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Nanometer particle size, pore size, and specific surface determination of colloidal suspensions and porous glasses by rayleigh light scattering
J O U R N A L OF
ELSEVIER
Journal of Non-CrystallineSolids 189 (1995)50-54
Nanometer particle size, pore size, and specific surface determination of colloidal suspensions and porous glasses by Rayleigh light scattering H. Gratz, A. Penzkofer *, P. Weidner Naturwissenschaftliche Fakultiit H-Physik, UniversitiitRegensburg, D-93040Regensburg, Germany
Received31 May 1994; revisedmanuscriptreceived5 January 1995
Abstract Rayleigh scattering of a He-Ne laser is applied to determine the particle size of colloidal silicon dioxide suspensions (Ludox CL-X and Ludox PG), and the effective sphere diameter of Vycor glass, and of silicate sol-gel glasses. The pore sizes and specific surfaces of the samples are estimated.
1. Introduction The size and surface area of micropores (diameter d~< 2 nm) and mesopores (2 nm ~< d~< 50 nm) of porous media are determined by gas adsorption studies [1,2], small-angle X-ray scattering (SAXS) [3-5], small-angle neutron scattering (SANS) [6], and transmission electron microscopy (TEM) [7]. Static light scattering [8] and dynamic light scattering [9] are used for particle size determination of colloidal suspensions. Rayleigh scattering has been used to discuss the transparency of silica gel-polymeric composites [10]. Small-angle light scattering (light diffraction in the forward direction) is restricted to macroporous media (particle size d > 50 nm) [11,12]. Here the Rayleigh scattering of a He-Ne laser is
* Corresponding author. Te1:+49-941 943 2107. Telefax: +49-941 943 2754. E-mail: [email protected] burg.de.
applied to determine the particle sizes of Ludox CL-X and Ludox PG (aqueous colloidal suspensions of silicon dioxide from DuPont [13]), of Vycor brand porous glass (code 7930 from Corning [14]), of Gelsil transparent porous silica glasses (from Geltech [15]), and self-prepared silicate xerogel [16]. The pore sizes and specific surfaces of the samples are estimated from the particle sizes and the porosities of the samples. The results are compared with the specifications of the manufacturers.
2. Theory The light scattering of particles is described by the Lorenz-Mie theory [17-22]. For particle diameters, d, small compared with the light wavelength, A, (d < 0.1A) [23] the Lorenz-Mie theory may be approximated by the Rayleigh scattering theory [24]. The scattering of linearly polarized light from a single sphere of diameter d is illustrated in Fig. l(a).
0022-3093//95//$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0022-3093(95)00206-5
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
Eo
.47)
I\
light polarized perpendicular to plane of scattering, s-polarization), the factor (sin 2 q~ + cos 2 0 cos 2 ~o) reduces to 1. A sample of particle number density, N (dimension c m - 3 ) , and thickness, l, is considered. It is irradiated by a light beam of power P0 = rio d A (integration over cross-sectional beam area). The scattered light is collected by a lens of radius, r~, at a distance, r. The collected power, P=I~rr21, of scattered light becomes
I
(b) ~ L I He-Ne Laser
51
A }2 m2_l ] 016
2
p = P°NI[ ~
× (sin 2 ~p + cos 2 0 cos 2 q~)
.......
1
X A a'-i + R ( 4 ' ) Fig. 1. (a) Illustration of the geometry of Rayleigh scattering from a single particle. E 0 is electric field strength of linearly polarized input light propagating in ~: direction. E is the electric field strength of light scattered an angle, 0, off the direction of the input light. (b) Schematic of experimental Rayleigh scattering arrangement: S is sample, L is lens, PM is photomultiplier tube, ¢ is angular collection direction outside sample, A0 is angular width of collected scattered light outside sample.
