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Size determinations of clay particles in water suspensions by use of low-angle x-ray diffraction
SIZE D E T E R M I N A T I O N S OF CLAY P A R T I C L E S I N W A T E R S U S P E N S I O N S BY U S E OF L O W - A N G L E X-RAY D I F F R A C T I O N 1,2 W. J. W e s t California Research Corporation, La Habra, California Received January 28, 1952 _~kBSTRACT
Low-angle x-ray diffraction permits studies of sizes of clay particles as they exist in clay-water suspensions. Change in the state of the suspension can be made, and sizes of the particles can be determined again as they exist in the new state. Particle-size determinations by other methods do not share this advantage. Supercentrifugal size analysis must be made with sample at high dilutions, and electron microscope size analysis must be made with sample in the dry state. The two-crystal x-ray spectrometer designed and built by Professor J. W. M. DuMond at the California Institute of Technology was used to measure the low-angle x-ray diffraction from clay-water suspensions. From these measurements, particle sizes were determined. Particle size determinations on several clays that are used as drilling fluids in the petroleum industry showed: (a) The size of hydrated clay particles in water suspension was larger than the size of the particles in the original dry powdered clay (approx. 0.3 t~). The increase in size was proportional to the clays' swelling abilities. This observation contradicts the common belief that highly swelling clays have smaller particles in water suspensions than poorly swelling clays. (b) Sizes of the hydrated particles in water suspensions were independent of dilution for concentrations less than 8%. (e) Viscosity lowering by agents such as tetrasodium pyrophosphate involved size reduction of hydrated particles. INTRODUCTION
Size of p/~rticles h a s b e e n c o n s i d e r e d a n i m p o r t a n t f a c t o r affecting t h e p r o p e r t i e s of swelling, v i s c o s i t y , a n d gel s t r e n g t h , of c l a y - w a t e r s u s p e n sions. H o w e v e r , m o s t m e t h o d s of m e a s u r i n g p a r t i c l e size h a v e b e e n u n a b l e t o d e t e r m i n e size of p a r t i c l e s us t h e y exist in c l a y - w a t e r s u s p e n s i o n s . T h e m e t h o d of l o w - a n g l e x - r a y d i f f r a c t i o n is u n i q u e in t h a t t h e size of t h e p a r t i c l e c a n b e m e a s u r e d as i t e x i s t s in t h e w a t e r s u s p e n s i o n . T h e s a m p l e s t u d i e d is large, 2 - 3 cc., a n d is n o t d i s t u r b e d in t h e s a m p l i n g or m e a s u r i n g process. C h a n g e in t h e s t a t e of t h e s u s p e n s i o n c a n b e m a d e a n d sizes of 1 Results reported here were obtained in a cooperative research project between the California Institute of Technology and the California Research Corporation. The experiments were performed in the laboratories of, and through the courtesy of, Professor J. W. IV[.DuMond of the Institute. Presented at the Annual Meeting of the Society of Rheology, Chicago, Illinois, October 24-27, 1951. 295
296
w . J . W~,ST
the particles can be determined again as they exist in the new state. Particle-size determinations by other methods do not share this advantage. Supercentrifugal size analysis must be made with sample at high dilutions, and electron microscope size analysis must be made with sample in the dry state. This paper presents the results of an investigation of the use of lowangle x-ray diffraction as a tool for studying sizes of particles in claywater suspensions. The method of using low-angle x-ray scattering for measuring particle size is discussed in detail. Results of size determinations on clay-water suspensions are presented and briefly discussed. This paper does not present a complete report on the subject, but presents results of initial investigations. Low-ANGLE X-RAY DIFFRACTION FROM SPHERES During the past few years, several articles (1-5) have been published describing low-angle x-ray diffraction from small particles. In these articles have been considered the problems of shapes of particles, particlesize distribution, and multiple scattering as well as the problems of measuring low-angle x-ray diffraction. In this paper, only the simple case of diffraction from spheres will be considered, and t h e investigation of the usefulness of the low-angle x-ray method f o r studying sizes of particles in clay-water suspensions will be presented.
INCIDENT
>
X-RAY.BEAM
FIG. 1. Geometrical construction for calculating the diffraction of x-rays by spherical particles. In Fig. 1, a spherical particle is shown being radiated by x-rays. Let a disk of thickness d z be cut in the sphere so that the surface of the disk makes equal angles with the incident and scattered beam. The phase of the scattered wave from all parts of the disk is the same but changes as the disk is moved along z. The fraction of the incident x-ray beam scattered b y disk and observed in direction e is given by: •d A ~
= A~(R
~ -
z2)ne-2'~l~dz
[-17
PARTICLE-SIZE
DETERMINATIONS
297
where A~ is the fraction of the total beam scattered by one electron; n is the electron density of the particle (if particle is surrounded by a liquid, then n is the difference between the electron density of the particle and the liquid) ; k is wavelength of incident x-rays; and R, z, and E are defined by Fig. 1. Integrating Eq. EI~ over the sphere and squaring the result gives the intensity of the diffracted beam observed in direction e.
