinitesimal point
(stagnation point) [19]. Therefore, the adsorption rates calculated
based on these signals could be significantly smaller than the
adsorption rates existing at the stagnation point. As a consequence,
similar experiments performed with diverse experimental setups
may render significantly different results (depending on the size
of the measured area), making quantitative interlaboratory com-
parisons rather challenging.
Considering the aforementioned points, this paper is intended
to address two important issues, it aims first, to demonstrate the
advantages of combining ellipsometry with stagnation point flow
conditions; and second, to provide a procedure to calculate a set-
up-specific correction function that allows for correcting experi-
mental kinetic data for errors associated with the size of the
measured spot. In order to verify the consistency of obtained data
in the proposed experimental setup with stagnation point flow
conditions, the liquid cell used here was characterized by studying
the adsorption kinetics of polyethylene glycol (PEG) to silica
according to a procedure developed by Dijt et al. [1]. Then, theoret-
ical calculations were performed by computational fluid dynamics
and compared with the experimental data in order to determine
the correction function needed for the geometry of our cell. Finally,
the adsorption kinetics of two proteins were studied to demon-
strate the applicability of the correction function to different
analytes.
2. Materials and methods
2.1. Cell description
Dynamic adsorption experiments were performed in a commer-
cial flow cell designed for spectroscopic ellipsometry (J.A. Woollam
Co; Lincoln, NE). In order to control the flux of adsorbate to the
substrate, the cell was modified by fixing an L-shaped stainless-
steel tube (R= 0.381 mm) to the cell (see Fig. 1). One end of the
tube faced the substrate at the same spot where the incident light
beam hits the surface. The other end of the tube was connected,
using Tygon tubing, to a peristaltic pump (Minipuls3, Gilson; Mid-
dleton, WI). A two-way valve (V100D, Upchurch Scientific; Oak Ar-
bor, WA) was also connected in series to enable rapid switching
between the background electrolyte and the solution containing
the adsorbate. The distance from the end of the tube to the surface
was kept constant (h= 1.1 mm) for experiments described herein.
Using this arrangement, the adsorbate solution impinged the sur-
face of the substrate with an angle of incidence equal to 90°(with
respect to the surface). The proposed setup provides an alternative
experimental design to those used by Arwin [33,37], Petri [38],
Pérez [39], or Logothetidis [34] and enables monitoring of the
adsorption process in situ and in real time. An additional advantage
of the proposed modification is that the cell allows the adjustment
of the distance between the inlet tube and the substrate providing
versatility to the design (see Fig. 1). Furthermore, the fixed position
Incident
beam
R
h
Stagnati on Poin t
To p u mp
Substrate
Fig. 1. Schematic drawing showing the described cell (left) and its main components: (1) inlet tube, (2) substrate holder, (3) substrate, (4) drain, (5) inlet tube positioner, and
(6) incident beam. Top view of the cell (right), highlighting the arrangement of the tube with respect to the substrate (right).
M.F. Mora et al. / Journal of Colloid and Interface Science 346 (2010) 208–215 209

of the inlet tube with respect to the substrate (which also has a
fixed position) after adjustment, facilitates the alignment of differ-
ent substrates.
2.2. Computational fluid dynamics
The cell was modeled using two parallel solid plates separated
by a distance h. One of the plates was assigned to the substrate
on which the adsorption process takes place, and used as a refer-
ence point (0, 0). The second plate (located at z=h) was assigned
to the end of the L-shaped tube (of radius R), which delivers the
solution containing the adsorbate. During all experiments de-
scribed in this paper, the flow was developed in the cell, and the
fluid moved to an outflow region, located away from the symmetry
axis. For all the calculations, a ratio of h/Requal to 2.8 was consid-
ered and all the fluids were assumed to be Newtonian and incom-
pressible. The ratio h/Ris an important parameter that determines
the flow distribution and the adsorption kinetics [40]. The flow of
fluid along the cell can be characterized using the Reynolds num-
ber (Re), a dimensionless parameter defined as:
Re ¼
q
RU
g
ð2Þ
where
q
and
g
are the density and the viscosity of the fluid respec-
tively, Ris the radius of the tube, and Uis the mean fluid velocity.
