Download Appendix 1 : Calculation of the wall density

This journal is © The Royal Society of Chemistry 2000
Appendix 1 : Calculation of the wall density
The relative pressure corresponding to the starting point of the capillary condensation
in the cylindrical mesopores is labelled P/Ps*. The volume adsorbed for P/Ps = P/Ps*
, V* , is the sum of the volume of nitrogen filling the microporosity of the silica walls
and of the volume of nitrogen adsorbed at the surface of the cylindrical mesopores,
Vt.
The wall density w is given by the relation:
w 
1
V  Vt 
*
(1)
1
s
where s is the skeletal density of the amorphous silica (s = 2.20 g.cm-3)
Determination of Vt :
The thickness t of the adsorbed layer at P/Ps = P/Ps* can be calculated from the
formula of Cranston and Inkley1 :
 5 

t  3 .43  
 ln P Ps* 
and
Vt  t  S t
(3)
1
3
(2)
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where St is the specific surface area developed by the cylindrical pores.
Iterative calculation of w :
The Luzzati's formula2 expresses the volume fraction of cylindrical mesopores, p, as
a function of the pore diameter, dc and of the lattice parameter a = 2d100/3 :
2
 dc 
p   
 2a 
3
2
(4)
A first value of p can be calculated with dc = d
BJH,
diameter determined by the BJH
method3 on the desorption isotherm curve.
Alternatively, the following relation can be written:
VP  ( V *  Vt )
p 
1
VP 
(5)
s
where VP is the total pore volume determined from the adsorbed volume for P/Ps
closed to 1.
From (5), a first value of Vt can be calculated and, using (1), a first value of w is
obtained.
The relation4 (6) is then applied to determine a first value of St :
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1




m


S t   .d c 
 3



a 2  d c2 
 2

4
(6)
From this first value of St and the relation (3), a second value of Vt is obtained. From
(1), (6) and (3), second values of w , St and Vt are successively calculated.
After ten iterative calculations, the final value of w is obtained. Using relations (5)
and (4), the final values of p and dc are then calculated.
Appendix 2 : Calculation of the surfactant volume fraction from the surfactant
weight percentage
The volume fraction of silica (Cn) in the material after the elimination of the
solvents and by-products is given by the following relation :
wt%Cn
C n  
Cn
wt%alk   M SiO2
w  M alk

wt%Cn
(7)
Cn
where,
wt %(alk) is the weight percentage of alkoxide in the starting sol,
wt % (Cn ) is the weight percentage of surfactant in the starting sol,
MSiO2 is the molar weight of SiO2,
Malk is the molar weight of silicon alkoxide,
This journal is © The Royal Society of Chemistry 2000
w is the density of the silica walls,
Cn is the density of the micellar cylinders (assumed equal to the density of the
surfactant).
1 R. W. Cranston and F. A. Inkley, in Advances in Catalysis and Related Subjects, vol. 9, ed.
A. Farkas, Academic Press, New York, 1955.
2 V. Luzzati, H. Mustacchi, A. Skoulios and F. Husson, Acta Crystallogr., 1960, 13, 660.
3 S. Lowell and J. E. Shields, Powder Surface Area and Porosity, Chapman and Hall,
London, 1984.
4 T. Dabadie, A. Ayral, C. Guizard, L. Cot and P. Lacan, J. Mater. Chem., 1996, 6, 1789.