.J--R
[ sin (27reR/X) -- (2~reR/X) cos (27reR/X)]~ [2? N~[ ] where N = 4/3~rRan and is equal to total number of electrons per particle. The intensity of the scattering from many particles in a sample is just the algebraic sum of the intensity of the scattering from each of the particles, provided interaction can be neglected as will be the ease for dilute suspensions. Equation [2-] can be approximated very closely by the Gaussian function of Eq. [-3~.
I~__=
L
N~e_4~2m~/3~2"
E3-]
Taking the logarithm of Eq. [-3-] gives: I. In I~N--2
[ 4~r2R2] ~2.
~
[47
If the logarithm of the diffracted intensity, I~, is plotted as a function of the square of the scattering angle, d; then the slope of this plot is proportional to the square of the radius of the particles, R 2. In this investigation, particle sizes were determined from the slopes of the measured I n /Ts VS. e2 curves. To illustrate why the term "low angle" is applied to this type of :(-ray diffraction, an example is given. If the radius of the sphere is 1000 A. (0.1 it), and the wavelength of the x-rays radiating the sphere is 1 A., then the "half width" of the diffraction pattern is 56 sec. of arc. Observations of the details of such a narrow diffraction pattern require special type instruments. Guinier (1), who first analyzed the details of low-angle x-ray diffraction, used a curved crystal x-ray spectrometer. The two-crystM spectrometer (6) designed and built b y Professor J. W. M. DuMond of the California Institute of Technology was used in this investigation to measure the x-ray diffraction from the clay-water suspensions. The instrument is described in the reference cited. Precision of the spectrometer is very good. Angular settings of the crystal holders can be made to within a quarter of a second of arc throughout 360 deg. of rotation.
298
w . ~. WEST ~
E
2 CRYSTAL SPECTROMETER
HOLDER'
U
~ X-RAY TUBE
FIG. 2. Schematic illustration of the two-crystal x-ray spectrometer as used for measuring low:angl e x-ray diffraction from clay-water suspensions. Thicknesses of crystals at positions A and B are exaggerated to show reflection from internal planes that are perpendicular'to crystal faces. The slits or stops are at S a n d S r. The xenon-filled counter is at C. Sample holder windows were of 0.003-in. Lucite. iODiDe
I~,0~
,,.:: z ~E z
I00~001
o~
x to
ooo
50,0C
DO0 ooo
IO,OC
~
,ooo
SCATTERINGANGLE(El IN SECONDSOF ARC
FIG. 3. Intensity detected by counter of tw0-crystal spectrometer as a function of the deviation of crystal B from parallelism with crystal A. Central curve is for no sample in place. Dashed and solid curves for wing intensities on expanded scale are for sample in place between crystals and sample in front of crystal A, respectively.
PARTICLE-SIZE
299
DETERMINATIONS
In Fig. 2, a schematic diagram is shown of the low-angle x-ray diffraction apparatus. The target of the x-ray tube was molybdenum. Crystal A was set so that only the Kal, a~ lines (k = 0.710 A.) of molybdenum were reflected by the crystal..All other wavelengths did not satisfy Bragg's law of reflection and so were rejected from the reflected beam. Crystal A served as a monochromator. The reflected beam incident on the sample was geometrically defined by the slit at S. Without a sample in place, the reflected beam from crystal A was reflected again from crystal B when
,o,ooc
~
~
~~
~ ~ - 2 ~
,.-?
z
Iz
.. ~/OLn_ ,,
~ 1,000
:)
Z L~J Z
I.--
I00 0
I00
200
300
400
500
600
SCATTERING ANGLE SQUARED ((~) IN SECONDS OF" ARC SQUARED
FIG. 4. Calibration of two-crystal spectrometer for particle-size determinations. Calibration particle-size measurements were made by electron microscope for carbon sample and optical microscope for biological spore sample. Particles of both samples were spheres.
crystal B was parallel to crystal A. As crystal B was turned away from parallelism with crystal A, the reflected intensity dropped quickly to zero, i.e., within a few seconds of arc. The intensity of the reflected beam was measured by a xenon-filled Geiger counter at C. With the sample in place, part of the x-ray beam was diffracted in passing through the sample. Because the wavelength remains unchanged in this diffraction, crystal B could be rotated away from parallelism with crystal A, where Bragg's law was satisfied for central beam, to positions where Bragg's law was satisfied by the diffracted beam. Consequently, the intensity of the dif-
300
w.J.
