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Structure of hard-sphere colloid observed in real space by spin-echo small-angle neutron scattering
ARTICLE IN PRESS
Physica B 357 (2005) 452–455 www.elsevier.com/locate/physb
Structure of hard-sphere colloid observed in real space by spin-echo small-angle neutron scattering$ Timofey Kruglova,, Wim G. Bouwmana, Jeroen Plompa, M. Theo Rekveldta, Gert Jan Vroegeb, Andrei V. Petukhovb, Dominique M.E. Thies-Weesieb a Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, University of Utrecht, Padualaan 8, 3508 TB, Utrecht, The Netherlands
b
Received 12 November 2004; received in revised form 29 November 2004; accepted 6 December 2004
Abstract Spin-echo small-angle neutron scattering (SESANS) opens a wide range of potential applications. One of them is the study of colloids. Using SESANS we measured the structure of hard-sphere colloidal suspensions at two concentrations. We show that SESANS measures real-space correlations due to the particle shape as well as interactions between them. The dilute suspension showed no correlations exceeding the particle diameter. For a concentrated colloid the effect of the excluded volume is observed as a minimum in correlation function. Experimental correlation functions are compared with theoretical ones for noninteracting spheres and a hard-sphere liquid. The scattering density, volume fraction and the radius of particles are also determined from SESANS correlation function. r 2005 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 82.70. Dd; 76.60.Lz Keywords: Spin-echo small-angle neutron scattering; SESANS; Correlation function; Colloid
1. Introduction
$
This paper was accepted to proceedings of the Third European Conference on Neutron Scattering, but was lost during the publishing process. Corresponding author. Tel.: +31 15 2784533; fax: +31 15 2788303. E-mail address: [email protected] (T. Kruglov). URL: http://www.iri.tudelft.nl/sfwww/sesans/.
The study of structure on the length scale of a hundred nanometers and bigger using neutron scattering is an extremely challenging task. For conventional scattering access to higher length scales requires beam collimation and loss of neutrons. Spin-echo small-angle neutron scattering (SESANS) [1–4] circumvents this problem.
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.12.032
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
2. Experimental The experiment was performed on the SESANS setup at Interfaculty Reactor Institute, Delft University of Technology, Delft, The Netherlands. The principles of the technique involved have already been discussed in a number of papers (see, for instance [7,8]). Let us consider a sample of thickness t illuminated by a polarized neutron beam of cross section S of a fixed wavelength l: The quantity measured by SESANS is the polarization of the scattered neutron beam P normalized to the polarization P0 measured without a sample. The detected polarization P of the neutron beam is a function of the spin echo length z: PðzÞ ¼ estðGðzÞ1Þ ; P0
(1)
where spin-echo length cl2 BL cot y0 ; (2) p where y0 is the foil inclination angle with respect to the x-axis, c ¼ 4:632 1014 T1 m2 is the Larmor
z¼
B
-
+
+
y
Sample
z
π flipper
SESANS is a novel real-space method to determine the structure of materials using small-angle neutron scattering (SANS). The idea of using spin echo for elastic scattering was initially proposed by Pynn [5]. The method is based on the Larmor precession of polarized neutrons in magnetic field, which encodes the transmission angle of the neutron through a precession device [2,4]. Advantages of this technique are the high intensity of the beam and the large length scales that can be reached, up to 7 mm: A recent study [4] showed that multiple scattering can easily be taken into account in SESANS. A significant difference of SESANS compared to conventional scattering techniques is that the former measures a realspace function. An explicit connection with the particle correlation function gðrÞ allows to calculate the SESANS correlation function for dilute spherical particles analytically [6]. Here we present an application of SESANS to dilute and concentrated hard-sphere colloidal suspensions.
