Determining Pathogen Adhesive Parameters on Heterogeneous

Determining Pathogen Adhesive Parameters on
Heterogeneous Surfaces
Scott A. Bradford1 and Saeed Torkzaban2
1US
Salinity Laboratory, USDA, ARS, Riverside, CA
2CSIRO,
Glen Osmond, Australia
Funding Sources: USDA NP 214
DLVO or XDLVO Interaction Energies
• Uses macroscopic zeta potentials
• Neglects microscope heterogeneities
Roughness
Absorption/desorption of ions, organics, and
clays
Energy Barrier
Sec. min.
Primary Min.
Microscale Heterogeneities
(Taboada-Serrano et al., 2005)
Transients in solution chemistry can induce
changes in chemical heterogeneity.
Surface Integration Techniques


Φ ( D) = ∑ ∑ [P(h )(e1 ⋅ ez )(n ⋅ ez )dA]
particle  wall

Duffadar and Davis (2007)
Trajectory Analysis
dvc
∑ Fi = mc
dt
i
V
dωc
∑ Ti = 0.5mc d c
i
dt
FL
FD
FB
FA
TApplied
TAdhesive FF
1o or 2o min
FA
• Computationally intensive (small spatial and temporal
grid sizes), random heterogeneities, flow field, repeated
surface integration, limited to pore-scale and constant
solution chemistry.
Continuum Scale Models
•
•
•
•
•
Average parameters over the pore-scale (REV)
Computationally efficient
Large spatial and temporal scales
Transients in chemistry
How to account for heterogeneities (α, Smax or Sf )?
J2
s
C2
Sf(IS)
α(IS)
J1
C1
∂ (θ w1C1 )
∂J
= − 1 − θ w1αk12 C1 +θ w 2 (1 − α )k 21C 2
∂t
∂z
∂ (θ w 2 C 2 )
∂J
= − 2 + θ w1αk12 C1 − θ w 2 (1 − α )k 21C 2 − θ w 2ψ 2 s k 2 s C 2 + ρ b k d (s − f c si )
∂t
∂z
∂ (ρ b s )
= θ w 2ψ 2 s k 2 s C 2 − ρ b k d (s − f csi )
∂t
Surface Integration
Az ≅
4πrc
κ
(Duffadar and Davis, 2008)
•
•
•
•
Discretize Az into Nt cells of size Ah and charge ζ1 or ζ2
Total number of cells with ζ2 is N2
Possible charge realization N2 =0, 1, … Nt
Mean Φ(h) within Az (Bendersky and Davis, 2011)
 N2 
N2
Φ1 (h ) +
Φ (h ) = 1 −
Φ 2 (h )
Nt
 Nt 
REV Scale
• Random distribution, and the fraction of N2 cells at the
REV scale is known (Ps)
• Prob(N2 ) within Az is given by the binomial distribution:
Nt !
N2
Nt − N 2
Prob( N 2 ) =
Ps (1 − P s )
N 2 !( N t − N 2 )!
• PDF of Φmax is given as plot of Prob(N2 ) and Φ max(N2)
• Mean value of Φmax at the REV scale is:
Φ max = 〈 Φ max 〉 =
*
Nt
∑ Prob(N )Φ (N )
N 2 =0
2
max
• Variance of Φmax at the REV scale <Φmax2>−<Φmax>2
2
Validation
Same approach can be applied to determine:
• Φ*(h)
• Φmax*
• Φ2min*
Bendersky and Davis (2011)
Φ*
100
0
-100
-200
0
1
2
3
4
5
6
κh
1 simulation
200 surface integrations simulations
Colloid Adhesive Parameters
• Sf= Sf1 + Sf2
• Sf1 occurs when Φmax=0 (N2>N2crit)
• Sf2 is determined from a balance of applied hydrodynamic and
resisting adhesive torques.
• α=α1+α2
• α1 and α2 is determined using the kinetic energy method
Heterogeneous SWI
Φ(h, N2) for N2=0, 1…Nt
Prob(N2), α(N2), and Sf(N2)
Homogeneous
porous media
Heterogeneous
porous media
α(N2) and Sf(N2)
α* and Sf *
Colloid
Φ(h, N2) and Prob(N2)
Sf2
1 1
 ln(Tadhesion ) − µ 
= + erf 

2 2
σ 2


• The CDF of applied hydrodynamic torque determined
from pore-scale water flow simulations for different
colloid sizes, water velocities, and grain sizes.
100
(b)
80
CDF
60
d50, µm
50 - simulated
50 - predicted
500 - simulated
500 - predicted
5000 - simulated
5000 - predicted
40
20
0
-22
-21
-20
-19
-18
Tapplied (N m)
• The resisting adhesive torque (Tadhesion) is derived from
information about Φ2min using JKR theory or an empirical
value for the friction coefficient.
Force Balance - 2o min
• KE of diffusing colloids at t=infinity
2
F (E) =
k bTk
α1 < ∫
∞
Φ 2 min + Φ bar
α2 = ∫
Φ 2 min
0
 E