The Reynolds number, which characterizes the tendency of a fluid
to develop turbulence or to flow with a laminar regime, was varied
in the 2–50 range. This range is significantly smaller than the value
of Re at which the flow changes from laminar to turbulent
(Re > 2000 [41,42]). Under the specified conditions, the laminar
flow can be described using the steady-state Navier–Stokes equa-
tion and the continuity equation, as previously described by Dabros
and van de Ven [10]. Due to the axisymmetrical design of the cell,
these equations can be expressed in cylindrical coordinates rand
z, with the form
@p
dr þ
g
1
r
@
@rr@u
r
@r

@
2
u
r
@z
2
u
r
r
2
!
¼0ð3Þ
@p
dz þ
g
1
r
@
@rr@u
z
@r

þ@
2
u
z
@z
2
!
¼0ð4Þ
1
r
@
@rðru
r
Þþ@u
z
@z¼0ð5Þ
where u
r
is the radial component of the fluid velocity, pis the pres-
sure, and u
z
is the axial component of the fluid velocity. Coordinates
were non-dimensionalized with respect to the radius Rand the
components of the fluid velocity were referred to the mean velocity
U. The fluid flow at the exit of the tube was assumed to show a par-
abolic profile, accordingly with a developed Poiseuille flow. With
this assumption, the boundary conditions required to solve Eqs.
(3)–(5) can be formulated as:
u
r
¼u
z
¼0 at the solid surfaces of the cell ð6Þ
u
r
¼0;u
z
¼Uð1r
2
Þat the surface of the plate
located at z¼0ð7Þ
@u
z
@r¼0 in the symmetry axis ð8Þ
u
z
¼0 at the outflow region of the cell ð9Þ
The numerical solutions for the fluid flow equations were ob-
tained using the finite element method, as implemented in the
COMSOL Multiphysics software. A mesh consisting of 4000 quadri-
lateral elements and 4141 points was constructed along the geom-
etry of the model cell. At each point, a solution was approximated
with a low order polynomial, and was further refined through a
number of iterations. Close to the surface of the plate located at
z= 0, the radial and axial components of the fluid velocity can be
approximated by the analytical expressions described in Eqs. (10)
and (11),
u
r
¼
a
rz ð10Þ
u
z
¼
a
z
2
ð11Þ
where
a
is a dimensionless flow intensity parameter that is con-
stant for a certain cell geometry and flow rate [43]. The proposed
approach was validated by comparing the results obtained using
different geometries [10]. In all cases, identical solutions were
achieved.
2.3. Reagents and solutions
All chemicals were analytical reagent grade and used as re-
ceived. All aqueous solutions were prepared using 18 MOcm water
(NANOpure Diamond, Barnstead; Dubuque, Iowa). Monodisperse
polyethylene glycol was purchased from Polymer Laboratories
(Amherst, MA) and used without further purification. Stock solu-
tions of PEG were prepared by dissolving a known amount of solid
material in DI water. Catalase (CAT) from bovine liver was pur-
chased as a lyophilized powder (2–5 Ug
1
) from Sigma Aldrich
(Saint Louis, MO) and kept at 20 °C until used. Bovine serum
albumin (BSA), fraction V (heat-shock treated) was purchased from
Fisher scientific (Fair Lawn, NJ) and kept at 4°C until used. Citrate
buffer (10 mM) was selected as the background electrolyte because
it provides high buffer capacity (pKa
1
= 3.13, pKa
2
= 4.76,
pKa
3
= 6.4) around the isoelectric point (IEP) of the proteins (IEP-
CAT
= 5.4 [44] and IEP
BSA
= 4.7 [45]). The pH of the solutions was ad-
justed using either 1 M NaOH or 1 M HCl (Fisher Scientific; Fair
Lawn, NJ) and measured using a glass electrode and a digital pH
meter (Orion 420A+, Thermo; Waltham, MA). Stock solutions of
proteins were prepared by dissolving a known amount of protein
in 10 mM citrate buffer. Other protein solutions were prepared
by diluting the corresponding amount of stock in citrate buffer. Ta-
ble 1 summarizes the corresponding molecular weights (M
w
), ratio
of weight to number molecular weight [46], and diffusion coeffi-
cients (calculated as described by Shao and Baltus [47]) of the se-
lected polymers and proteins. All experiments were performed
using h111iSi/SiO
2
wafers (Sumco; Phoenix, AZ) as substrates. Be-
fore each experiment, Si/SiO
2
substrates were cleaned in a 1:1 mix-
ture of HCl:H
2
O
2
for 10 min at 80 °C, rinsed thoroughly with water,
and immersed in DI water until used to avoid contamination.
