Modeling of adsorption isotherms of zinc nitrate on a thin layer of porphyrin

Article abstract of DOI:10.1016/j.molliq.2016.07.011

Adsorption isotherms of zinc nitrate are achieved at four temperatures using Tetraphenylporphyrin (TPP) coated onto Quartz crystal used as biosensor. These isotherms are fitted by models established through statistical physics formalism. It is found that

Full text of DOI:10.1016/j.molliq.2016.07.011

Modeling of adsorption isotherms of zinc nitrate on a thin layer of porphyrin
Mohamed Ben Yahia, Layla Ben Haj Hsan, Salah Knani, Manel Ben
Yahia, Hbib Nasri, Abdelmottaleb Ben Lamine
PII:
DOI:
Reference:
S0167-7322(16)30829-7
doi: 10.1016/j.molliq.2016.07.011
MOLLIQ 6025
To appear in:
Journal of Molecular Liquids
Received date:
Revised date:
Accepted date:
6 April 2016
29 June 2016
3 July 2016
Please cite this article as: Mohamed Ben Yahia, Layla Ben Haj Hsan, Salah Knani, Manel
Ben Yahia, Hbib Nasri, Abdelmottaleb Ben Lamine, Modeling of adsorption isotherms
of zinc nitrate on a thin layer of porphyrin, Journal of Molecular Liquids (2016), doi:
1
0.1016/j.molliq.2016.07.011
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ACCEPTED MANUSCRIPT
Modeling of Adsorption Isotherms of Zinc nitrate on a Thin Layer of
Porphyrin
1
2
1,3
1
2
Mohamed Ben Yahia , Layla Ben Haj Hsan , Salah Knani , Manel Ben Yahia , Hbib Nasri ,
1
Abdelmottaleb Ben Lamine
1
Unité de Recherche de Physique quantique 11 ES 54, Faculté des Sciences de Monastir,
Tunisia.
2
Laboratoire Physico-chimique des Materieaux (LPCM), Faculté des Sciences de Monastir,
Tunisia.
3
Faculty of Science, Northern Borders University (NBU), Arar, Saudi Arabia
*
Corresponding author: Salah KNANI, Faculty of Sciences of Monastir, 5000 Monastir, Tunisia.
Email address: knanisalah@yhoo.fr
Phone: 0021695666600
Fax: 00216 73 500 278
1

ACCEPTED MANUSCRIPT
Abstract
Adsorption isotherms of zinc nitrate are achieved at four temperatures using porphyrin
TTP coated onto Quartz crystal used as biosensor. These isotherms were fitted by models
established through statistical physics formalism. It was found that a multilayer model with
two energies is able to give good approximation of the experimental data. In this model, six
parameters affecting the adsorption process were introduced and interpreted. Such parameters
are namely the number of ions per site, the density of receptor sites, the two energetic
parameters and the numbers of layers. It seems that the modeling of experimental isotherms
using statistical formalism is a powerful method to explain and interpret the adsorption
process at microscopic level. The adsorption energies were computed and their negative
values indicate that the process is a physical adsorption.
Keywords: Adsorption Isotherm; QCM; Porphyrin; Modeling; Statistical Physics
1
. Introduction
Porphyrins, aromatic tetrapyrrole macrocycles, are molecules very widely represented in
living environments. The study of porphyrins has more interest and increased in recent years
1, 2]. The applications of tetrapyrrole macrocycles is essentially as catalysts organic substrate
[
oxygenation, degradation of organic matter and for the exploration of the mechanisms of
biologically important reactions such as photosynthesis [3].
The most important property is the possibility of forming with certain cations (magnesium,
fer, zinc, iron, cobalt ...) very stable chelate complexes, metalloporphyrins [4]. Therefore
porphyrins can be used as a sensor for the zinc cationic metal. This metallopophyrin (complex
zinc-porphyrin) participates at many biological processes [5, 6]. These compounds represent a
large family of functional molecular materials with high chemical and thermal stability [7].
They have recently gained increasing interest in the fabrication of sensors, electroluminescent,
and photovoltaic devices [8, 9].
The mechanism of the metallization is a fundamental step which needs to be studied and
educated. The goal of this work is to study the adsorption of zinc in porphyrin via adsorption
isotherm achieved using QCM.
The zinc can be used as guest in the porphyrin corities which induces the metallization as
presented in figure 1 [10-12].
2