WEST
fraction from the sample could be measured at several scattering angles by setting crystal B at these angles. The procedure used for determining the size of the particles in a sample was as follows: The spectrometer was aligned so that crystals A and B were parallel and their axis of rotation vertical. The optimum thickness of the sample was .determined so that the x-ray intensity was reduced by 1/e in passing through the sample. When this condition was satisfied, the maximum scattering was obtained from the sample. This optimum thickness Was about 1 cm. After these adjustments were made,
~C~ ~ "• Z ¢4
P, P z
O ¢,) >t.z
X W¥O'dEL 0
WYO'JEt,-
SAMPLE SAMPLEZ
PiT-SS
w IIo
0
100
200 300 400 500 S C A T T E R I N G ANGLE S Q U A R E D (~.=) IN SECONDS OF ARC S Q U A R E D
600
FIG. 5. Low-anglex-ray diffractionfrom small particles of powdered dry clay samples. Sizes lie between 0.3# and 0.7~. the instrument was ready to be used in measuring low-angle diffraction from particles of clay-water suspensions. For each setting of crystal B, the intensity was recorded with the sample behind slit S and then with the sample placed in front of crystal A. The difference between the two readings was proportional to the diffracted beam from the sample observed at a diffraction angle E. The angle Ewas the difference in angular readings for crystal B when set parallel to crystal A and when set away from parallelism with crystal A for a diffraction measurement. The diffraction curve was obtained by repeating the readings for several settings of crystal B. In Fig. 3 is shown two plots of intensity of the diffracted beam as a
PARTICLE-SIZE
301
DETERMINATIONS
function of the setting of crystal B. One plot is for no sample in place, and the other plot is for a sample of colloidal carbon in place. The extremely narrow curve of Fig. 3 for no sample in place is the feature t h a t makes this instrument useful for measuring certain types of low-angle diffraction. The dashed curve is the diffraction from a sample of colloidal carbon. The diameter of the carbon particles was approximately 1000 A.
(0.1 ~). I0~000 T .6 5 Z IE
o\
z 0 o I,z W jZ
I-.3
,I
0
I00
200 300 400 500 SCATTERING ANGLE SQUARED (Ez) IN SECONDS OF ARC SQUARED
600
FIG. 6. Low-angle x-ray diffraction from hydrated particles of four 8% clay-water suspensions. Increasing particle size corresponds to increasing slopes of these curves. Particle size of the Pit-95 sample remained same on forming suspension. Particles of other samples increased roughly proportionate to their swelling abilities. In Fig. 4, the logarithm of the diffracted intensity from the carbon sample is plotted as a function of the square of the diffraction angle. This curve is a straight line as predicted b y the theory. For particles larger t h a n 1000 A., the t h e o r y ceases to hold exactly because of refraction effects. However, b y calibrating the instrument with particles of known size, unknown sizes can be determined in the range of calibration. T h e plot of the diffracted intensity from biological spores shown in Fig. 4 was for calibration.
302
w . z . WEST Low-ANGLE X-RAY STUDIES
o F C L A Y - W A T E R SUSPENSIONS
Five clays were selected for this low x-ray investigation, some of which are u s e d in drilling fluids in the petroleum industry. F o u r of the clays are of the montmorillonite t y p e and cover a wide range of swelling abilities. These four clays have the trade names of Pit-95, Blue Wyoming, Wyo-Jel, and ttectorite, whose swelling abilities measured b y the American Colloid C o m p a n y test 3 are 4, 9, 16, a n d 22, respectively. The fifth clay has the trade name Otay. This c l a y is a good example of a group of 100oo
z
O 8 % CLAY-WATER
3E
X 4.% CLAY-WATER SUSPENSION
SUSPENSION
oJ
I-z
o >-
I-
\ \4\
1,00
z L~ Z t-i LU bJ I-I-.<
X
0
aoo
20O 3OO 4OO 5OO SCATTERING ANGLE SQUARED ((~,z.) tN SECONDS OF ARC SQUARED
sO0
FIG. 7. Low-angle x-ray diffraction from 8% and 4% Wyo-Jel clay-water suspensions. These two curves are essentially the same, except for reduction in intensity, showing that particle size remained constant on dilution. clays called bentonites; however, this particular clay is a poorly swelling clay. The Otay sample was selected because when tetrasodium pyrophosphate was added to Otay clay-water suspension essentially no viscosity change occurred, whereas the same additive caused a viscosity reduction to water suspension of the first four clays selected. The first set of measurements on clay particles was made to determine the particle size of the powdered dry clay samples. In Fig. 5 is shown the a The swelling test used was the one publicized by the American Colloid Company, 363 West Superior Street, Chicago 10, Illinois, "Data No. 251."
