Supplementary data
1. Adsorption kinetics modeling
1.1 Model A: rate limiting step
Virus adsorption is limited only by an energy barrier, since the diffusion of the viruses is
much faster than their adsorption. The surrounding film (concentration gradient zone) does
not exist, and the integrated rate of the adsorption reaction gives:
N ads  N ads.eq (1  ekobst )
eq. S1
The expected curve has, in this case, a monoexponential shape. It is usually considered
that equilibrium is reached for t values higher than 5/kobs (it is easilly calculated with eq.2 that
t>5/kobs impliesNads> 0.99 Nads.eq). The conditions in which the pseudo first order model may
be used are still being discussed in the literature (Liu 2008, Haerifar 2012). Clearly, under
hydrostatic conditions, this model can only describe the cases for which the adsorption step of
the process is much slower than the diffusion one.
1.2 Model B: pure mass transfer limitation and high affinity of the viruses for the
surface and/or irreversible adsorption.
In molecular chemistry, a very similar situation is often encountered when all the
molecules or ions arriving by diffusion onto a planar electrode are immediately oxidized or
reduced. Here, the integrated flux of viruses (or of solutes) is given by the Ilkovic equations
(Bard and Faulkner 2001):
N ads  A
2N0
V
D

t
eq. S2
It can be seen from Equation 3 that, according to mathematics, the viruses would
adsorb indefinitely. D is the diffusion coefficient.
1.3 Model C: pure mass transfer limitation and high affinity of the viruses for the
surface and/or irreversible adsorption with an available “surface area” decreasing with
an increase in the number of adsorbed viruses
According to us, at the macroscopic scale, the available surface decrease concept is
equivalent to the one of the decrease in the number of available adsorption sites. Indeed, the
surface area apparently available may also decrease if adsorption takes place on a limited
number of localized sites.
The Langmuir approach (one site occupied by one adsorbed virus) gives (Hibbert
2002):
N ads  N ads.eq (1  e

2 AN0
V
Dt

)
eq. S3
The C’ model refers to Equation S3 in the following.
If more than one site is occupied by one adsorbed virus, we propose to introduce an
exponential surface area variation with time:
dN ads
N
D
  0 A(t ) 0
dt
V t
eq. S4
α0 is a statistical factor potentially taking into account the efficiency of  the characteristic
time of site occupancy. The higher the  value is, the slower the available surface decrease is:
N ads   0
 t 
N0 A

 D erf 

V
 
eq. S5
The C model refers to Equation S5.
1.4 Model D: pure mass transfer limitation and rapid reversible equilibrium at
the interface
In the D model only diffusion limits the arrival of virus to the biofilm, the adsorption
step is very fast, partial desorption occurs significantly immediately after the beginning of the
adsorption. At any time the equilibrium constant can be written as:
K ads 
N ads V0
N x 0 A
eq. S6
In other words, when considering the concentration of viruses at the closest vicinity of the
surface, the adsorption constant describing equilibrium is valid throughout the experiment.
Since no reference standard system exists for viruses, Kads is expressed in cm-1 units in this
work.Then, integration of Fick’s law gives (Delahay 1957):
N ads  N ads.eq (1  exp(
D
Dt
t )erfc(
)
2
K ads
K ads
eq. S7
In this case the equilibrium time (for which N ads  0.99 N ads.eq ) is given by the relation:
t
56.5K ads 2
D
eq. S8
1.5 Model E: limit boundaries model
The most simple way to deal with adsorption experiments is to consider at the same
time all the phenomena that may happen in such experiments: a rate limiting step on the same
time scale as the mass transport limit, a non-zero adsorbed quantity a t=0 due to non-ideal
experimental conditions, and a statistic factor (hereafter α0) accounting for the adsorption
efficiency of the contact virus/surface. This is the simplest way for the mind, nevertheless as
may be seen by its complicated mathematical equation (eq. S11), it is also the model using the
greatest number of adjustable parameters.
Adapting the equations from the Brownian particle transport (Bird 1966)in the case of
interfacial transfer resistance, we propose the following modification of the diffusion
equation:
dN ads 
N V N
D
 1  ads  ads
dt
 N 0 A1   01  t
eq. S9
1 is the distance parameter, usually called diffusion boundary layer thickness under
hydrodynamic conditions. Roughly speaking, it allows us to take into account the adsorption
reaction kinetics and the diffusion when, contrary to what happens in the A and B models,
they occur over the same time range. The smaller it is, the more significant the rate limitation
is.
N ads 
1C0 A