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In the first we detailed the experimental method used to obtain the adsorption isotherms of
Zinc onto thin layer porphyrin at four temperatures. We also propose a study of the influence
of temperature on the change of the adsorption curves.
Secondly, throughout the statistical physics formalism, we will propose modeling of these
isotherms and we interpret the behavior of different parameter versus temperature.
2
. Materials and Methods
The adsorbate used is the Zinc nitrate. This white, crystalline solid is highly deliquescent
and is typically encountered as a hexahydrate Zn (NO ) •6H O. It is soluble in both water and
3
2
2
alcohol.
The adsorbent used is the Tetraphenylporphyrin (TPP or H TPP) as represented in
2
figure.1.The compound is a dark purple solid that dissolves in nonpolar organic solvents such
as chloroform and benzene. Tetraphenylporphyrin is hydrophobic, symmetrically substituted,
and synthesized as basis of porphyrins. Among the chemical properties we have to quote the
basic character of porphyrines. The overall structure of these nitrogenous bases is quite rigid
due to pyrrole rings but it folds to include a small metal atom or becomes dome to accept
bigger metal atom [13, 14].
2
.1. Synthesis of the 5, 10, 15, 20-tetraphenylporphyrin (H TPP)
2
All reagents and solvents employed were commercially available and were used as
received without further purification. The H TPPwas synthesized according to the standard
2
literature method [15]. All reactions and manipulations for the preparation of theH TPP were
2
carried out under aerobic conditions. A UV–Vis spectrum was recorded with a Win ASPECT
PLUS (validation for SPECORD PLUS version 4.2) spectrometer. Accurate mass
measurements (high-resolution mass spectrometry) were obtained on an Orbitrap
spectrometer. The measurements were made at the Pôle Chimie Moléculaire. An IR spectrum
was registered on a FT-IR Nexus (Nicolet) spectrometer with a micro-ATR accessory (Pike).
H TTPP was synthesized by the reaction of 4-tolualdehyde and pyrrole, using the Adler-
2
Longo method [15].In a two-necked flask surmounted by a condenser and an addition funnel,
we introduced a 40 mL (39.47 mmol) of benzaldehydeand 2 L of the propanoic acid. This
solution is heated under reflux and then is added drop wise 28 mL (40.47mmol) of distilled
pyrrole. This reaction was allowed to reflux for 30 min and then the heat was turned off and
allowed to cool and then made a filtration. The purple solid was dried under vacuum for 3 h.
The mass of dry solid obtained is equal to 9.76 g (Rd ~ 36%).
3

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Anal. Calc. C H N (614.73 g/mol): C, 85.97; H, 4.92; N, 9.11%. Found: C, 85.94;
4
4
30
4
H, 4.90; N, 9.07%. UV-visible [(CHCl : λmax in nm (log)]: 423(5.54), 519(4.96), 557(4.86),
3
-1
5
98(4.74), 654(4.68).FT-IR [solid, cm ]: ν(NH)pyrrolic: 3327, ν(CH)porphyrinic: 3065-
3
030, ν(C=C)(C=N)porphyrinic: 1564, ν(CN)porphyrinic: 1075, δ(CCH): 966. MS (MALDI-
+
TOF) m/z: 615.3[H TPP] .
2
A number of features make porphyrins eligible as good sensing materials. Furthermore, the
organic chemistry of these compounds is well developed, and a wide range of different
substituents can be introduced at their peripheral positions [16]. Almost all metals have been
coordinated to the porphyrins [17]. The coordinated metal, peripheral substituents, and
macrocyclic skeleton influence the related sensing properties of these compounds [18]. All
these characteristics increase the versatility of these molecules for porphyrin based sensors
[
19, 20, 21], showing excellent properties in terms of stability, sensitivity, and reproducibility.
We use the quartz microbalance (QCM) [22] to realize adsorption isotherms of zinc nitrate on
a thin layer of porphyrin.
2
.2. Preparation of gold surface
AT-cut quartz crystals with optically flat polished gold/titanium electrodes with a
fundamental resonant frequency of 5MHz were used in experiments [23]. The crystals were
2

2.54cm in diameter with the upper electrode area of 1.37cm and lower electrode area of
2
0
.40cm . The lower electrode area is the area of overlap between the upper and lower
electrodes, and represents the part of the crystal which is piezoelectrically most active [24,
5]. Before modification, the crystals were first cleaned with Piranha solution (1 part of 30%
2
hydrogen peroxide in 3 parts of 98% sulfuric acid) at room temperature for 1-5 min. After this
cleaning treatment, the crystals were rinsed thoroughly with ultra-pure water and ethanol,
followed by drying with high purity nitrogen to remove any remaining water and dried again. The
resonant frequency of the blank crystal was recorded.
For film deposition, a volume of 30L of TPP was spin coated at 3000 rpm for 1min onto the
larger gold electrode to cover the overlapping electrodes portion. The coated crystals were
then dried at 100°C for at least 2 h. After drying, the resonant frequency of the polymer coated
crystal was measured. The difference between the resonant frequencies of the uncoated and
coated crystal was determined. The mass of the polymer deposited on the surface was
calculated from the shift in the crystal resonant frequency.
4