The scattered intensity, I, at a distance, r, from the sphere is given by [23]
, 0( )206 m2+2m21 =I
A
x (sin 2
2
+ cos 2 0 cos 2
(1)
where I 0 is the incident light intensity, A is the light wavelength, o~ = ~ d / A is the Mie parameter, q~ is the input polarization angle in the (7, ~ ) plane, and 0 is the scattering angle in the (~:, r I) plane, m, the complex refractive index ratio, is given by n 2 -- i K2 m . . . . hi n 1 - i Ka h2
n - iK,
(2)
where the subscript 2 indicates the sphere and the subscript 1 indicates the surrounding medium, n is the refractive index and K is the extinction coefficient. For transparent spheres in a transparent medium Eq. (2) reduces to m = n2/n r For q~ = 90 ° (input
=
'rr4d 6 rn 2 - 1 2(sin 2
PoNI-~ - - -
m2+2
~ "Jr-c o s 2 0 c o s 2 q))
1
xan~l +R(4') '
(3)
where A Oi = rr A 0i2/4 is the solid angle of light collection inside the sample and R(4') is the reflectance of the scattered light at the exit surface. Eq. (3) neglects multiple scattering. The solid angle of light collection, A/2 = "rr A 02/4, outside the sample is given by A~(-2='trr~/r 2 [25]. Considering the reduction of the collection angles due to refraction in the plane of scattering and perpendicular to the plane of scattering, one finds [25] COS 4'
AOi
n2(1-
(sin 24')/n2) c o s 4,
= n2(1-(
sin2
1/2"A~(2 " rl
4')/n2)1/2 r2 ,
(4)
where n s is the refractive index of the sample (suspension or porous glass monolith). 4' is the angle between the external scattering direction and the normal of the surface of refraction (4' = 0 for exit surface normal to excitation light direction, 4' = 90 ° 0 if the exit surface is parallel to the excitation light direction).
52
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
Neglecting scattering losses, the intensity in the sample is I0,in =I0( 1 -R)
+I0(1 -R)R +I0(1 - R ) R 2
-'[- ... ~ I 0 , i.e., the scattering efficiency is not reduced by reflection of the incident light (intensity loss at the entrance surface, but intensity increase due to multiple internal reflections). The detected scattered light at an angle 4} to the direction of the exit surface normal is reduced by the factor
For s-polarized excitation (~0 = 90°), the particle diameter is obtained by insertion of Eqs. (4), (6), (8) into Eq. (3) and solving Eq. (3) for d:
1 [ P P2-Pl 2A4 m 2 + 2 2 d=-~[-~o P-P1 31 n2[1 - (sin 2 cos
r2
[ 1 - R ( { b ) ] + R 2 ( { b ) [ 1 - R(~b)]
+g4(t~)[l
-R(q~)] +...
cos
-(2ns/n"2 -
sin 2 {b) 1/2 ]2,
J
cos4,+( ns/n 2 a2 __ sin 2 ~,~11/2
(5)
-p)/'rrd 3,
(6)
where p is defined as the ratio of solvent volume to total volume for suspensions and as the ratio of pore volume to total volume for porous monoliths. The porosity is determined by the density relation
P =PPl + (1 -P)P2,
(7)
giving
P = ( P 2 - P ) / ( P 2 - Pl),
(8)
where p is the average density, Pl is the density of the solvent or the void and P2 is the density of the particles. The refractive index of the samples may be derived from the Clausius-Mossotti formula (LorenzLorenz relation). One finds [27]
n -I n2+2
n -I =pnz-~+(1-P)
(lO)
.
-~ [1 + R ( 4 } ) ] - 1
where n a is the refractive index of air. The number density of particles, N, may be be expressed by the porosity, p, [2] and the single-particle volume, "rrd3/6, according to N = 6(1
]1/3
× ~r---T[1 + R ( ~ ) ]
For s-polarization ( E perpendicular to the plane of scattering) the reflectance is given by [26]
=
ck)/n2] 1/2
×
n -I n2~2.