453
+
x
+ -
L
Fig. 1. Schematic side view diagram of a SESANS setup built with magnetized foils. The spin-echo length z can be varied by changing the sample position L or by varying the magnetic field B. The definition of the x, y and z direction is given in the diagram.
precession constant. The parameter z can be tuned experimentally by changing the magnetic field B or the distance L of the sample to the last magnet (see Fig. 1). In the experiment a few values of the magnetic field were chosen and for every value of the magnetic field a fine scan over z was performed by varying L. The SESANS correlation function GðzÞ in terms of the differential scattering cross section dsðQÞ=dO of a sample is 1 GðzÞ ¼ 2 sk0
Z
Z
þ1
þ1
dQy 1
dQz 1
dsðQÞ cosðQz zÞ; dO (3)
where Q ¼ ð0; Qy ; Qz Þ is a scattering vector. The total scattering probability is given by [6,9] st ¼
2NðDrÞ2 V 34 Rl2 3 ¼ fV ð1 fV ÞðDrÞ2 l2 tR; 2 S (4)
where V ¼ 43 pR3 is the volume of a sphere, N the number of spheres with scattering contrast Dr forming a suspension with volume fraction fV ¼ NV =tS: Eq. (3) presents the SESANS correlation function via the scattering cross section. It can also be presented directly via the conventional correlation function in real space: Z
Z
þ1
GðzÞ ¼
dxgðx; 0; zÞ ¼ 2 1
z
þ1
gðrÞr dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; r 2 z2 (5)
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
454
where the conventional density correlation function Z gðrÞ ¼ rðr0 Þrðr0 þ rÞ dr0 ; (6)
1.0
0.9
P (z)/P0
O
where rðrÞ is the scattering density of the sample and brackets stand for orientational averaging. The behavior of GðzÞ has a lot of similarities with that of gðrÞ; so the analysis of SESANS signal basically reduces to the analysis of the corresponding spatial correlation function. As a sample we used sterically stabilized silica spheres suspended in deuterated cyclohexane. The radius of the particles determined by the dynamic light scattering was 160 nm. The particles were covered by a sterically stabilizing layer of a few nanometers thick polyisobutene.
0.8
0.7
0.6
0.5 0
100
200
300
400
500
600
800
1000
z [nm]
1.0
In order to observe noninteracting particles the dilute solution was prepared. As can be seen from Fig. 2 a saturation level is reached at z ¼ 2R: This constitutes the fact that there is no correlation between particles and they can be considered noninteracting. The top curve in Fig. 2 was fitted using Eq. (1), where function GðzÞ for isolated spherical particles is known analytically (see Ref. [6]). GðzÞ is normalized and its only fitting parameter is the radius R of the particles. The polarization given by Eq. (1) depends also on the total scattering probability st: The s contains unknown product ð1 fV ÞfV ðDrÞ2 ; which is used as the second fitting parameter. In order to separate ð1 fV ÞfV from ðDrÞ2 data were also taken from the concentrated solution (see below). The combined data yield fV ¼ 0:055: In order to observe the appearance of pair correlations between particles the semidilute solution was also used. The curve on the bottom graph of Fig. 2 is calculated using the Percus–Yevick solution for the structure factor (see, for instance Ref. [10]). Unlike the dilute solution, not only the scattering probability s; but also GðzÞ depends on the volume fraction fV : The scattering contrast Dr affects only s; but not GðzÞ: This gives the possibility to determine fV and Dr independently,
P (z)/P0
0.8
3. Results and discussion
0.6
0.4
0.2
0
200
400 z [nm]
Fig. 2. Silica particles in deuterated cyclohexane. Open circles—experiment. Error bars are approximately equal to the size of the circles. Top: dilute solution. Solid line is the fit for isolated spherical particles with fV ¼ 0:055; R ¼ 149 nm: Bottom: concentrated solution. Solid line—calculated curve for hard-sphere liquid using Percus–Yevick solution with volume fraction fV ¼ 0:27 and radius R ¼ 149 nm:
which is not possible in case of the dilute solution. We fixed the value of the radius R obtained from the fit of the dilute solution and determined fV ¼ 0:27 and Dr ¼ 3:6 1010 cm2 : The last value is consistent with value Dr ¼ 3:4 1010 cm2 expected assuming a mass density of porous silica of 1:5 g=cm3 : A small deviation could be due to slightly different density of porous silica, which the colloidal particles are made of. In principle, all three parameters R, fV and Dr can be determined independently using the semidilute solution only.