 πk bTk



0.5
f det = 1 − α 2 − α 1



FA
1.0
F ( E )dE
F ( E )dE

E
exp −
 k bTk
FB
0.8
α2
0.6
0.4
0.2
0.0
-10
-8
-6
Φ 2min
-4
Shen et al. (2010) corrected for velocity as αSf
-2
0
0.8
Ps
Sf1*
0.6
0.1
0.2
0.4
0.4
0.2
0.6
0
0
10
20
30
40
50
ζ2 (mV)
Values of Sf1* as a function of ζ2 for a homogeneously charged colloid
(-30 mV) on various heterogeneously charged porous media.
Simulation properties included the following: ζ1=-70 mV, IS=10 mM,
rc=500 nm, Nt=Az/Ah=10, a Hamaker constant of 4.04E-21 J, and
Ps=0.1, 0.2, 0.4, and 0.6.
0.6
Ps
0.1
0.4
Sf1*
0.2
0.4
0.2
0.6
0
0
2
4
6
8
10
Nt=Az/Ah
Values of Sf1* as a function of Nt for a homogeneously charged
colloid (-30 mV) on various heterogeneously charged porous media.
Simulation properties included the following: ζ1=-70 mV, ζ2=0 mV, a
Hamaker constant of 4.04E-21 J, and Ps=0.1, 0.2, 0.4, and 0.6.
0.4
IS (mM)
0.3
2
Sf1*
25
0.2
100
0.1
0
0
200
400
600
800
1000
rc (nm)
Values of Sf1* as a function of rc for a homogeneously charged colloid
(-30 mV) on various heterogeneously charged porous media.
Simulation properties including the following: Ah=2500 nm2, ζ1=-70
mV, ζ2=20 mV, Ps=0.1, a Hamaker constant of 4.04E-21 J, and IS=2,
25, and 100 mM.
1
rc (nm)
0.8
5000
α2 ∗
0.6
2500
0.4
1250
500
0.2
250
0
0
20
40
60
80
100
80
100
IS (mM)
0.1
0.08
sf2*
0.06
0.04
0.02
0
0
20
40
60
IS (mM)
Predicted values of α2* and Sf2* as a function of IS and rc. Simulation
properties included Ah=500 nm2, ζ1=-70 mV, ζ2=20 mV, Ps=0.1, and a
Hamaker constant of 4.04E-21 J.
Implications – Larger Pathogens
• Bacteria and protozoa parasites have a large value of Az, therefore
little primary minima interaction is expected (α1=Sf1 =0).
• α2 is large, but Sf2 is close to zero
• Grain-grain contacts and large surface roughness locations are
expected to play a dominant role in immobilization due their
influence on hydrodynamic (lower) and adhesive (increases) forces
and torques.
E. coli O157:H7
Bradford et al. (2006)
Choi et al. (2007)
Implications – Virus
• Attachment of virus in primary and secondary
minima is likely.
• Small scale chemical heterogeneity (adsorbed
multi-valent cations) can have a strong influence
on retention of viruses.
• Factors that influence cation exchange will
influence release of virus.
Sadeghi (2012)
Implications - Hysteresis in Retention
DI
4
Initial IS
(b)
1.1 mm CML
6 mM
0.11 mm CML
1.6
31 mM
2
C/C0
C/C0
3
DI
2
106 mM
56 mM
31 mM
6 mM
56 mM
106 mM
56 mM
31 mM
6 mM
6 mM
1.2
31 mM
0.8
DI after FI
Initial IS
56 mM
1
0.4
0
0
400
800
Time (min)
1200
0
0
400
800
Time (min)
1200
Other Modifications
Ellipsoidal pathogens
Chemical heterogeneity on SWI and colloid
Physical heterogeneity (roughness) on SWI
*Physical and chemical heterogeneity on SWI
Physical heterogeneity on SWI and colloid
Physical and chemical heterogeneity on SWI and
colloid
60
20
Φ*
•
•
•
•
•
•
-20
-60
-100
0
1
2
3
κh
4
5
6
Conclusions
• A simple model has been developed to determine pathogen adhesive
parameters on heterogeneous surfaces at the REV scale.
• Results are consistent with surface integration models.
• Values of α* and Sf* were demonstrated to be very complex functions
that depend on the charge heterogeneity properties (size, amount, and
zeta potential), the colloid size, the solution chemistry, and the
system hydrodynamics.
• Irreversible colloid immobilization was predicted for smaller colloids
and higher IS when Az≤Ah.
• Larger colloids and smaller IS tended to yield reversible interactions
via a secondary minimum when Az>Ah, and colloid immobilization
could only be explained by consideration of other factors (e.g.,
surface roughness, eddy zones, and grain-grain contacts).
• Hysteresis in Sf* with IS can occur because of the different
dependence of reversible and irreversible colloid immobilization on
solution chemistry.

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