PARTICLE-SIZE
303
DETERMINATIONS
low-angle x-ray diffraction from three powdered clays. Two of the curves are for two different grinds of the Wyo-Jel sample, and the other curve is for the Pit-95 sample. The particle size was determined by comparing the slopes of the curves of Fig. 5 with the calibration curves of Fig. 4. The particle size of these samples was found to be between 0.3 and 0.7 ~. Eight per cent clay-water suspensions were made from the clay samples. The low-angle x-ray diffraction measured by the two-crystal I0,000
X 8 % WYO-JEL CLAY-WATER SUSPENSION Z
O 8 % WYO-JEL CLAY-WATER SUSPENSION WITH 2 . 5 0 R P B , TETRASODIUM PYROPHOSPHATE ADDED
4
O} I-o
z LU z
bA I-t-<~
7
6
X\%
5
3
ioo
6 5
"~-
~':o
"~'X 0
I00
200 SCATTERtNG IN
300
400
ANGLE
SQUARED
SECONDS
500
600
(•=)
OF A R C S Q U A R E D
FIG. 8. Low-angle x-ray diffraction from two 8% Wyo-Jel clay-water suspensions. Addition of tetrasodium pyrophosphate reduced the viscosity of suspension. The scattered intensity falls off less rapidly with scattering angle for the treated sample than for the untreated sample, indicating smaller particle size for treated sample.
spectrometer is shown in Fig. 6. The slopes of these curves are widely different. The increase in the slopes of the curves of Fig. 6 over those of Fig. 5 is greatest for Hectorite, next for Blue W y o m i n g and Wyo-Jel,
and least for Pit-95. The slope of the' curve for Pit-95 in clay-water suspension increased v e r y little over the curve for Pit-95 in dry powdered
state. Comparing the curves of Fig. 6 with the calibration curves of Fig. 6 shows that particle-size increase in forming a clay-water suspension was
304
w . J . WEST
greatest for Hectorite, and particle-size increase was least for Pit-95. This same trend is found in the swelling ability, viscosity, and gel strength of these clays. The swelling ability, viscosity, and gel strength were greatest for the Hectorite clay-water suspension and least for the Pit-95 clay-water suspension. This experiment shows that the sizes of clay particles increase in forming water suspensions and the increase is proportional to the swelling ability of the clay. An absolute particle-size determination could not be made because the diffraction curves for the clay=water suspensions fell outside the calilOgO0
0 7 % OTAY-WATER SUSPENSION X 7%OTAY-WATER SUSPENSION PLUS TETRASODIUM PYROPHOSPHATE
Z =E N Iz 0 ...,. >., Iz W kz w bl I< 0 ffl
10
0
IO0
20o
300
4o0
.500
6oo
SCATTERING ANGLE SQUARED (E =) iN SECONDS OF ARC S Q U A R E D
FIG. 9. Low-angle x-ray diffraction h'om two 7% Otay clay-water suspensions. Addition of tetrasodiumpyrophosphatedid not reduce viscosityof suspension.These two curves show no appreciable change in particle size between treated and untreated suspensions. brated range of the instrument. Further calibration of the instrument will be made when larger particles of known sizes can be obtained. The data obtained are useful because they show relative changes in particle size when water suspensions are formed. In the next experiment, the results of which are shown in Fig. 7, an 8% Wyo-Jel suspension was diluted to a 4% suspension. No appreciable change in particle size was observed. This experiment indicates that the size of the hydrated particles is essentially independent of further dilution.
PARTICLE-SIZE DETERMINATIONS
305
In the concluding set of experiments, changes in particle sizes were measured when tetrasodium pyrophosphate was added to clay-water suspensions. Figure 8 shows that the size of particles of the Wyo-Jel sample decreased when tetrasodium pyrophosphate was added. The decrease in size is indicated by the decrease in the slope of the logarithmic plot of the diffraction curve. Figure 9 shows that the size of the particles of the Otay suspension changed very little. Viscosity measurements show that the decrease in viscosity of the Wyo-Jel suspensions is greater than the decrease in the viscosity of the Otay suspensions with the addition of tetrasodium pyrophosphate. This experiment shows that viscosity lowering by adding tetrasodium pyrophosphate is accompanied by size reduction of the hydrated particles. CONCLUSIONS
This study has shown that low-angle x-ray diffraction can be used to measure sizes of particles in clay-water suspensions. The principal advantage of the method is that the sample studied is relatively large and it is not disturbed in the sampling or measuring process. From these measurements of size of particles in clay-water suspensions a better picture of the physical nature of the clay-water system has been obtained. These experiments show that the clay particle size increases in forming the water suspension. The increase in size is proportional to the swelling ability of the clay. The best swelling clays had the largest particle size in the clay-water suspension. This is contrary to the theory that particles in a water suspension are smaller for a highly swelling clay than for a poorly swelling clay. The clay-water systems with the largest particles were found to have the largest viscosity and the largest gel strength. The experiments also showed that viscosity lowering by additives is accompanied by size reduction of the particles. t~EFERENCES 1.~ GVlNIER, A., Ann. phys. 12, 161 (1939). 2. WARaEN, B. E., J. Applied Phys. 20, 96 (1949). 3. BOLDUAlV,O. E. A., AND BEAR, R. S., J. Applied Phys. 20, 983 (1949). 4. KA~SB]~RG,P., Phys. Rev. 74, 71 (1948); RITLAND,H. N., I~_~SBERG,P., ANDBEEMAN, W. W., J. Chem. Phys. 18, 1237 (1950) ; J. Applied Phys. 21, 838 (1950). 5. FANKUCtIEN,I., AND JELLINEK, M. H., Phys. Rev. 67, 201 (1945); Ind. Eng. Chem. 41, 2259 (1949); JELLINEK, M. H., SOLOMON,E., AND FANKUCtIEN,I., Ind. Eng. Chem., Anal. Ed. 18, 172 (1946); BARTON H. M., AND BRILL, R., d. Applied Phys. 21, 783 (1950). 6. DuiVIoND~J. W. live., AND 1VJ[ARLOW,D., Rev. Sci. Instruments 8, 112 (1937).