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
4. Conclusion The SESANS correlation function provides information about correlations directly in real space and easily takes into account multiple scattering, which was high in all measured cases. The dilute suspension showed no correlations exceeding the particle diameter. The concentrated solution shows minima in its correlation function due to the excluded volume of hard spheres. The scattering density, volume fraction and the radius of particles were determined. SESANS measurements on hard-sphere colloid demonstrated very good agreement with the theoretical curves for noninteracting spheres and a hard-sphere liquid. References [1] T. Keller, R. Ga¨hler, H. Kunze, R. Golub, Neutron News 6 (1995) 16. [2] M. Theo Rekveldt, Nucl. Instrum. Methods B 114 (1996) 266.
455
[3] W.G. Bouwman, M.T. Rekveldt, Physica B 276–278 (2000) 126. [4] M.T. Rekveldt, W.G. Bouwman, W.H. Kraan, O. Uca, S. Grigoriev, S. Habicht, T. Keller, Neutron Spin Echno, Lecture Notes in Physics, vol. 601, 2003, p. 87. [5] R. Pynn, J. Phys. E: Sci. Instrum. 11 (1978) 1133. [6] T. Krouglov, I. de Schepper, W.G. Bouwman, M.T. Rekveldt, J. Appl. Crystallogr. 36 (2003) 117. [7] M.T. Rekveldt, W.G. Bouwman, W.H. Kraan, O. Uca, S. Grigoriev, S. Habicht, T. Keller, Neutron Spin Echno, Lecture Notes in Physics, vol. 601, 2003, p. 100. [8] T. Krouglov, W.H. Kraan, J. Plomp, M.T. Rekveldt, W.G. Bouwman, J. Appl. Crystallogr. 36 (2003) 816. [9] W.G. Bouwman, O. Uca, S. Grigoriev, W.H. Kraan, J. Plomp, M. Theo Rekveldt, J. Appl. Phys. A 74 (2002) S115. [10] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, San Diego, 1991.
Physica B 357 (2005) 452–455 www.elsevier.com/locate/physb
Structure of hard-sphere colloid observed in real space by spin-echo small-angle neutron scattering$ Timofey Kruglova,, Wim G. Bouwmana, Jeroen Plompa, M. Theo Rekveldta, Gert Jan Vroegeb, Andrei V. Petukhovb, Dominique M.E. Thies-Weesieb a Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, University of Utrecht, Padualaan 8, 3508 TB, Utrecht, The Netherlands
b
Received 12 November 2004; received in revised form 29 November 2004; accepted 6 December 2004
Abstract Spin-echo small-angle neutron scattering (SESANS) opens a wide range of potential applications. One of them is the study of colloids. Using SESANS we measured the structure of hard-sphere colloidal suspensions at two concentrations. We show that SESANS measures real-space correlations due to the particle shape as well as interactions between them. The dilute suspension showed no correlations exceeding the particle diameter. For a concentrated colloid the effect of the excluded volume is observed as a minimum in correlation function. Experimental correlation functions are compared with theoretical ones for noninteracting spheres and a hard-sphere liquid. The scattering density, volume fraction and the radius of particles are also determined from SESANS correlation function. r 2005 Elsevier B.V. All rights reserved. PACS: 61.12.Ex; 82.70. Dd; 76.60.Lz Keywords: Spin-echo small-angle neutron scattering; SESANS; Correlation function; Colloid
1. Introduction
$
This paper was accepted to proceedings of the Third European Conference on Neutron Scattering, but was lost during the publishing process. Corresponding author. Tel.: +31 15 2784533; fax: +31 15 2788303. E-mail address: [email protected] (T. Kruglov). URL: http://www.iri.tudelft.nl/sfwww/sesans/.
The study of structure on the length scale of a hundred nanometers and bigger using neutron scattering is an extremely challenging task. For conventional scattering access to higher length scales requires beam collimation and loss of neutrons. Spin-echo small-angle neutron scattering (SESANS) [1–4] circumvents this problem.