ACCEPTED MANUSCRIPT
2
.3. QCM apparatus
The experimental setup for the adsorption measurements with the quartz crystal
microbalance is depicted in figure 3.
The QCM [22, 26] setup consisted of the coated electrode, the QCM instrument (model PM
7
00), a crystal holder, and adsorption cell. In the calculation of the adsorbed amount of Zinc
nitrate, a linear relationship between the change of the areal mass density and the frequency
shift was assumed. This assumption implies that the mass change is small and that the
adsorbed layer is rigidly bound onto the surface. Indeed, owing to the piezoelectric properties
of quartz and its crystalline orientation, application of an external alternating electric potential
through the metal electrodes produces an internal mechanical stress in the crystal. This is due
to its shear deformation which results in the vibrational motion of the crystal at its resonant
frequency [27]. This resonant frequency is sensitive to any mass change on the crystal surface
as well as any change in the viscosity–density of the surrounding environment. In 1959,
Sauerbrey [28] derived a relationship between the shifts in the crystal resonant frequency and
elastic mass bound to the crystal surface [28]:
2
2
F0
F  
m
(1)
A 
Where F is the frequency shift (Hz), F is the resonant frequency of the crystal (Hz), m is
0
2
the adsorbed amount (g), A is the active electrode area (cm ),  is the density of quartz
-3
11
-1 -2
(
2.648 g.cm ), and  is the shear modulus of quartz (2.947×10 g.cm .s ).
Thus, the constant terms can be combined together to give a crystal sensitivity constant, C,
which is specific to crystal [29, 30, 31]:

C
Fm
(2)
m  
-1
2
Where C≈56.6 μg .cm .Hz for a 5MHz AT-cut quartz crystal, i.e., when 56.6 μg of mass is
2
deposited on 1cm of area, it will produce a shift of negative 1Hz in the resonant frequency of
the crystal .
The Sauerbrey equation introduces linear sensitivity factor, C, which is a fundamental
property of the QCM crystal. Thus, in theory, the QCM mass sensor does not require
calibration. However, it must be kept in mind, that the Sauerbrey equation is only strictly
applicable to uniform, rigid, thin-film deposits [30, 31]. Vacuum and gas phase thin-film
5

ACCEPTED MANUSCRIPT
depositions which fail to fulfill any of these conditions actually exhibit more complicated
frequency-mass correlations and often require some calibration to yield accurate results.
The resonant frequency of a 5 MHz quartz crystal can be measured with a precision of
about 0.01 Hz in vacuum and 0.1 Hz in liquids . Accordingly, very small masses on the
nanogram scale could be measured by the QCM.
The QCM crystal holder was immersed in the adsorption cell filled with V = 100 mL of pure
s
water. After stabilization of the QCM frequency change, which usually took around 2 h, 100
-4

L of concentrated zinc nitrate solution (10 M) was injected with syringe while the solution
was stirred with a magnetic stirrer.
The quartz microbalance displays the change in the frequency after each injection. The
-2
adsorbed amount per unit of area expressed in μg.cm is conducted directly from the
measured frequency. We proceed in the same way to get the adsorbed amounts for the
increasing concentration solutions. We give in figure.4 the mass of Zinc adsorbed per unit of
mass adsorbant.
We note that the research of an adequate model is very useful to get information about the
adsorption process. We will give in the following a detailed treatment of adsorption models
using statistical physics formalism.
3
. Theory
Adsorption involves an exchange of particles from the free state to the adsorbed state. Its
investigation in statistical physics cannot be performed without employing the grand
canonical ensemble to take account of the particle number variation. Therefore, a chemical
potential is introduced in the adsorption process [32].
Theoretical modeling of adsorption isotherms in gas or liquid phase is a powerful
technique used for surface characterization. Thus, the correlation of equilibrium data by either
theoretical or empirical equations is essential to the practical design and operation of
adsorption systems. Moreover, in contrast to the empirical methods, the use of statistical
physics development gives a physical meaning to the model parameters and allows the
establishment of significant analytical expressions.
In order to treat adsorption problems using statistical physics methods, we will proceed
with some assumptions as the basis of an initial postulate.
As a first approximation, we neglect the specific interactions between the adsorbate molecules
6