(9)
In the experiments, P/Po is measured and the particle diameter, d, is determined by use of Eq. (10). The parameters Pl, /92, ,)t, m = n2/n 1 and n s are generally known with high precision. The parameters l, r, rl, p are easily measured with a relative accuracy of better than 1%. The reflection factor, 1 +R(th), varies only slightly with th, so a precise measurement of 4} is not necessary. The ratio, P/Po, of scattered power to incident power may be measured with an accuracy of about 1% by averaging over some measurements. The one-third power dependence of the right-hand side of Eq. (10) reduces the relative error of d to one-third of the error of the root term. A strong attenuation of the incident light beam by scattering should be avoided by sample thickness selection (attenuation along propagation direction ~< 20%, attenuation not included in Eq. (3)) in order to reduce significant multiple scattering. The beam attenuation would lead to an asymmetry of scattering efficiency along the propagation direction. It would cause a prominent scattering into the entrance hemisphere and a reduced scattering into the exit hemisphere. The specific surface, S, (surface area per unit mass of particles [1,2]) for spherical particles is given by S.
S2 .
m2
.
S2
.
.
"iTd 2 .
P2V2 pz~d3/6
6
p2 d'
(11)
where S 2 is the surface, P2 is the density and V2 is the volume of the particles.
1t. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
53
Table 1 Parameters of investigated media Parameter
Ludox CL-X
Ludox PG
Vycor 7930
Gelsil 25 nm
Gelsil 50 nm
Silicate xerogel
p(g/cm3): p: ns: m: 0 (deg): I (cm):
1.342 0.715 1.36724 1.09436 90 1 1.8 x I0-3 21 20 26 130 137
1.168 0.86 1.34849 1.09436 90 1 2.3 X 10-3 22 27 48 100
1.5 [14] 0.28 [14] 1.3161 1.45674 45 0.159 1.3 X 10-4 4.5 4 3.3 250 600
1.2 [15] 0.48 [15] 1.235 1.45674 31 0.5 7.9 X 10-5 2.6 2.5 2.5 610 1040
0.9 [15] 0.63 [15] 1.17 1.45674 31 0.6 1.9 X 10-4 3.5 5 4.2 580 770
0.52 0.76 1.098 1.45674 31 0.25 4>(10-4
P/Po: d (nm): dp (nm): S (m2/g):
6.8 6 10 970 400
Comments
Eq. (8) Eq. (9) Eq. (2)
reported this work reported this work reported this work
Fixed parameters are p(air) = 0.001205 g / c m 3, P(H20) = 1 g / c m 3, p(SiO z) = 2.2 g / c m 3, n(air) = 1.0003, n(H20) = 1.33154, n(SiO 2) = 1.45718, r = 10 cm, r 1 = 2 cm, A = 632.8 nm.
An effective pore volume, Vp, and an effective pore diameter, d p = [6Up/'lr] 1/3, may be defined by assuming the same number density of pores as of particles. This setting gives vp = V J ( N V ) = p / N = xr d 3 p / [ 6 ( 1 - p)] and dp = ( p / ( 1
- p ) ) l/a d.
(12)
its pores are filled with formamide [16,28] (used as drying control chemical additive [29]). Before particle size measurement, the residual formamide was replaced by methanol (putting the sample in a tube filled with methanol) and then the methanol was removed by drying at 65°C. In the Rayleigh scattering experiments the pores of the Vycor, Gelsil and silicate xerogel glasses were only filled with air. The entrance and exit surfaces of the samples were of optical quality.