2.4. Adsorption experiments
Both the characterization of the substrates as well as the dy-
namic adsorption experiments were performed at room tempera-
ture using a variable angle spectroscopic ellipsometer (WVASE,
J.A. Woollam Co; Lincoln, NE). Spectroscopic ellipsometry has pro-
ven suitable to study adsorption processes, and provides useful
information about the optical constants and structure of the ad-
sorbed film [37,48–51]. Dynamic adsorption experiments were
Table 1
Molecular weights, ratio of weight to number molecular weight, and calculated
diffusion coefficients of PEG and the selected proteins.
Molecular weight M
W
/M
N
D(10
7
cm
2
s
1
)
PEG 278,100 1.05 1.36
PEG 93,000 1.06 2.50
PEG 8730 1.05 9.34
Catalase 250,000 – 4.00 [74]
Bovine serum albumin 66,000 – 5.93 [74]
210 M.F. Mora et al. / Journal of Colloid and Interface Science 346 (2010) 208–215

performed in the modified cell which was mounted directly on the
vertical base of the ellipsometer. In all experiments, the variation
of
W
and
D
as a function of time was determined at an angle of
incidence of 70°, as defined by the inlet/outlet of the UV fused-sil-
ica windows. Considering that the adsorption of PEG onto silica is a
rather fast process, only one wavelength (650 nm) was used in or-
der to increase the data acquisition rate and to determine the ini-
tial adsorption rate ((d
C
/dt)
t?0
) more accurately. The collected
data were modeled using the WVASE software package (J.A. Wool-
lam Co; Lincoln, NE). Experimental results were interpreted using a
model consisting of three uniaxial layers with optical axes parallel
to the substrate surface. The dielectric functions of the substrates
were described by a layer of Si (bulk; d= 1 mm) and a layer of
SiO
2
(d= 2.5 ± 0.5 nm). The adsorbed layers were described using
a Cauchy parameterization model, according to Eq. (12), where A,
B, and Care computer-calculated fitting parameters, and kis the
wavelength of the incident light beam. The values of A,B, and C
for each analyte were selected in such a way that the refractive in-
dexes match the values reported in the literature [35]. The Cauchy
constants as well as the refractive indexes are summarized in Ta-
ble 2. In agreement with previous reports, the extinction coeffi-
cient of all the adsorbates used in this study was considered to
be zero [1,52–54]. The mean square error (MSE) was used to quan-
tify the difference between the experimental and model generated
data.
n
ðkÞ
¼AþB
k
2
þC
k
4
ð12Þ
Prior to each adsorption experiment, a spectroscopic scan was
performed for a clean strip of silicon wafer in order to determine
the thickness of the SiO
2
. This procedure allowed verifying the
thickness of each substrate; therefore improving the accuracy of
the calculation of the adsorbed layer. Then, the dynamic experi-
ment was initiated by pumping water through the cell (5 min)
to measure the baseline. Next, the valve was switched, adsorbate
solution was introduced, and the adsorption process started. After
the process reached the corresponding equilibrium and no signifi-
cant change in the signal was observed, the dynamic scan was
stopped, and a more accurate spectroscopic scan was collected in
the 300–850 nm range (with 10 nm steps). This scan was used to
verify the thickness of the adsorbed layer.
The adsorbed amount (
C
, expressed in mg m
2
) was calculated
using Eq. (13),
C
¼dðnn
0
Þ
ðdn=dcÞð13Þ
where nand n
0
are the refractive index of the adsorbed layer (see
Table 2) and the ambient (H
2
O), respectively [55]. In accordance
with previous reports [1,55–58], a constant refractive index incre-
ment (dn/dc) was considered for the adsorbed layers (see Table 2).
Experiments performed in this way provided data for calculating
the initial adsorption rate ((d
C
/dt)
t?0
) and the saturation adsorbed
amount (
C
SAT
). These values were calculated for the adsorption of
PEG for different polymer concentrations, Reynolds numbers, and
molecular weights, as well as for the adsorption of proteins.