ACCEPTED MANUSCRIPT
(in free state) [33, 34, 35, 36] since the used concentration of adsorbate is very weak.
The internal degrees of freedom of the studied molecules are neglected and we consider
only the translation freedom degrees. Indeed, electronic degrees of freedom cannot be
thermally excited and the rotational freedom degrees are frozen in the solution. The
vibrational freedom degrees can be neglected compared to those of translation.
In liquid phase, we consider that a variable number of ions are adsorbed onto N receptor
M
sites located on a unit mass of the adsorbent. The adsorption reaction of liquide ions (A) onto
receptor sites (S) should include a stoichiometric coefficient n as shown in the following
equation:
(3)
nA  S  AnS
In our treatment, our starting point is the grand canonical partition function describing the
microscopic details of the adsorption process. Therefore, we define the state of occupation
number N that expresses the situation in which this receptor site is placed [34, 36].
i



  Ni

i
z  e
gc

(4)
N
i
Where ε is the receptor site adsorption energy, μ is the chemical potential, and β is defined
i
as 1/k × T, where k is the Boltzmann constant and T is the absolute temperature. The total
B
B
grand canonical partition function related to N receptor sites per surface unit, which is
M
assumed to be independent and identical, is written as [37]:
NM
NM
(5)
Z 
z (T,)  (z (T,))
gc

gc
gc
i1
This total grand canonical partition function led to the determination of the average-site
occupation number, which can be written as follows [34]:
NM
ln

zgc

ln(zgc )
(6)
N  k T
 N k T
o
B
M
B
μ

And therefore the average number of the adsorbed quantity is written as [37, 38]:
Q  nN0
(7)
7

ACCEPTED MANUSCRIPT
The adsorbed quantity (Q) is given per unit surface, because the density of receptor sites
(
N ) is written per unit surface. This formalism can be applied to any adsorption isotherms of
M
liquids or gases.
The adsorption isotherms profiles show a saturation phenomenon at high concentration but
the adsorption doesn’t stop after the formation of a limited number of layers suggesting that
the molecules of the N first layer were supposed to be adsorbed with  energy, and the
1
1
molecules of other layers were adsorbed on N with  energy ( > >0). Four models were
2
2
1
2
developed in order to simulate our experimental isotherms.
.1. Monolayer model onto two independent types of sites
3
To develop the expression of the adsorbed quantity according to the concentration, we
consider that a variable number of Zn ions are adsorbed onto two types of independent sites.
The number of the first type is N1M sites per unit of surface and the number of the second type
is N . We suppose that the adsorption onto the first type is carried out with the energy (- )
2
M
1
and onto the second type with the energy (- ). The partition functions of the two types of site
2
are z1gc and z ; they are expressed by the relations:
2gc



 

Ni
 (  )
(8)
(9)
1
1
1
1
z  e
1 e
1
gc

N 0,1
i



 2

Ni
 (  )
2 2
2
z  e
1 e
2
gc

N 0,1
i
In this case, we can write the total grand canonical partition function as follows [38]:
N
N
2M
1
M
Z  ((z ) (z ) )
(10)
gc
1gc
2gc
Therefore the adsorbed quantity of molecules is written:
n1N
1M
n2 N2M
Q 

Model (1)
c
c
2
1
 ( )
n1
n2
1
1 ( )
c
c

E1a
RT

Where n1 and n2 are the number of ions per site, c  c .e
and
1
s
a
E2


RT
c  c .e
are the concentrations at half saturation corresponding to the first and
the second type of sites, respectively.
.2 Double-layer model with two energies
2
s
3
8

ACCEPTED MANUSCRIPT
For this model we suppose that the adsorbate molecules are anchored onto receptor sites in
two successive layers with different energy levels ε and ε , respectively. Each receptor site
1
2
can be empty or occupied once or twice. The first layer is used as a receptor site for the
second one.
According to these assumptions, the grand canonical partition function of one receptor site
may be expressed as [37]:



 
i

Ni
 ( )
1
 (  2)
1 2
(11)
zgc
e
N 0,1,2
1 e
 e

i
With the number of occupations of identical N receptor sites at equilibrium being:
M

( )
 (  2)
1 2
1

ln(zgc )  NM

e
 2e
N  N k T
(12)
o
M
B
 ( )
2 (  2)
1 2
1
1 e
 e
Therefore, the adsorbed quantity is written as a function of adsorbate concentration [39]:
c
c
n
2n
(
)  2( )
c1
c
2
Q  nNM
Model (2)
c
c
n
2n
1
 ( )  ( )
c
c
2
1
E1a
RT
a
E2
RT



Where: c  c .e
[39] and c  c .e
[39].
1
s
2
s
With c is the solubility of adsorbate in aqueous solution, R is the ideal gas constant and T is
s
a
a

E1
 E
the isotherm temperature. (
and the second one, respectively.
.3 Multi-layer models with two energies
) and (
) are the adsorption energies at the first layer
2
3
We suppose that the adsorption can accede two layers. So, there is complete formation of
monomolecular layers before progressing to a subsequent layer. We assume that the layers are
formed one after the other with different energy. L layers of adsorbed molecules are formed
1
with the first energy ε and L layers with the energy ε The grand canonical partition function
1
2
2.
is written as [40]:
L1
L2