3. Experimental The experimental arrangement is shown schematically in Fig. l(b). A 2 mW linearly polarized H e - N e laser is used as excitation source. Part of the scattered light is collected with a lens, L, and detected with a photomultiplier tube (Valvo type PM2254B with $20 spectral characteristic). The photomultiplier is calibrated by measuring directly the photocurrent caused by the H e - N e laser light after appropriate attenuation. In all measurements the photomultiplier dark current is subtracted. The Ludox samples were used as delivered. They were investigated in a 1 cm fused silica fluorescence cell. The porous glass samples were kept in a desiccator to avoid water vapor condensation in the pores. The method of preparation of our self-prepared silicate xerogel is given in Ref. [16]. After preparation
4. Results The particle size, pore size, and specific surface of Ludox CL-X, Ludox PG [13], Vycor brand porous glass code 7930 [14], two Gelsil transparent porous silica glass samples of different pore size [15] and a porous silicate xerogel sample have been determined. For the purchased samples porous data are known from data sheets. The self-prepared xerogel has been investigated by dye adsorption studies [30]. Reported parameters and our own Rayleigh scattering results are given in Table 1. The reported particle diameters of Ludox CL-X and Ludox PG are in good agreement with our measurements. The effective pore diameters obtained by application of Eq. (12) are in
54
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
reasonable agreement with the specified values for the Vycor and Gelsil glasses.
5. Discussion
The particle size determination by Rayleigh scattering is very accurate for spherical particles of diameter d ~< 0.1 A. Therefore the results obtained for the diameters and specific surfaces of the Ludox colloidal silicon dioxide suspensions are in good agreement with the reported data. For the Vycor glasses, Gelsil glasses and silicate xerogels, the skeletal structure is approximated by spheres of effective diameters d. This approach gives specific surfaces which agree within a factor of two with reported values determined by gas adsorption porosimetry. The effective pore diameters determined by the use of Eq. (12) agree reasonably well with the reported pore diameters. For more precise specific surface determinations, gas adsorption studies are required [1,2]. Dye adsorption studies may be used after calibration to gas adsorption results have been made [30]. For an accurate pore size analysis, transmission electron microscopy [7] or scanning force microscopy [31] might be used. For porous media with certain particle size distributions, large particles contribute more strongly to the Rayleigh scattering signal since the Rayleigh scattering power depends on the third power of the particle diameter. Therefore Rayleigh scattering would give larger particles diameters in size-dispersive systems than the real mean diameter, d m = ~ , d i / ( V N ) , where the sum runs over all VN particles in volume V.
6. Conclusions
The Rayleigh light scattering theory has been applied to determine the average particle diameter and specific surface of colloidal suspensions. The effective spherical particle size, specific surface and pore size of nanometer porous transparent glasses has been calculated from measured Rayleigh scattering and porosity data. The simple Rayleigh scattering method is well suited for particle size determination of nanometer-size particles in gaseous, liquid or solid transparent matrix.
References [1] S.D. Gregg and K.S.W. Sing, Adsorption, Surface Area and Porosity (Academic Press, London, 1967). [2] S. Lowell and J.E. Shields, Powder Surface Area and Porosity, 3rd Ed. (Chapman and Hall, London, 1991). [3] L.L. Hench and J.K. West, Chem. Rev. 90 (1990) 33. [4] A. Guinier and G. Fournet, Small-angle Scattering of X-rays (Wiley, New York, 1955). [5] O. Kratky, Angew. Chem. 72 (1960) 467. [6] B. Jacrot, Rep. Prog. Phys. 39 (1976) 911. [7] C.A.M. Mulder, G. van Leeuwen-Stienstra, J.G. van Lierop and J.P. Woerdman, J. Non-Cryst. Solids 82 (1986) 148. [8] A.K. van Helden and A. Vrij, J. Colloid Interf. Sci. 78 (1980) 