2
 
1   0     0 1

e
VT 1  
   0 

Dt 

 




eq.S10
0 is the distance parameter taking into account the initial adsorption at experimental t=0; the
smaller it is, the less initial adsorption there is.
1.6 Model F: the pseudo second order equation
The most often used equation to fit adsorption kinetics data is the pseudo second order
equation.
N ads 
2
k 2 N ads
.eq t
eq. S11
1 k 2 N ads.eq t
Although its mathematical expression is rather simple, 30 years after its first use, the
physical phenomenonssupporting this equation are still discussed (Plasinski 2013). It is often
considered that this equation takes into account the case in which the surface reaction is rate
limiting step. In the experimental conditions considered here, this case is taken into account
by the A model. Intraparticle diffusion of the adsorbing solute may also be taken into account
by this equation. In this work, since all the viruses in the biofilm (even those eventually
diffusing in the biofilm) are counted as adsorbed, the diffusion in the biofilm cannot be the
limiting step. For these reasons, although this equation fits quite well our data we are not
discussing it further in the text.We only give the numerical result for the sake of comparison
with other results in the literature.
Table S1: Results of the various modeling attempts to describe the GA and
Qßphagesvirusesadsorption onto drinking water biofilms versus time.
For the description of the parameters given in the table please refer to the text.
2 is the sum of squared errors between the best fit with the considered model and the
experimental points.
Model
Parameters
Number
of
adjustable
Qβ
GA
parameter
A
kobs(s-1)
~
N ads.eq (cm)
2
2
B
C
~
N ads.eq (cm)

0
Β
2
2
0
2
~
N ads.eq (cm)
D
-1
Kads(cm )
2
2
0
E
0(cm)
1(cm)
F
2
k2
~
N ads.eq

2
3
2
8.55±2.8510-5
3.97±0.5310-2
1.21±0.2110-4
7.57±0.43 10-2
3.24 10-4
4.71 10-4

3.9 10-2
0.458±0.084
1.08±2.55 106
4.44 10-4

5.10 10-4
1.09±0.12
6.04±5.410-4
6.98 10-4
4.83±0.48 10-2
0.45±0.55
-2
5.59±1.16 10
7.81 10-3
0,309±0.106
0.41±0.57
7.54 10-4
0.159±0.003
3.81±0.43 10-2
3.56±0.46 10-3
1.06±0.899 10-3
7.58±0.461 10-2
3.089 10-4
4.47 10-4
1.31±0.899 10-3
5.43±1.12 10-2
3.43 10-4
1.21±0.40 10-4
9.54±8.52 10-2
4.85 10-4
Obviously among the 6 ways of modeling, the question is to choose the right one. The
most common criterion in such a study is the sum of the squared errors between model and
experiment. In this study, this cannot be the only one: obviously, the fact that a modeling
using 3 adjustable parameters fits the experimental data better than one without any adjustable
parameter does not bring any evidence of consistency. The difficulty of choosing a model is
increased when the considered data is measurement of virus numbers with some intrinsic
experimental uncertainty and, owing to their biological nature, a limited number of
experiments.
Fitting of GA data by the A and D models gave adjusted parameters without any physical
meaning, the 2 obtained with C model (2 adjusted parameters) is higher than with B model
(0 adjusted parameter), the E model taking into account empiric parameters would have only
been considered if no othermodel could have been used. The B model is retained to describe
GA sorption kinetic data.
Describing Qβ data with B model is not possible. The A model gives the smallest 2,
even when considering the E model (3 adjusted parameters), A is retained.
2. Thickness of the diffusion layer in static conditions
The concentration profiles (x being the distance from the surface) reported in figure S1 are
calculated according to equation S10 (Bard and Faulkner 2001):
C ( x, t )
 erf (
C0
1s
1min
1.0
0.8
x
2( Dt )
1
2
)
eq. S10
2 min
10 min
1 hour
C/C0
0.6
0.4
0.2
0.0
0
500
1000
distance (m)
Fig. S1. Theoretical concentration profilesof GA adsorbing onto the drinking water
biofilm according to distance and time.
References
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& Sons, second edition, 2001.
R.B. Bird, W.E. Stewart, E.N. Lightfoot. Transport phenomena. J. Wiley&Sons, 1966.
P. Delahay, I. Trachtenberg. Adsorption kinetics and electrode processes. J. Am. Chem. Soc.
1957, 79, 2355-2362.
M. Haerifar, S. Azizian. Fractal-like adsorption kinetics at the solid/solution interface. J.
Phys. Chem. C116 (2012) 13111.
D.B. Hibbert ,J.J. Gooding, P. Erokhin. Kinetics of irreversible adsorption with diffusion:
application to biomolecules immobilization. Langmuir 18(2002) 1770.
Y. Liu, L. Shen. From Langmuir kinetics to first- and second-order rate equations for
adsorption. Langmuir24 (2008) 11625.
W. Plasinski, J. Dziuba, W. Rudzinski Adsorption 19 (2013) 1055