 
1

Ni


L  (N L ) N 
2

1 1
i
1
i
z  e
gc

 e

(13)
N 0
i
N L 1
1
i
9

ACCEPTED MANUSCRIPT
E1a
RT
a
E2
RT



Using eq (7) and if we take c  c .e
and c  c .e
two energetic
1
s
2
s
a
a
parameters, with E1 and E2 are the molar adsorption energies and c is the solubility of
s
adsorbate in aqueous solution, the adsorbed quantity Q has the following expression [41]:
nN
n

f1


c,c1

 f
c,c ,c 
n
1
M
L1 1
2
1
2
Q 
Model (3)


nL
nL2






c 

 c   c 
 c 
    1  
     




1
  





c
c
c
c
2

1

1
 
2







n
n


c 

 c 
1  
 
1
  

c
c
2
1



With f and f are functions of the concentration and the parameters c and c such as [41]:
1
2
1
2
n
n

L1 1

n

L1 1





c 
 c 
 1 
 

c 
 





L1 1

 

c

c
1

c

1



n

1



f1 

(14)
n
2

c 



c 
1
 


1 
 



c
1

 c 
1




nL1
c   c 
n
nL2


nL1
n
nL2

 c 

  c   c   c 


  
  

L1 1

1  
 
c
      L


n

     
2
c1
c

 
2


2

c
c
c
 

1 2 2
 
 

n
f2 
(15)

c 
 c 
1  
 
1
  


c
c
2

2



nL1
c   c 
2n
nL2




 c 



   1  
  
 



c
c

c1
 
2


2
2



n



c 


1
  






c

1

Such model is the first time used by our group to model the adsorption isotherms.
In the expression of adsorbed quantity given by the established Model there are six
parameters which are divided in two classes, the first is a steric aspect such as the number of
molecules per site n and the density of receptor sites N , and the number of layers L , L , the
M
1
2
second aspect is energetic one given by the parameters c and c which are function of the
1
2
adsorption energy.
1
0

ACCEPTED MANUSCRIPT
If we suppose in particular that L layers of adsorbed molecules are formed with the first
1
energy ε and infinite layers with the energy ε the grand canonical partition function will be
1
2;
written as [40]:
L1




 
1

Ni


L  (N L ) N 
1 1

(16)
i
1
2
i
zgc

T,

 e


e

N L 1
N 0
i
i
1
In the expression of the adsorbed quantity of molecules there are five parameters fitting:
nNM

c 
f

c,c
L 1

 f2

c,c ,c2 
1
n
1
1
Q 
Model (4)


1
nL1
n




  c   c 
1
  

  




   
1 2
c

1

c
c

 

 

n
n

c 

 c 
1
  
1  
 

c
c
2

1



Where:
n
n L1 1
 
n

L1 1





 
c 
 c 
 1 
 

c 
 






L1 1


c

c

c
1
1


1

n




f1 

(17)
n
2

c 
 



 c 
 c 
 
1


1 
1

c
1





nL1

nL1
c   c 
n
2n

 c   c 
   
   

  
2

L1 1


  
c1
 
c

c 

c
1
 
c
2


f2 

(18)
n
2
n




c 
  
 
1
  


1


c





2

c

2

4
. Fitting results and interpretation
As represented previously, the four models present different mode of adsorption. So
selecting the adequate model means properly the interaction of the adsorbate in the adsorption
area.
4
.1 Fitting results
We use a fitting computer program to adjust the experimental data with the different
models. The mathematical fitting method utilized to contrast the experimental data versus the
1
1