2.5. Validation of experimental results
The rate of adsorption at the solid/liquid interface is generally
considered to comprise three steps: transport of the solute mole-
cules from the bulk to the interface, attachment to the surface,
and relaxation on the surface. When perfect sink boundary condi-
tions are present (concentration at the surface is zero) and in the
absence of a barrier, the flux of solute (J) towards the surface can
be described by Eq. (14),
J¼0:776
m
1=3
R
1
ð
a
ReÞ
1=3
D
2=3
Cð14Þ
where
m
is the kinematic viscosity of the solvent, Rthe inner radius
of the tube through which the solution enters the cell (0.381 mm), D
the diffusion coefficient of the studied adsorbate and, Cis the con-
centration of the solute in the bulk [1,59].
It is also worth considering that, depending on the interactions
involved (van der Waals, electrostatic, polymer bridging, or steric
repulsion), solutes have a certain probability of attaching to the
surface. This probability is represented by the efficiency factor
(b) which relates the flux of adsorbate (J) to the adsorption rate
((d
C
/dt)
t?0
) according to Eq. (15). Other denominations commonly
used in the literature to refer to this factor are retardation factor
[43] or free surface area fraction [60].
d
C
dt

t!0
¼bJð15Þ
Under perfect sink boundary conditions, which can exist when
the adsorbate has a high affinity for the sorbent surface, this prob-
ability equals one [19]. This will likely be the case for adsorption of
neutral polymers such as PEG [1]. As a consequence, in these sys-
tems, the initial rate of adsorption should be equal to the maxi-
mum rate of mass transfer from the bulk (J)[1].
3. Results and discussion
In order to demonstrate that the experimental setup presented
in this study enables the formation of a stagnation point and that
the adsorption process can be described by Eq. (14), two different
approaches (one theoretical and one experimental) were taken.
The results of each strategy are herein discussed and compared.
Subsequently, a methodology to correct the obtained kinetic
parameters to account for the size of the measured spot is
proposed.
3.1. Computational fluid dynamics
Initially, the development of the flow fields in the cell was
investigated. Fig. 2 shows the velocity profiles for Re of 2.8, 12.0,
24.7, and 49.2. The streamlines can be interpreted as the path of
fluid on the cell so the line tangent to the streamline represents
the direction of the fluid velocity vector at a given point in space
[40]. As can be observed in Fig. 2, characteristic hyperbolic field
lines around the point where the flow intersects with the substrate
are developed, supporting the presence of a stagnation point. It is
also noted that the streamlines slightly deviate from hyperbolas
at high Re. This behavior observed at high Re leads to a vortex for-
mation away from the stagnation point. However, its presence
should not affect the adsorption process because it is relatively
far from the interface. Similar flow patterns have been previously
reported in the literature for other impinging-jet geometries
[10,40,61].
As noted by Dabros and van der Ven [10], the main advantage of
performing adsorption experiments under stagnation point condi-
tions is that the influence of hydrodynamic disturbances on the
flow near the stagnation point can be eliminated. Although it has
Table 2
Optical parameters used to calculate the thicknesses and adsorbed amounts of the
selected adsorbates. A,B, and Care the Cauchy parameters, nis the refractive index,
and dn/dc is the variation of refractive index with concentration. The reported
refractive indexes are the values obtained with the Cauchy parameters for 650 nm.
The references included here reported the values of nfor 633 nm.
ABCn
650 nm
dn/dc
PEG 1.34 0.01 0 1.362 [1] 0.136 [1,56]
Proteins 1.44 0.01 0 1.465 [35] 0.180 [55,57]
M.F. Mora et al. / Journal of Colloid and Interface Science 346 (2010) 208–215 211

been generally reported that
a
is roughly proportional to Re [1,10],
no information was available for the specific setting of our cell de-
sign (h/R= 2.8). Consequently, to demonstrate that the described
cell allows the development of stagnation flow conditions, the
dependence of the dimensionless parameter
a
with respect to Re
was calculated (see Fig. 3) for two different cell designs featuring
h/R= 2.8 and h/R= 5 and compared to previously reported data
for h/R= 1 and h/R= 1.7 [10]. As can be observed, the model herein
described follows the same general trend that others previously re-
ported. Although designs with large h/Rvalues suffer from low
mass-transfer efficiencies [62], the value selected for our cell lays
in the range of 1.5–4, which is considered optimum for adsorption
kinetic measurements [10]. More details regarding the character-
ization of cells with other geometries can be found elsewhere
[1,10,62,63].