312. [9] B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). [10] E.J.A. Pope, M. Asami and J.D. Mackenzie, J. Mater. Res. 4 (1989) 1018. [11] B. Wolff, GIT Fachz. Lab. 2/93 (1993) 107. [12] D.W. Schaefer, J.E. Martin, P. Wiltzius and D.S. Cannell, Phys. Rev. Lett. 52 (1984) 2371. [13] Ludox CL-X Colloidal Silicon Dioxide, and Ludox PG, data sheets of DuPont Chemical and Pigments Department, Genf, Switzerland. [14] Vycor Brand Porous Glass, Product Information, Materials Business, Coming Glass Works, Coming, New York 14831, USA. [15] Gelsil Porous Glass and Lenses, data sheet, Geltech, Alachua, FL 32615, USA. [16] P. Weidner and A. Penzkofer, Chem. Phys. 191 (1995) 303. [17] G. Mie, Ann. Phys. (Leipzig) 25 (1908) 377. [18] P. Debye, Ann. Phys. (Leipzig) 30 (1909) 57. [19] H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957). [20] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York, 1969). [21] L.P. Bayvel and A.R. Jones, Electromagnetic Scattering and Applications (Applied Science, London, 1981). [22] C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983). [23] R. Weichert, 'Untersuchungen zur photometrischen charakterisierung von partikelkollektiven,' habilitationsschrift, Universit~it Karlsruhe (1984). [24] Lord Rayleigh, Philos. Mag. 41 (1871) 274 and 447. [25] Th. F6rster, Fluoreszenz Organischer Verbindungen (Vandenhoeck and Ruprecht, G6ttingen, 1951). [26] R.W. Ditchburn, Light (Academic Press, London, 1976). [27] A. Penzkofer, H. Glas and J. Schmailzl, Chem. Phys. 70 (1982) 47. [28] D. Lo, J.E. Parris and J.L. Lawless, Appl. Phys. B55 (1992) 365. [29] D.R. Ulrich, J. Non-Cryst. Solids 100 (1988) 174. [30] A. Penzkofer, H. Gratz and P. Weidner, J. Non-Cryst. Solids, this issue, p. 55. [31] D. Sarid, Scanning Force Microscopy (Oxford University, Oxford, 1991).
ELSEVIER
Journal of Non-CrystallineSolids 189 (1995)50-54
Nanometer particle size, pore size, and specific surface determination of colloidal suspensions and porous glasses by Rayleigh light scattering H. Gratz, A. Penzkofer *, P. Weidner Naturwissenschaftliche Fakultiit H-Physik, UniversitiitRegensburg, D-93040Regensburg, Germany
Received31 May 1994; revisedmanuscriptreceived5 January 1995
Abstract Rayleigh scattering of a He-Ne laser is applied to determine the particle size of colloidal silicon dioxide suspensions (Ludox CL-X and Ludox PG), and the effective sphere diameter of Vycor glass, and of silicate sol-gel glasses. The pore sizes and specific surfaces of the samples are estimated.
1. Introduction The size and surface area of micropores (diameter d~< 2 nm) and mesopores (2 nm ~< d~< 50 nm) of porous media are determined by gas adsorption studies [1,2], small-angle X-ray scattering (SAXS) [3-5], small-angle neutron scattering (SANS) [6], and transmission electron microscopy (TEM) [7]. Static light scattering [8] and dynamic light scattering [9] are used for particle size determination of colloidal suspensions. Rayleigh scattering has been used to discuss the transparency of silica gel-polymeric composites [10]. Small-angle light scattering (light diffraction in the forward direction) is restricted to macroporous media (particle size d > 50 nm) [11,12]. Here the Rayleigh scattering of a He-Ne laser is
* Corresponding author. Te1:+49-941 943 2107. Telefax: +49-941 943 2754. E-mail: [email protected] burg.de.
applied to determine the particle sizes of Ludox CL-X and Ludox PG (aqueous colloidal suspensions of silicon dioxide from DuPont [13]), of Vycor brand porous glass (code 7930 from Corning [14]), of Gelsil transparent porous silica glasses (from Geltech [15]), and self-prepared silicate xerogel [16]. The pore sizes and specific surfaces of the samples are estimated from the particle sizes and the porosities of the samples. The results are compared with the specifications of the manufacturers.
2. Theory The light scattering of particles is described by the Lorenz-Mie theory [17-22]. For particle diameters, d, small compared with the light wavelength, A, (d < 0.1A) [23] the Lorenz-Mie theory may be approximated by the Rayleigh scattering theory [24]. The scattering of linearly polarized light from a single sphere of diameter d is illustrated in Fig. l(a).