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.12.032
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
2. Experimental The experiment was performed on the SESANS setup at Interfaculty Reactor Institute, Delft University of Technology, Delft, The Netherlands. The principles of the technique involved have already been discussed in a number of papers (see, for instance [7,8]). Let us consider a sample of thickness t illuminated by a polarized neutron beam of cross section S of a fixed wavelength l: The quantity measured by SESANS is the polarization of the scattered neutron beam P normalized to the polarization P0 measured without a sample. The detected polarization P of the neutron beam is a function of the spin echo length z: PðzÞ ¼ estðGðzÞ1Þ ; P0
(1)
where spin-echo length cl2 BL cot y0 ; (2) p where y0 is the foil inclination angle with respect to the x-axis, c ¼ 4:632 1014 T1 m2 is the Larmor
z¼
B
-
+
+
y
Sample
z
π flipper
SESANS is a novel real-space method to determine the structure of materials using small-angle neutron scattering (SANS). The idea of using spin echo for elastic scattering was initially proposed by Pynn [5]. The method is based on the Larmor precession of polarized neutrons in magnetic field, which encodes the transmission angle of the neutron through a precession device [2,4]. Advantages of this technique are the high intensity of the beam and the large length scales that can be reached, up to 7 mm: A recent study [4] showed that multiple scattering can easily be taken into account in SESANS. A significant difference of SESANS compared to conventional scattering techniques is that the former measures a realspace function. An explicit connection with the particle correlation function gðrÞ allows to calculate the SESANS correlation function for dilute spherical particles analytically [6]. Here we present an application of SESANS to dilute and concentrated hard-sphere colloidal suspensions.
453
+
x
+ -
L
Fig. 1. Schematic side view diagram of a SESANS setup built with magnetized foils. The spin-echo length z can be varied by changing the sample position L or by varying the magnetic field B. The definition of the x, y and z direction is given in the diagram.
precession constant. The parameter z can be tuned experimentally by changing the magnetic field B or the distance L of the sample to the last magnet (see Fig. 1). In the experiment a few values of the magnetic field were chosen and for every value of the magnetic field a fine scan over z was performed by varying L. The SESANS correlation function GðzÞ in terms of the differential scattering cross section dsðQÞ=dO of a sample is 1 GðzÞ ¼ 2 sk0
Z
Z
þ1
þ1
dQy 1
dQz 1
dsðQÞ cosðQz zÞ; dO (3)
where Q ¼ ð0; Qy ; Qz Þ is a scattering vector. The total scattering probability is given by [6,9] st ¼
2NðDrÞ2 V 34 Rl2 3 ¼ fV ð1 fV ÞðDrÞ2 l2 tR; 2 S (4)
where V ¼ 43 pR3 is the volume of a sphere, N the number of spheres with scattering contrast Dr forming a suspension with volume fraction fV ¼ NV =tS: Eq. (3) presents the SESANS correlation function via the scattering cross section. It can also be presented directly via the conventional correlation function in real space: Z
Z
þ1
GðzÞ ¼
dxgðx; 0; zÞ ¼ 2 1
z
þ1
gðrÞr dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; r 2 z2 (5)
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
454
where the conventional density correlation function Z gðrÞ ¼ rðr0 Þrðr0 þ rÞ dr0 ; (6)
1.0
0.9
P (z)/P0
O
where rðrÞ is the scattering density of the sample and brackets stand for orientational averaging. The behavior of GðzÞ has a lot of similarities with that of gðrÞ; so the analysis of SESANS signal basically reduces to the analysis of the corresponding spatial correlation function. As a sample we used sterically stabilized silica spheres suspended in deuterated cyclohexane. The radius of the particles determined by the dynamic light scattering was 160 nm. The particles were covered by a sterically stabilizing layer of a few nanometers thick polyisobutene.