ACCEPTED MANUSCRIPT
proposed model was based on the Levenberg-Marquardt iterating algorithm using a
multivariable nonlinear regression program.
The best fitting result is established once the residuals between the experimental data and
adsorbed quantities value predicted by the model are minimized according to a determined
level of confidence. In our case, the level of confidence was set at 95%.
2
The fitting criterion in the present work is the well-known coefficient of determination R
which is a measure of the goodness of the fit. A very common procedure for choosing among
alternative models is to select the one that fits the best, that is, the one that has the highest
2
2
coefficient of determination R . The ideal fitting is obtained when the value of R is close to
the unit. Indeed, the standard way to obtain the best fit is to choose the parameters with the
condition to get the minimum of the sum of squared deviations between the theoretical and
the experimental curves.
2
In table 1 we present the R values deduced from the fit of the four adsorption isotherms of
the Zinc nitrate to Porphyrin.
Based on the values of adjustment coefficients given by each model we can gather some
information on the variation of these coefficients. We noticed that model (3) and model (4)
2
present the highest R values, which vary from 0.997 to 0.998. Never the less the variation of
the correlations coefficients is small between the two models.
So, we conclude that the adsorption of the Zinc nitrate to Porphyrin can be very well
explained by the Multi-layer adsorption with two energies.
In figure 5 we illustrate the fit of adsorption isotherms at four temperatures by model (3).
The fitting parameters of model (3) are indicated in table. 2.
4
.2. Discussions
As proposed below, the adsorption of Zinc nitrate on a thin layer of Porphyrin can be
simulated by multi-layer adsorption model; the layers are formed one after the other with
different energy. In fact the Tetraphenylporphyrin could function as a sensor for the cationic
2
+
metal Zn but a question is how these layers are adsorbed by the biosensor?
The work elaborated by Katsuhiko Ariga et al. [42] can answer this question. The
formation of adsorbate is achieved layer by layer based on the electrostatic force between the
adsorbate. This approach is founded on the alternating adsorption of materials containing
complementary charges. Strong electrostatic attraction occurs between a charged surface and
an oppositely charged molecule in solution [42, 43].These facts will be approved in the
calculation of energetic bound. In general, the adsorption of molecules carrying more than one
equal charge allows for charge reversal on the surface, which has two important
1
2

ACCEPTED MANUSCRIPT
consequences: (i) repulsion of equally charged molecules and thus self-regulation of the
adsorption and restriction to a single layer, and (ii) the ability of an oppositely charged
molecule to be adsorbed in a second step on top of the first one [44].
The Layer by Layer method (LBL) is dependent on charge neutralization between
polycations and polyanions. This procedure of LBL can be regarded as a template controlled
growth process. First, the solid surface acts as a template, and after the adsorption, it is
refunctionalized and acts as a new template for the following adsorption.
The LBL technique has many potential applications in the biosensor area [44, 45]. Data about
the structure of these multilayer assembly constructed using LBL methods have been
evaluated by various kinds of instrumental techniques, such as QCM [45, 46], ultraviolet
adsorption and atomic force microscopy (AFM).
Lvov et al. [42, 45] by the use of QCM showed that the immobilization of adsorbate during
the LBL procedure resulted in a multilayer conformation in one adsorption step.
In our case the isotherms are produced by LBL in the TPP coated onto gold electrode of
2
+
-
Quartz crystal is the biosensor, Zn are the cationic ions and NO are the anionic ions.
3
Figure 6 shows the multi-layer adsorption of Zinc nitrate solution on a thin layer of Porphyrin.
In the expression of adsorbed quantity given by the established Model there are six
parameters which are divided in two classes, the first is a steric aspect such as the number of
adsorbates per site n, the density of receptor sites N , and the number of layers L and L , the
M
1
2
second aspect is energetic one given by the parameters c and c which are function of the
1
2
adsorption energies. These parameters will allow us to well describe and understand this
mechanism of adsorption.
Before the interpretation of these parameters we give theoretical study of the effect of each
parameter on the adsorbed quantity [41].
Figure 7 shows the influence of the number of adsorbates per site n and the density of
receptor sites N on the plots of adsorption isotherms.
M
We note that the parameter density of receptor sites N is relevant in the behavior of the
M
isotherm. Indeed, the greater the value of N , the greater the adsorbed quantity. The value of
M
constant N influences the mechanism of monolayer formation, because this constant is
M
responsible for the receptor sites accessible to adsorbate ions in the first layer.
The influence of the number of molecules per site n parameter is similar as that observed
for N because this constant is also related to the monolayer and multilayer adsorption range
M
and is responsible for the interactions in the first and subsequent layers.
1
3