3.2. Adsorption of PEG
It is well-known that PEG readily adsorbs to a variety of mate-
rials with high affinity and with no significant energy barriers
[9,64,65]. Therefore, the adsorption of PEG is determined by the
mass transfer rate from the bulk solution, making it reasonable
to assume b=1 [1]. Consequently, the initial adsorption rate
((d
C
/dt)
t?0
) of PEG (as measured in our cell) should be equal to
the flux (J) of PEG to the surface, following Eq. (14).
A typical dynamic adsorption experiment is shown in Fig. 4.In
agreement with previous reports [1,59,66], an initial fast adsorp-
tion process, followed by a slower one was always observed. Dy-
namic measurements (see Fig. 4) enabled calculating the initial
adsorption rate ((d
C
/dt)
t?0
) under different experimental condi-
tions. The results are summarized in Fig. 5.
As can be observed in Fig. 5A, the initial rate of adsorption of
PEG was proportional to the concentration of polymer in the
impinging jet. Fig. 5B shows that a linear dependence between
the initial adsorption rate and D
2/3
was obtained when polymers
with different molecular weights (8000–280,000 g mol
1
) were
adsorbed (c= 1 mg L
1
).
C
SAT
values of 0.16 ± 0.02 mg m
2
,
0.33 ± 0.02 mg m
2
, and 0.40 ± 0.05 mg m
2
were obtained for
PEG with molecular weights of 8730, 93,000, and 278,100 g mol
1
,
respectively. These results follow the general theory behind poly-
mer adsorption which predicts an increase in
C
SAT
for larger
adsorbing molecules [1,16]. Furthermore, as shown in Fig. 5C,
(d
C
/dt)
t?0
increases with an increase in Re
1/3
in the concentration
range (0.5–5 mg L
1
) studied.
In general, the initial adsorption rates obtained for our system
were smaller than those previously reported with different cell set-
ups (same polymer and h/R= 1.6) [1]. This can be in part explained
by the difference in h/Rvalue. The higher h/Rvalue selected for our
cell implies a lower
a
(see Fig. 3) and, consequently, a lower rate of
mass transfer. Furthermore, despite the fact that the experimental
data obtained with our system followed the trends predicted by Eq.
(14), the initial adsorption rates were significantly lower than the
flux of PEG towards the surface at the stagnation point (as pre-
dicted by Eq. (14)). It is necessary to note that the ellipsometer,
as many other optical techniques, acquires data over a relatively
large area compared to the infinitesimal stagnation point. In fact,
it has been previously reported [19] that initial adsorption rates
obtained by reflectometry (under stagnation point flow conditions
Fig. 2. Fluid streamlines from the inlet tube to the substrate determined by
computational fluids dynamics for Reynolds numbers used in this study. In all cases
h/R= 2.8.
0 5 10 15 20 25 30 35 40 45 50 55
0
5
10
15
20
25
30
5
2.8
1.7
α
Re
h/R = 1
Fig. 3. Dependence of the flow intensity parameter (
a
) on the Reynolds number
(Re) for several values of h/R. The data for h/R= 1 and 1.7 was used with permission
from Ref. [10].
0 200 400 600 800 1000 1200
0.0
0.1
0.2
0.3
0.4
ΓSAT
(dΓ/dt)t→0
Γ (mg/m2)
Time (sec)
Injection
of PEG
Fig. 4. Example of a typical curve for adsorption of PEG onto Si/SiO
2
substrate.
Conditions: M
w
= 93,000 g mol
1
, concentration 3 mg L
1
and Re = 24.7.