Low-angle x-ray diffraction permits studies of sizes of clay particles as they exist in clay-water suspensions. Change in the state of the suspension can be made, and sizes of the particles can be determined again as they exist in the new state. Particle-size determinations by other methods do not share this advantage. Supercentrifugal size analysis must be made with sample at high dilutions, and electron microscope size analysis must be made with sample in the dry state. The two-crystal x-ray spectrometer designed and built by Professor J. W. M. DuMond at the California Institute of Technology was used to measure the low-angle x-ray diffraction from clay-water suspensions. From these measurements, particle sizes were determined. Particle size determinations on several clays that are used as drilling fluids in the petroleum industry showed: (a) The size of hydrated clay particles in water suspension was larger than the size of the particles in the original dry powdered clay (approx. 0.3 t~). The increase in size was proportional to the clays' swelling abilities. This observation contradicts the common belief that highly swelling clays have smaller particles in water suspensions than poorly swelling clays. (b) Sizes of the hydrated particles in water suspensions were independent of dilution for concentrations less than 8%. (e) Viscosity lowering by agents such as tetrasodium pyrophosphate involved size reduction of hydrated particles. INTRODUCTION
Size of p/~rticles h a s b e e n c o n s i d e r e d a n i m p o r t a n t f a c t o r affecting t h e p r o p e r t i e s of swelling, v i s c o s i t y , a n d gel s t r e n g t h , of c l a y - w a t e r s u s p e n sions. H o w e v e r , m o s t m e t h o d s of m e a s u r i n g p a r t i c l e size h a v e b e e n u n a b l e t o d e t e r m i n e size of p a r t i c l e s us t h e y exist in c l a y - w a t e r s u s p e n s i o n s . T h e m e t h o d of l o w - a n g l e x - r a y d i f f r a c t i o n is u n i q u e in t h a t t h e size of t h e p a r t i c l e c a n b e m e a s u r e d as i t e x i s t s in t h e w a t e r s u s p e n s i o n . T h e s a m p l e s t u d i e d is large, 2 - 3 cc., a n d is n o t d i s t u r b e d in t h e s a m p l i n g or m e a s u r i n g process. C h a n g e in t h e s t a t e of t h e s u s p e n s i o n c a n b e m a d e a n d sizes of 1 Results reported here were obtained in a cooperative research project between the California Institute of Technology and the California Research Corporation. The experiments were performed in the laboratories of, and through the courtesy of, Professor J. W. IV[.DuMond of the Institute. Presented at the Annual Meeting of the Society of Rheology, Chicago, Illinois, October 24-27, 1951. 295
296
w . J . W~,ST
the particles can be determined again as they exist in the new state. Particle-size determinations by other methods do not share this advantage. Supercentrifugal size analysis must be made with sample at high dilutions, and electron microscope size analysis must be made with sample in the dry state. This paper presents the results of an investigation of the use of lowangle x-ray diffraction as a tool for studying sizes of particles in claywater suspensions. The method of using low-angle x-ray scattering for measuring particle size is discussed in detail. Results of size determinations on clay-water suspensions are presented and briefly discussed. This paper does not present a complete report on the subject, but presents results of initial investigations. Low-ANGLE X-RAY DIFFRACTION FROM SPHERES During the past few years, several articles (1-5) have been published describing low-angle x-ray diffraction from small particles. In these articles have been considered the problems of shapes of particles, particlesize distribution, and multiple scattering as well as the problems of measuring low-angle x-ray diffraction. In this paper, only the simple case of diffraction from spheres will be considered, and t h e investigation of the usefulness of the low-angle x-ray method f o r studying sizes of particles in clay-water suspensions will be presented.
INCIDENT
>
X-RAY.BEAM
FIG. 1. Geometrical construction for calculating the diffraction of x-rays by spherical particles. In Fig. 1, a spherical particle is shown being radiated by x-rays. Let a disk of thickness d z be cut in the sphere so that the surface of the disk makes equal angles with the incident and scattered beam. The phase of the scattered wave from all parts of the disk is the same but changes as the disk is moved along z. The fraction of the incident x-ray beam scattered b y disk and observed in direction e is given by: •d A ~
= A~(R
~ -
z2)ne-2'~l~dz
[-17
PARTICLE-SIZE
DETERMINATIONS
297
where A~ is the fraction of the total beam scattered by one electron; n is the electron density of the particle (if particle is surrounded by a liquid, then n is the difference between the electron density of the particle and the liquid) ; k is wavelength of incident x-rays; and R, z, and E are defined by Fig. 1. Integrating Eq. EI~ over the sphere and squaring the result gives the intensity of the diffracted beam observed in direction e.