0.8
0.7
0.6
0.5 0
100
200
300
400
500
600
800
1000
z [nm]
1.0
In order to observe noninteracting particles the dilute solution was prepared. As can be seen from Fig. 2 a saturation level is reached at z ¼ 2R: This constitutes the fact that there is no correlation between particles and they can be considered noninteracting. The top curve in Fig. 2 was fitted using Eq. (1), where function GðzÞ for isolated spherical particles is known analytically (see Ref. [6]). GðzÞ is normalized and its only fitting parameter is the radius R of the particles. The polarization given by Eq. (1) depends also on the total scattering probability st: The s contains unknown product ð1 fV ÞfV ðDrÞ2 ; which is used as the second fitting parameter. In order to separate ð1 fV ÞfV from ðDrÞ2 data were also taken from the concentrated solution (see below). The combined data yield fV ¼ 0:055: In order to observe the appearance of pair correlations between particles the semidilute solution was also used. The curve on the bottom graph of Fig. 2 is calculated using the Percus–Yevick solution for the structure factor (see, for instance Ref. [10]). Unlike the dilute solution, not only the scattering probability s; but also GðzÞ depends on the volume fraction fV : The scattering contrast Dr affects only s; but not GðzÞ: This gives the possibility to determine fV and Dr independently,
P (z)/P0
0.8
3. Results and discussion
0.6
0.4
0.2
0
200
400 z [nm]
Fig. 2. Silica particles in deuterated cyclohexane. Open circles—experiment. Error bars are approximately equal to the size of the circles. Top: dilute solution. Solid line is the fit for isolated spherical particles with fV ¼ 0:055; R ¼ 149 nm: Bottom: concentrated solution. Solid line—calculated curve for hard-sphere liquid using Percus–Yevick solution with volume fraction fV ¼ 0:27 and radius R ¼ 149 nm:
which is not possible in case of the dilute solution. We fixed the value of the radius R obtained from the fit of the dilute solution and determined fV ¼ 0:27 and Dr ¼ 3:6 1010 cm2 : The last value is consistent with value Dr ¼ 3:4 1010 cm2 expected assuming a mass density of porous silica of 1:5 g=cm3 : A small deviation could be due to slightly different density of porous silica, which the colloidal particles are made of. In principle, all three parameters R, fV and Dr can be determined independently using the semidilute solution only.
ARTICLE IN PRESS T. Kruglov et al. / Physica B 357 (2005) 452–455
4. Conclusion The SESANS correlation function provides information about correlations directly in real space and easily takes into account multiple scattering, which was high in all measured cases. The dilute suspension showed no correlations exceeding the particle diameter. The concentrated solution shows minima in its correlation function due to the excluded volume of hard spheres. The scattering density, volume fraction and the radius of particles were determined. SESANS measurements on hard-sphere colloid demonstrated very good agreement with the theoretical curves for noninteracting spheres and a hard-sphere liquid. References [1] T. Keller, R. Ga¨hler, H. Kunze, R. Golub, Neutron News 6 (1995) 16. [2] M. Theo Rekveldt, Nucl. Instrum. Methods B 114 (1996) 266.
455
[3] W.G. Bouwman, M.T. Rekveldt, Physica B 276–278 (2000) 126. [4] M.T. Rekveldt, W.G. Bouwman, W.H. Kraan, O. Uca, S. Grigoriev, S. Habicht, T. Keller, Neutron Spin Echno, Lecture Notes in Physics, vol. 601, 2003, p. 87. [5] R. Pynn, J. Phys. E: Sci. Instrum. 11 (1978) 1133. [6] T. Krouglov, I. de Schepper, W.G. Bouwman, M.T. Rekveldt, J. Appl. Crystallogr. 36 (2003) 117. [7] M.T. Rekveldt, W.G. Bouwman, W.H. Kraan, O. Uca, S. Grigoriev, S. Habicht, T. Keller, Neutron Spin Echno, Lecture Notes in Physics, vol. 601, 2003, p. 100. [8] T. Krouglov, W.H. Kraan, J. Plomp, M.T. Rekveldt, W.G. Bouwman, J. Appl. Crystallogr. 36 (2003) 816. [9] W.G. Bouwman, O. Uca, S. Grigoriev, W.H. Kraan, J. Plomp, M. Theo Rekveldt, J. Appl. Phys. A 74 (2002) S115. [10] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, San Diego, 1991.