0022-3093//95//$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0022-3093(95)00206-5
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
Eo
.47)
I\
light polarized perpendicular to plane of scattering, s-polarization), the factor (sin 2 q~ + cos 2 0 cos 2 ~o) reduces to 1. A sample of particle number density, N (dimension c m - 3 ) , and thickness, l, is considered. It is irradiated by a light beam of power P0 = rio d A (integration over cross-sectional beam area). The scattered light is collected by a lens of radius, r~, at a distance, r. The collected power, P=I~rr21, of scattered light becomes
I
(b) ~ L I He-Ne Laser
51
A }2 m2_l ] 016
2
p = P°NI[ ~
× (sin 2 ~p + cos 2 0 cos 2 q~)
.......
1
X A a'-i + R ( 4 ' ) Fig. 1. (a) Illustration of the geometry of Rayleigh scattering from a single particle. E 0 is electric field strength of linearly polarized input light propagating in ~: direction. E is the electric field strength of light scattered an angle, 0, off the direction of the input light. (b) Schematic of experimental Rayleigh scattering arrangement: S is sample, L is lens, PM is photomultiplier tube, ¢ is angular collection direction outside sample, A0 is angular width of collected scattered light outside sample.
The scattered intensity, I, at a distance, r, from the sphere is given by [23]
, 0( )206 m2+2m21 =I
A
x (sin 2
2
+ cos 2 0 cos 2
(1)
where I 0 is the incident light intensity, A is the light wavelength, o~ = ~ d / A is the Mie parameter, q~ is the input polarization angle in the (7, ~ ) plane, and 0 is the scattering angle in the (~:, r I) plane, m, the complex refractive index ratio, is given by n 2 -- i K2 m . . . . hi n 1 - i Ka h2
n - iK,
(2)
where the subscript 2 indicates the sphere and the subscript 1 indicates the surrounding medium, n is the refractive index and K is the extinction coefficient. For transparent spheres in a transparent medium Eq. (2) reduces to m = n2/n r For q~ = 90 ° (input
=
'rr4d 6 rn 2 - 1 2(sin 2
PoNI-~ - - -
m2+2
~ "Jr-c o s 2 0 c o s 2 q))
1
xan~l +R(4') '
(3)
where A Oi = rr A 0i2/4 is the solid angle of light collection inside the sample and R(4') is the reflectance of the scattered light at the exit surface. Eq. (3) neglects multiple scattering. The solid angle of light collection, A/2 = "rr A 02/4, outside the sample is given by A~(-2='trr~/r 2 [25]. Considering the reduction of the collection angles due to refraction in the plane of scattering and perpendicular to the plane of scattering, one finds [25] COS 4'
AOi
n2(1-
(sin 24')/n2) c o s 4,
= n2(1-(
sin2
1/2"A~(2 " rl
4')/n2)1/2 r2 ,
(4)
where n s is the refractive index of the sample (suspension or porous glass monolith). 4' is the angle between the external scattering direction and the normal of the surface of refraction (4' = 0 for exit surface normal to excitation light direction, 4' = 90 ° 0 if the exit surface is parallel to the excitation light direction).
52
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
Neglecting scattering losses, the intensity in the sample is I0,in =I0( 1 -R)
+I0(1 -R)R +I0(1 - R ) R 2
-'[- ... ~ I 0 , i.e., the scattering efficiency is not reduced by reflection of the incident light (intensity loss at the entrance surface, but intensity increase due to multiple internal reflections). The detected scattered light at an angle 4} to the direction of the exit surface normal is reduced by the factor
For s-polarized excitation (~0 = 90°), the particle diameter is obtained by insertion of Eqs. (4), (6), (8) into Eq. (3) and solving Eq. (3) for d:
1 [ P P2-Pl 2A4 m 2 + 2 2 d=-~[-~o P-P1 31 n2[1 - (sin 2 cos
r2
[ 1 - R ( { b ) ] + R 2 ( { b ) [ 1 - R(~b)]
+g4(t~)[l
-R(q~)] +...