.J--R
[ sin (27reR/X) -- (2~reR/X) cos (27reR/X)]~ [2? N~[ ] where N = 4/3~rRan and is equal to total number of electrons per particle. The intensity of the scattering from many particles in a sample is just the algebraic sum of the intensity of the scattering from each of the particles, provided interaction can be neglected as will be the ease for dilute suspensions. Equation [2-] can be approximated very closely by the Gaussian function of Eq. [-3~.
I~__=
L
N~e_4~2m~/3~2"
E3-]
Taking the logarithm of Eq. [-3-] gives: I. In I~N--2
[ 4~r2R2] ~2.
~
[47
If the logarithm of the diffracted intensity, I~, is plotted as a function of the square of the scattering angle, d; then the slope of this plot is proportional to the square of the radius of the particles, R 2. In this investigation, particle sizes were determined from the slopes of the measured I n /Ts VS. e2 curves. To illustrate why the term "low angle" is applied to this type of :(-ray diffraction, an example is given. If the radius of the sphere is 1000 A. (0.1 it), and the wavelength of the x-rays radiating the sphere is 1 A., then the "half width" of the diffraction pattern is 56 sec. of arc. Observations of the details of such a narrow diffraction pattern require special type instruments. Guinier (1), who first analyzed the details of low-angle x-ray diffraction, used a curved crystal x-ray spectrometer. The two-crystM spectrometer (6) designed and built b y Professor J. W. M. DuMond of the California Institute of Technology was used in this investigation to measure the x-ray diffraction from the clay-water suspensions. The instrument is described in the reference cited. Precision of the spectrometer is very good. Angular settings of the crystal holders can be made to within a quarter of a second of arc throughout 360 deg. of rotation.
298
w . ~. WEST ~
E
2 CRYSTAL SPECTROMETER
HOLDER'
U
~ X-RAY TUBE
FIG. 2. Schematic illustration of the two-crystal x-ray spectrometer as used for measuring low:angl e x-ray diffraction from clay-water suspensions. Thicknesses of crystals at positions A and B are exaggerated to show reflection from internal planes that are perpendicular'to crystal faces. The slits or stops are at S a n d S r. The xenon-filled counter is at C. Sample holder windows were of 0.003-in. Lucite. iODiDe
I~,0~
,,.:: z ~E z
I00~001
o~
x to
ooo
50,0C
DO0 ooo
IO,OC
~
,ooo
SCATTERINGANGLE(El IN SECONDSOF ARC
FIG. 3. Intensity detected by counter of tw0-crystal spectrometer as a function of the deviation of crystal B from parallelism with crystal A. Central curve is for no sample in place. Dashed and solid curves for wing intensities on expanded scale are for sample in place between crystals and sample in front of crystal A, respectively.
PARTICLE-SIZE
299
DETERMINATIONS
In Fig. 2, a schematic diagram is shown of the low-angle x-ray diffraction apparatus. The target of the x-ray tube was molybdenum. Crystal A was set so that only the Kal, a~ lines (k = 0.710 A.) of molybdenum were reflected by the crystal..All other wavelengths did not satisfy Bragg's law of reflection and so were rejected from the reflected beam. Crystal A served as a monochromator. The reflected beam incident on the sample was geometrically defined by the slit at S. Without a sample in place, the reflected beam from crystal A was reflected again from crystal B when
,o,ooc
~
~
~~
~ ~ - 2 ~
,.-?
z
Iz
.. ~/OLn_ ,,
~ 1,000
:)
Z L~J Z
I.--
I00 0
I00
200
300
400
500
600
SCATTERING ANGLE SQUARED ((~) IN SECONDS OF" ARC SQUARED
FIG. 4. Calibration of two-crystal spectrometer for particle-size determinations. Calibration particle-size measurements were made by electron microscope for carbon sample and optical microscope for biological spore sample. Particles of both samples were spheres.
crystal B was parallel to crystal A. As crystal B was turned away from parallelism with crystal A, the reflected intensity dropped quickly to zero, i.e., within a few seconds of arc. The intensity of the reflected beam was measured by a xenon-filled Geiger counter at C. With the sample in place, part of the x-ray beam was diffracted in passing through the sample. Because the wavelength remains unchanged in this diffraction, crystal B could be rotated away from parallelism with crystal A, where Bragg's law was satisfied for central beam, to positions where Bragg's law was satisfied by the diffracted beam. Consequently, the intensity of the dif-
300
w.J.