212 M.F. Mora et al. / Journal of Colloid and Interface Science 346 (2010) 208–215

and for transport limited adsorption processes) are only 50–80% of
the theoretical values predicted by Eq. (14). Because the local mass
flux decreases significantly at interfacial points away from the
stagnation point, the average mass transfer, i.e. measured mass
transfer, to the substrate decreases with increasing the area of
measurement (compared to the stagnation point) [19]. Theoretical
calculations performed by Adamczyk et al. [67] using a circular
spot rather than a point, demonstrated that the measurement over
an area rather than at a point is indeed the source of deviation of
experimental values from the theoretical ones for the stagnation
point. In fact, it was shown that when the radius of the measured
spot (r) equals to the radius of the inlet tube (r/R= 1), the experi-
mentally obtained adsorption rates are only 90% and 70% of the
flux at the stagnation point for Re of 1 and 48, respectively. The
mentioned underestimation of the initial adsorption rates should
be accounted for in our system because the area of the measured
spot is significantly larger (approximately 7 mm
2
) than the values
normally reported in the literature for reflectometry (1mm
2
). In
fact, it was determined by computational dynamics that for our
experimental setup,
a
reaches a value of approximately zero when
the distance from the stagnation point (r) is greater than three
times the radius of the impinging tube (see Supplementary infor-
mation). It should be considered here that the estimated area of
the measured spot in our ellipsometer comprises interfacial points
that yield r/R> 3 where
a
is actually very low. It is also worth not-
ing that because
a
depends on Re, the errors in the experimental
data are also Re-dependent for a fixed area around the stagnation
point [67]. This fact complicates the comparison of the results ob-
tained for different Re numbers even within the same experimen-
tal setup and, in turn, highlights the necessity of finding a setup-
specific correction function. Such function would allow researchers
to reasonably estimate the adsorption kinetics at the stagnation
point and quantitatively compare their results even when using
different cell setups and techniques.
In order to find such a function, the experimental initial adsorp-
tion rates ((d
C
/dt)
t?0
) were analyzed in relation to the theoretical
values for flux predicted by Eq. (14) (J
THEO
). It was observed (see
Supplementary information) that, in agreement with previous re-
ports from the literature [67], the correction needed to account
for the size of the measured spot is a function of Re. Subsequently,
the ratio J
THEO
/(d
C
/dt)
t?0
was assumed to be a simple power func-
tion of Re and a correction function was empirically obtained (Eq.
(16)).
J
THEO
d
C
dt

Corrected
¼d
C
dt

t!0
2:3406 Re
0:4094
ð16Þ
The corrected initial adsorption rates ((d
C
/dt)
Corrected
) provide a
more realistic estimation of the adsorption kinetics at the stagna-
tion point. It is worth noting that only experiments performed
0123456
0.000
0.002
0.004
0.006
0.008 Re = 49.2
24.7
(dΓ/dt)t → 0 (mg/m2sec)
[PEG] (mg/L)
12.0
0.5 1.0 1.5 2.0
0.000
0.001
0.002
0.003
Re = 49.2
24.7
12.0
(dΓ/dt)t → 0 (mg/m2sec)
D2/3 (10-7 m4/3/s2/3)
2.8
051015
0.000
0.002
0.004
0.006
0.008
5 mg/L
3
1
(dΓ/dt)t → 0 (mg/m2sec)
Re1/3
0.5
(A)
(B)
(C)
Fig. 5. Initial adsorption rate of PEG as a function of: (A) the concentration for three
Re numbers (93,000 g mol
1
), (B) the diffusion coefficient (c= 1 mg L
1
), and (C)
the cube root of the Reynolds number for different PEG concentrations (M
w
=
93,000 g mol
1
).
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.02
0.04
0.06
0.08
0.10
CAT
IEP
(dΓ/dt)Corrected (mg/m2sec)
JTHEO (mg/m2sec)
BSA
Fig. 6. Corrected experimental values ((d
C
/dt)
Corrected
) calculated from Eq. (16)
versus the theoretical flux (J
THEO
) calculated from Eq. (14) for PEG. All of the
experiments were included. The solid line represents the linear fit (slope =
1.00 ± 0.02, R
2
= 0.9948). Dashed lines represent the prediction bands for a 95%
confidence level. (d
C
/dt)
Corrected
for catalase (CAT) and bovine serum albumin (BSA)
below (h), at (s), and above (4) the IEP (c= 0.005 mg mL
1
).
M.F. Mora et al. / Journal of Colloid and Interface Science 346 (2010) 208–215 213

using adsorbates with well-known behavior (such as PEG, (d
C
/
dt)
t?0
=J) would allow the calculation of this function for a given
set of experimental conditions. Fig. 6 shows the correlation be-
tween experimental values after applying the correction function
((d
C
/dt)
Corrected
) and the theoretical initial adsorption rates (J
THEO
).