ACCEPTED MANUSCRIPT
In figure 8 we illustrate the Influence of energetic parameters c and c on the adsorbed
1
2
quantity.
As one can observe, the rise in the value of c changes the shape of isotherms at low
1
concentrations. This constant is responsible for solid–liquid interactions in the first layer. On
the other hand, the rise in the value of the constant c , which is responsible for interaction
2
between layers, does not influence the shape of isotherms at low concentrations but increases
the adsorbed quantity in the multilayer region.
The influence of the number of layers L and L on the adsorbed quantity is reported in
1
2
figure 9.
We can note that the adsorbed quantity increases with rise in L and L . The rise in the value
1
2
of L changes the shape of isotherms at low concentrations because this parameter influences
1
layers which are in direct contact with the surface instead the L parameter which influence
2
the adsorbed quantity in the multilayer region [47].
5
. Interpretation of the parameters
The parameters deduced from the fitting of the experimental adsorption isotherms by
numerical simulation were discussed and interpreted.
5
.1. Number of molecules per site
The adsorption reaction (Model (3)) expresses that the, parameter n is a stoichiometric
coefficient which represents the number of adsorbate per site [36, 40]. In earlier works, Knani
et al. [49] suggests that this parameter give information on the position of the adsorbate at the
adsorbent surface. In fact, the adsorbate has several manners to be anchored on the receptor
site according to its geometry and its angle of incidence with the adsorbent surface.
For electric reasons the porphyrin cannot complex more than one zinc ion which gives this
number n lower or equal to the unit [1, 7].Table.2 shows that the fitting value of n varies
between 0.60 and 0.66. This parameter must be an integer but its fitted values are not, because
they are an average of some successive integer values.
We can also note that the increase of temperature does not affect the n values.
5
.2. Density of receptors sites NM
The parameter N gives information about the number of receptor sites accessible to the
M
adsorbate. It represents the ability of the adsorbent surface to retention of the adsorbate.
The variation versus temperature of the receptor sites density is reported in figure 10.
1
4

ACCEPTED MANUSCRIPT
It has been noted that N increases versus temperature which implies that the temperature
M
allow the formation of supplementary active sites that are hidden at low temperatures.
It is interesting to differentiate between the active receptor sites occupied by the adsorbate
(
The N in our case) and the existing number of receptor sites which represents the number of
M
receptor sites available in the adsorbent surface.
It can be noticed that when the temperature increases, the density N notably increases
M
which leads to an increase in the amount of zinc nitrate adsorbed as we will show due
probably to energetic reasons.
As proposed by Knani et al. [37, 48] using this parameter, we can deduce the specific surface
of adsorption which represents the total surface area available for adsorbate binding:
N N A
M
A
(19)
S 
M
2
with S being expressed by cm /µg solid, N is the number of receptor sites given by µg
M
adsorbate/µg solid, M is the molecular mass of adsorbate ( 297.48g/µg mol), N is
A
2
3
Avogadro’s number (6.02 × 10 ions/mol) and A is the area of adsorbate ion which is
-20
2
approximately 1.71×10 m .
We note that this surface also increase versus temperature which can be probably due to
the expansion of porphyrin [13].
5
.3. Numbers of Layers L and L2
1
The L layers of adsorbate are formed with the energy ε whereas the L layers are formed
1
1
2
with the energy ε .Table.2 shows that the fitting of L valuesare between 1.16 and 1.98.
2
1
We depict in figure.7 the evolution of L versus temperature.
2
These values are in principle integer but the fitted values are decimal because they are average
values. So we can access to the formation of the second and other layers without the
saturation of the first layer.
The effect of temperature on the number of multilayer L parameters is evident. It was
2
observed that the increase in temperature of adsorption caused a decrease in L2.
The reason for the decrease could be explained by the thermal agitation, which acts as a
surface force between the adsorbate. The increase in temperature alters the forces of adsorbate
bound in the multilayer region and a decrease in L2.
5
.4. Adsorption energies
Using the values of the dimensionless parameters c , c and the solubility of adsorbate in
1
2
a
a
aqueous solution c we calculate the molar adsorption energies E1 and E2
:
s,
1
5

ACCEPTED MANUSCRIPT


c1,2



a
1,2
(20)

E  RTLn

cs
The first energy gives information about the interaction between the zinc ion and the
porphyrin and the second energy represents the interaction adsorbate-adsorbate.
a
a
Table.3. shows the values of the adsorption energy E1 and E2 at the four temperatures.
a
As one can see that the adsorption energy values E1 do not exceed 40 kJ/ mol which
imply that the process takes place with physical adsorption. Such binding is a Vander Waals
type or hydrogen link [49]. The second energy corresponding to the interaction ion-ion is less
strong than the first energy, so the adsorbate-adsorbate interaction is lower than the adsorbate-
adsorbent interaction.
6
. Conclusion
Using the grand canonical ensemble in statistical physics, we have established different
theoretical expressions to fit the adsorption isotherms of zinc nitrate on a biosensor (system
formed by porphyrin TTP coated onto Quartz crystal). These models are thermodynamically
consistent and the physicochemical parameters involved in the adsorption process are taken
into account in the theoretical treatment. The appropriate model for fitting the different
adsorption isotherms was the multilayer layer model with two energies. It was found that the
proposed statistical treatment is sufficiently flexible to give a good representation of
experimental data. Our model confirmed the LBL technique for the possibility of forming
several layers adsorbed on a solid thin layer.
Thanks to the theoretical modeling, we can well understand the behavior of experimental
isotherms and thus the corresponding adsorption phenomenon at microscopic level. On the
basis of these isotherms we determined the physicochemical parameters related to the
adsorption process. The fitting values of n are always lower than the unit. Therefore, our
model confirms that every site of TPP is occupied by only one zinc ion. The study of the
density of receptor sites indicates that when the temperature increases new hidden sites appear
and participate in the adsorption then the adsorbed amount increases. The fitting value of
L shows that approximately one layer is formed with ε because the first layer is in direct
1
1
contact with the surface. We show also that an increase in temperature of adsorption caused a
decrease in L due to the thermal agitation. The values of the adsorption energies obtained
2
through the model were found to be typical of a physisorption.
1
6