WEST
fraction from the sample could be measured at several scattering angles by setting crystal B at these angles. The procedure used for determining the size of the particles in a sample was as follows: The spectrometer was aligned so that crystals A and B were parallel and their axis of rotation vertical. The optimum thickness of the sample was .determined so that the x-ray intensity was reduced by 1/e in passing through the sample. When this condition was satisfied, the maximum scattering was obtained from the sample. This optimum thickness Was about 1 cm. After these adjustments were made,
~C~ ~ "• Z ¢4
P, P z
O ¢,) >t.z
X W¥O'dEL 0
WYO'JEt,-
SAMPLE SAMPLEZ
PiT-SS
w IIo
0
100
200 300 400 500 S C A T T E R I N G ANGLE S Q U A R E D (~.=) IN SECONDS OF ARC S Q U A R E D
600
FIG. 5. Low-anglex-ray diffractionfrom small particles of powdered dry clay samples. Sizes lie between 0.3# and 0.7~. the instrument was ready to be used in measuring low-angle diffraction from particles of clay-water suspensions. For each setting of crystal B, the intensity was recorded with the sample behind slit S and then with the sample placed in front of crystal A. The difference between the two readings was proportional to the diffracted beam from the sample observed at a diffraction angle E. The angle Ewas the difference in angular readings for crystal B when set parallel to crystal A and when set away from parallelism with crystal A for a diffraction measurement. The diffraction curve was obtained by repeating the readings for several settings of crystal B. In Fig. 3 is shown two plots of intensity of the diffracted beam as a
PARTICLE-SIZE
301
DETERMINATIONS
function of the setting of crystal B. One plot is for no sample in place, and the other plot is for a sample of colloidal carbon in place. The extremely narrow curve of Fig. 3 for no sample in place is the feature t h a t makes this instrument useful for measuring certain types of low-angle diffraction. The dashed curve is the diffraction from a sample of colloidal carbon. The diameter of the carbon particles was approximately 1000 A.
(0.1 ~). I0~000 T .6 5 Z IE
o\
z 0 o I,z W jZ
I-.3
,I
0
I00
200 300 400 500 SCATTERING ANGLE SQUARED (Ez) IN SECONDS OF ARC SQUARED
600
FIG. 6. Low-angle x-ray diffraction from hydrated particles of four 8% clay-water suspensions. Increasing particle size corresponds to increasing slopes of these curves. Particle size of the Pit-95 sample remained same on forming suspension. Particles of other samples increased roughly proportionate to their swelling abilities. In Fig. 4, the logarithm of the diffracted intensity from the carbon sample is plotted as a function of the square of the diffraction angle. This curve is a straight line as predicted b y the theory. For particles larger t h a n 1000 A., the t h e o r y ceases to hold exactly because of refraction effects. However, b y calibrating the instrument with particles of known size, unknown sizes can be determined in the range of calibration. T h e plot of the diffracted intensity from biological spores shown in Fig. 4 was for calibration.
302
w . z . WEST Low-ANGLE X-RAY STUDIES
o F C L A Y - W A T E R SUSPENSIONS
Five clays were selected for this low x-ray investigation, some of which are u s e d in drilling fluids in the petroleum industry. F o u r of the clays are of the montmorillonite t y p e and cover a wide range of swelling abilities. These four clays have the trade names of Pit-95, Blue Wyoming, Wyo-Jel, and ttectorite, whose swelling abilities measured b y the American Colloid C o m p a n y test 3 are 4, 9, 16, a n d 22, respectively. The fifth clay has the trade name Otay. This c l a y is a good example of a group of 100oo
z
O 8 % CLAY-WATER
3E
X 4.% CLAY-WATER SUSPENSION
SUSPENSION
oJ
I-z
o >-
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z L~ Z t-i LU bJ I-I-.<
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aoo
20O 3OO 4OO 5OO SCATTERING ANGLE SQUARED ((~,z.) tN SECONDS OF ARC SQUARED
sO0
FIG. 7. Low-angle x-ray diffraction from 8% and 4% Wyo-Jel clay-water suspensions. These two curves are essentially the same, except for reduction in intensity, showing that particle size remained constant on dilution. clays called bentonites; however, this particular clay is a poorly swelling clay. The Otay sample was selected because when tetrasodium pyrophosphate was added to Otay clay-water suspension essentially no viscosity change occurred, whereas the same additive caused a viscosity reduction to water suspension of the first four clays selected. The first set of measurements on clay particles was made to determine the particle size of the powdered dry clay samples. In Fig. 5 is shown the a The swelling test used was the one publicized by the American Colloid Company, 363 West Superior Street, Chicago 10, Illinois, "Data No. 251."