cos
-(2ns/n"2 -
sin 2 {b) 1/2 ]2,
J
cos4,+( ns/n 2 a2 __ sin 2 ~,~11/2
(5)
-p)/'rrd 3,
(6)
where p is defined as the ratio of solvent volume to total volume for suspensions and as the ratio of pore volume to total volume for porous monoliths. The porosity is determined by the density relation
P =PPl + (1 -P)P2,
(7)
giving
P = ( P 2 - P ) / ( P 2 - Pl),
(8)
where p is the average density, Pl is the density of the solvent or the void and P2 is the density of the particles. The refractive index of the samples may be derived from the Clausius-Mossotti formula (LorenzLorenz relation). One finds [27]
n -I n2+2
n -I =pnz-~+(1-P)
(lO)
.
-~ [1 + R ( 4 } ) ] - 1
where n a is the refractive index of air. The number density of particles, N, may be be expressed by the porosity, p, [2] and the single-particle volume, "rrd3/6, according to N = 6(1
]1/3
× ~r---T[1 + R ( ~ ) ]
For s-polarization ( E perpendicular to the plane of scattering) the reflectance is given by [26]
=
ck)/n2] 1/2
×
n -I n2~2.
(9)
In the experiments, P/Po is measured and the particle diameter, d, is determined by use of Eq. (10). The parameters Pl, /92, ,)t, m = n2/n 1 and n s are generally known with high precision. The parameters l, r, rl, p are easily measured with a relative accuracy of better than 1%. The reflection factor, 1 +R(th), varies only slightly with th, so a precise measurement of 4} is not necessary. The ratio, P/Po, of scattered power to incident power may be measured with an accuracy of about 1% by averaging over some measurements. The one-third power dependence of the right-hand side of Eq. (10) reduces the relative error of d to one-third of the error of the root term. A strong attenuation of the incident light beam by scattering should be avoided by sample thickness selection (attenuation along propagation direction ~< 20%, attenuation not included in Eq. (3)) in order to reduce significant multiple scattering. The beam attenuation would lead to an asymmetry of scattering efficiency along the propagation direction. It would cause a prominent scattering into the entrance hemisphere and a reduced scattering into the exit hemisphere. The specific surface, S, (surface area per unit mass of particles [1,2]) for spherical particles is given by S.
S2 .
m2
.
S2
.
.
"iTd 2 .
P2V2 pz~d3/6
6
p2 d'
(11)
where S 2 is the surface, P2 is the density and V2 is the volume of the particles.
1t. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
53
Table 1 Parameters of investigated media Parameter
Ludox CL-X
Ludox PG
Vycor 7930
Gelsil 25 nm
Gelsil 50 nm
Silicate xerogel
p(g/cm3): p: ns: m: 0 (deg): I (cm):
1.342 0.715 1.36724 1.09436 90 1 1.8 x I0-3 21 20 26 130 137
1.168 0.86 1.34849 1.09436 90 1 2.3 X 10-3 22 27 48 100
1.5 [14] 0.28 [14] 1.3161 1.45674 45 0.159 1.3 X 10-4 4.5 4 3.3 250 600
1.2 [15] 0.48 [15] 1.235 1.45674 31 0.5 7.9 X 10-5 2.6 2.5 2.5 610 1040
0.9 [15] 0.63 [15] 1.17 1.45674 31 0.6 1.9 X 10-4 3.5 5 4.2 580 770
0.52 0.76 1.098 1.45674 31 0.25 4>(10-4
P/Po: d (nm): dp (nm): S (m2/g):
6.8 6 10 970 400
Comments
Eq. (8) Eq. (9) Eq. (2)
reported this work reported this work reported this work
Fixed parameters are p(air) = 0.001205 g / c m 3, P(H20) = 1 g / c m 3, p(SiO z) = 2.2 g / c m 3, n(air) = 1.0003, n(H20) = 1.33154, n(SiO 2) = 1.45718, r = 10 cm, r 1 = 2 cm, A = 632.8 nm.