As can be observed, a very good agreement between the two
parameters was obtained (slope = 1.00 ± 0.02, R
2
= 0.9948) indicat-
ing that the correction is valid for all data points. It is also worth
mentioning that this function would only be valid for the experi-
mental setup described in this paper. Any other cell with a specific
geometry (h/R) and size of the measuring spot will generate data
with different ratios and consequently, another expression of the
correction function will be necessary.
In order to verify the applicability of the correction function to
other analytes, the adsorption of two proteins (catalase and bovine
serum albumin) with different molecular weights was studied (see
Table 1). For these proteins, the initial adsorption rates obtained at
pH values above, at, and below the isoelectric point were measured
and corrected using Eq. (16). It is well-known that proteins exhibit
different adsorption behavior depending on the pH of the solution
because their net charge (and therewith the attachment probabil-
ity) is affected by the pH [43,68–70]. When the adsorbing analyte
and the surface are both neutral or have opposite charges, there are
no electrostatic barriers for the adsorption and, therefore, the ini-
tial adsorption rate should be larger than in the case when similar
charges are present. In the latter case, the energy barrier is pro-
duced by the repulsion between the adsorbate and the sorbent sur-
face. Accordingly, the (d
C
/dt)
Corrected
for both proteins at the IEP fall
close to the 95% confidence interval predicted by Eq. (16) (see
Fig. 6), resembling the adsorption rates obtained for PEG. These re-
sults suggest that under these conditions, the efficiency factor (b)is
close to one. These results can be explained by considering that at
the IEP, proteins do not have a net charge and therefore the electro-
static repulsion between the molecules and the surface is minimal.
On the other hand, the (d
C
/dt)
Corrected
obtained above the IEP was
smaller than at the corresponding IEP likely due to electrostatic
repulsion between the protein and the surface. Under these condi-
tions, the adsorption rate after correction (d
C
/dt)
Corrected
is much
smaller than the J
THEO
indicating a reduction in the efficiency factor
(b1). At pH values below the IEP the results were different for
the two proteins. For BSA, when the protein is positively charged,
the initial adsorption rate is similar to the one obtained at the
IEP because there is an attraction between the protein and the sur-
face, so there is no barrier for the adsorption. These results are in
agreement with previous reports [69,71]. For catalase, at low pH
values the (d
C
/dt)
Corrected
was smaller than the corresponding
adsorption rate at the IEP. This result can be due to electrostatic
repulsion between the protein molecules due to their high positive
charge which also reduces the attachment efficiency. Similar re-
sults have been reported for other proteins on different surfaces
[70,72].
It is evident from the results discussed here that the correction
of the initial adsorption rates with the function obtained from PEG
provides reasonable results. Thus, the correction function depends
only on the cell setup and the instrumentation used, and does not
depend specifically on the adsorbate/substrate under study.
4. Conclusions
The paper describes a simple modification to a commercial cell
that enables the investigation of adsorption processes under stag-
nation point flow conditions using spectroscopic ellipsometry.
Theoretical calculations executed by computational fluid dynamics
as well as adsorption experiments performed with PEG on SiO
2
support the existence of stagnation point flow in the cell design de-
scribed in this paper. In all cases, these results are consistent with
previous reports stating that the adsorption of PEG to silica sur-
faces can be considered a fast process controlled by the transport
of the adsorbate to the surface, rather than the transfer from the
subsurface to the interface [73]. In addition, the use of a setup-spe-
cific correction function that should be determined and applied to
the experimental data in order to correct for the size of the mea-
suring spot and for variations in mass transfer away from the stag-
nation point was proposed. This correction should allow
quantitative comparisons of results obtained by different research
groups using various techniques and experimental conditions. The
validity of the correction function for other adsorbates was also
demonstrated. In addition, the application of the mentioned cor-
rection allowed to distinguish between situations where b1
and where b1.
Acknowledgments
Financial support for this project was provided in part by The
University of Texas at San Antonio, and the National Institute of
General Medical Sciences (NIGMS)/National Institutes of Health
(1SC3GM081085).
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.jcis.2010.02.019.
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