ACCEPTED MANUSCRIPT
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Figure Captions
Fig. 1. Metallization of porphyrin (TPP) with zinc ion.
Fig. 2. Synthesis of H TPP.
2
Fig. 3. Experimental setup of adsorption measurements with QCM.
Fig. 4. SEM images of (a) TPP , (b) Zn-TPP.
Fig. 5. Experimental data of adsorption isotherms of zinc nitrate fitted by statistical model
illustrated by different lines with different colors) at different temperatures.
Fig. 6. Layer-by-layer adsorption (LBL) technique.
(
Fig. 7. Influence of parameters n and N on the shape of adsorption isotherms generated
M
by model (3): (a) Effect of n (N =19.38, c =0.0036, c =0.073, L =1.9, L =80), (b) Effect
M
1
2
1
2
of N (n=0.61, c =0.0036,c =0.073, L =1.9, L =80).
M
1
2
1
2
Fig. 8. Influence of parameters c and c on the shapes of adsorption isotherms given by
1
2
model (3): (c) Effect of c (n=0.61, N =19.38,c =0.073, L =1.9, L =80), (d) Effect of c
2
1
M
2
1
2
(
n=0.61, N =19.38, c =0.0036, L =1.9, L =80).
M 1 1 2
Fig. 9. Influence of parameters L and L on the shape of adsorption isotherms given by
1
2
model (3): (e) Effect of L (n=0.61, N =19.38, c =0.0036, c =0.073, L =80), (f) Effect of
1
M
1
2
2
L (n=0.61, N =19.38, c =0.0036,c =0.073, L =1.9).
2
M
1
2
1
Fig. 10. Behavior of the effectively occupied receptor sites density as a function of
temperature.
Fig. 11. Surface area versus temperature.
Fig. 12.Variation in the number of layers L versus temperature.
2
2
1

ACCEPTED MANUSCRIPT
Zn (NO3)2
Fig. 1.
Fig. 2.
2
2

ACCEPTED MANUSCRIPT
Fig. 3.
2
3

ACCEPTED MANUSCRIPT
(a)
(b)
Fig. 4.
2
4

ACCEPTED MANUSCRIPT
340
320
300
280
260
240
220
200
180
160
140
120
100
T= 298 K
T= 303 K
T= 308 K
T= 313 K
Fit at 298 K
Fit at 303 K
Fit at 308 K
Fit at 313 K
80
60
40
20
0
0
,00
0,01
0,02
0,03
0,04
0,05
C(mol/l)
Fig. 5.
2
5

ACCEPTED MANUSCRIPT
Fig. 6.
2
6

ACCEPTED MANUSCRIPT
350
300
250
200
150
100
(
a)
n=0.6
n=1.0
n=1.4
n=1.8
50
0
0
.00
0.02
0.04
0.06
0.08
C(mol/L)
350
(b)
300
250
200
150
100
N =19.3
M
N =25.3
M
N =31.3
M
N =38.3
M
50
0
0
.00
0.02
0.04
0.06
0.08
C(mol/L)
Fig. 7.
2
7

ACCEPTED MANUSCRIPT
350
300
250
200
150
100
(
c)
c1=0.003 mol/L
c1=0.015 mol/L
c1=0.040 mol/L
c1=0.080 mol/L
50
0
0
.00
0.02
0.04
0.06
0.08
C(mol/L)
3
3
2
2
1
1
50
(d)
00
50
00
50
00
c2=0.07 mol/L
c2=0.08 mol/L
c2=0.09 mol/L
c2=0.10 mol/L
50
0
0
.00
0.02
0.04
0.06
0.08
0.10
C(mol/L)
Fig. 8.
2
8

ACCEPTED MANUSCRIPT
350
300
250
200
150
100
(e)
L =1.9
1
L =2.9
1
L =3.9
1
L =4.9
1
50
0
0
.00
0.02
0.04
0.06
0.08
C(mol/L)
3
3
2
2
1
1
50
(
f)
00
50
00
50
00
L =30
2
L =40
2
L =50
2
L =60
2
50
0
0
.00
0.02
0.04
0.06
0.08
0.10
C(mol/L)
Fig. 9.
2
9