An effective pore volume, Vp, and an effective pore diameter, d p = [6Up/'lr] 1/3, may be defined by assuming the same number density of pores as of particles. This setting gives vp = V J ( N V ) = p / N = xr d 3 p / [ 6 ( 1 - p)] and dp = ( p / ( 1
- p ) ) l/a d.
(12)
its pores are filled with formamide [16,28] (used as drying control chemical additive [29]). Before particle size measurement, the residual formamide was replaced by methanol (putting the sample in a tube filled with methanol) and then the methanol was removed by drying at 65°C. In the Rayleigh scattering experiments the pores of the Vycor, Gelsil and silicate xerogel glasses were only filled with air. The entrance and exit surfaces of the samples were of optical quality.
3. Experimental The experimental arrangement is shown schematically in Fig. l(b). A 2 mW linearly polarized H e - N e laser is used as excitation source. Part of the scattered light is collected with a lens, L, and detected with a photomultiplier tube (Valvo type PM2254B with $20 spectral characteristic). The photomultiplier is calibrated by measuring directly the photocurrent caused by the H e - N e laser light after appropriate attenuation. In all measurements the photomultiplier dark current is subtracted. The Ludox samples were used as delivered. They were investigated in a 1 cm fused silica fluorescence cell. The porous glass samples were kept in a desiccator to avoid water vapor condensation in the pores. The method of preparation of our self-prepared silicate xerogel is given in Ref. [16]. After preparation
4. Results The particle size, pore size, and specific surface of Ludox CL-X, Ludox PG [13], Vycor brand porous glass code 7930 [14], two Gelsil transparent porous silica glass samples of different pore size [15] and a porous silicate xerogel sample have been determined. For the purchased samples porous data are known from data sheets. The self-prepared xerogel has been investigated by dye adsorption studies [30]. Reported parameters and our own Rayleigh scattering results are given in Table 1. The reported particle diameters of Ludox CL-X and Ludox PG are in good agreement with our measurements. The effective pore diameters obtained by application of Eq. (12) are in
54
H. Gratz et al. /Journal of Non-Crystalline Solids 189 (1995) 50-54
reasonable agreement with the specified values for the Vycor and Gelsil glasses.
5. Discussion
The particle size determination by Rayleigh scattering is very accurate for spherical particles of diameter d ~< 0.1 A. Therefore the results obtained for the diameters and specific surfaces of the Ludox colloidal silicon dioxide suspensions are in good agreement with the reported data. For the Vycor glasses, Gelsil glasses and silicate xerogels, the skeletal structure is approximated by spheres of effective diameters d. This approach gives specific surfaces which agree within a factor of two with reported values determined by gas adsorption porosimetry. The effective pore diameters determined by the use of Eq. (12) agree reasonably well with the reported pore diameters. For more precise specific surface determinations, gas adsorption studies are required [1,2]. Dye adsorption studies may be used after calibration to gas adsorption results have been made [30]. For an accurate pore size analysis, transmission electron microscopy [7] or scanning force microscopy [31] might be used. For porous media with certain particle size distributions, large particles contribute more strongly to the Rayleigh scattering signal since the Rayleigh scattering power depends on the third power of the particle diameter. Therefore Rayleigh scattering would give larger particles diameters in size-dispersive systems than the real mean diameter, d m = ~ , d i / ( V N ) , where the sum runs over all VN particles in volume V.
6. Conclusions
The Rayleigh light scattering theory has been applied to determine the average particle diameter and specific surface of colloidal suspensions. The effective spherical particle size, specific surface and pore size of nanometer porous transparent glasses has been calculated from measured Rayleigh scattering and porosity data. The simple Rayleigh scattering method is well suited for particle size determination of nanometer-size particles in gaseous, liquid or solid transparent matrix.
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