PARTICLE-SIZE
303
DETERMINATIONS
low-angle x-ray diffraction from three powdered clays. Two of the curves are for two different grinds of the Wyo-Jel sample, and the other curve is for the Pit-95 sample. The particle size was determined by comparing the slopes of the curves of Fig. 5 with the calibration curves of Fig. 4. The particle size of these samples was found to be between 0.3 and 0.7 ~. Eight per cent clay-water suspensions were made from the clay samples. The low-angle x-ray diffraction measured by the two-crystal I0,000
X 8 % WYO-JEL CLAY-WATER SUSPENSION Z
O 8 % WYO-JEL CLAY-WATER SUSPENSION WITH 2 . 5 0 R P B , TETRASODIUM PYROPHOSPHATE ADDED
4
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7
6
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ioo
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200 SCATTERtNG IN
300
400
ANGLE
SQUARED
SECONDS
500
600
(•=)
OF A R C S Q U A R E D
FIG. 8. Low-angle x-ray diffraction from two 8% Wyo-Jel clay-water suspensions. Addition of tetrasodium pyrophosphate reduced the viscosity of suspension. The scattered intensity falls off less rapidly with scattering angle for the treated sample than for the untreated sample, indicating smaller particle size for treated sample.
spectrometer is shown in Fig. 6. The slopes of these curves are widely different. The increase in the slopes of the curves of Fig. 6 over those of Fig. 5 is greatest for Hectorite, next for Blue W y o m i n g and Wyo-Jel,
and least for Pit-95. The slope of the' curve for Pit-95 in clay-water suspension increased v e r y little over the curve for Pit-95 in dry powdered
state. Comparing the curves of Fig. 6 with the calibration curves of Fig. 6 shows that particle-size increase in forming a clay-water suspension was
304
w . J . WEST
greatest for Hectorite, and particle-size increase was least for Pit-95. This same trend is found in the swelling ability, viscosity, and gel strength of these clays. The swelling ability, viscosity, and gel strength were greatest for the Hectorite clay-water suspension and least for the Pit-95 clay-water suspension. This experiment shows that the sizes of clay particles increase in forming water suspensions and the increase is proportional to the swelling ability of the clay. An absolute particle-size determination could not be made because the diffraction curves for the clay=water suspensions fell outside the calilOgO0
0 7 % OTAY-WATER SUSPENSION X 7%OTAY-WATER SUSPENSION PLUS TETRASODIUM PYROPHOSPHATE
Z =E N Iz 0 ...,. >., Iz W kz w bl I< 0 ffl
10
0
IO0
20o
300
4o0
.500
6oo
SCATTERING ANGLE SQUARED (E =) iN SECONDS OF ARC S Q U A R E D
FIG. 9. Low-angle x-ray diffraction h'om two 7% Otay clay-water suspensions. Addition of tetrasodiumpyrophosphatedid not reduce viscosityof suspension.These two curves show no appreciable change in particle size between treated and untreated suspensions. brated range of the instrument. Further calibration of the instrument will be made when larger particles of known sizes can be obtained. The data obtained are useful because they show relative changes in particle size when water suspensions are formed. In the next experiment, the results of which are shown in Fig. 7, an 8% Wyo-Jel suspension was diluted to a 4% suspension. No appreciable change in particle size was observed. This experiment indicates that the size of the hydrated particles is essentially independent of further dilution.
PARTICLE-SIZE DETERMINATIONS
305
In the concluding set of experiments, changes in particle sizes were measured when tetrasodium pyrophosphate was added to clay-water suspensions. Figure 8 shows that the size of particles of the Wyo-Jel sample decreased when tetrasodium pyrophosphate was added. The decrease in size is indicated by the decrease in the slope of the logarithmic plot of the diffraction curve. Figure 9 shows that the size of the particles of the Otay suspension changed very little. Viscosity measurements show that the decrease in viscosity of the Wyo-Jel suspensions is greater than the decrease in the viscosity of the Otay suspensions with the addition of tetrasodium pyrophosphate. This experiment shows that viscosity lowering by adding tetrasodium pyrophosphate is accompanied by size reduction of the hydrated particles. CONCLUSIONS
This study has shown that low-angle x-ray diffraction can be used to measure sizes of particles in clay-water suspensions. The principal advantage of the method is that the sample studied is relatively large and it is not disturbed in the sampling or measuring process. From these measurements of size of particles in clay-water suspensions a better picture of the physical nature of the clay-water system has been obtained. These experiments show that the clay particle size increases in forming the water suspension. The increase in size is proportional to the swelling ability of the clay. The best swelling clays had the largest particle size in the clay-water suspension. This is contrary to the theory that particles in a water suspension are smaller for a highly swelling clay than for a poorly swelling clay. The clay-water systems with the largest particles were found to have the largest viscosity and the largest gel strength. The experiments also showed that viscosity lowering by additives is accompanied by size reduction of the particles. t~EFERENCES 1.~ GVlNIER, A., Ann. phys. 12, 161 (1939). 2. WARaEN, B. E., J. Applied Phys. 20, 96 (1949). 3. BOLDUAlV,O. E. A., AND BEAR, R. S., J. Applied Phys. 20, 983 (1949). 4. KA~SB]~RG,P., Phys. Rev. 74, 71 (1948); RITLAND,H. N., I~_~SBERG,P., ANDBEEMAN, W. W., J. Chem. Phys. 18, 1237 (1950) ; J. Applied Phys. 21, 838 (1950). 5. FANKUCtIEN,I., AND JELLINEK, M. H., Phys. Rev. 67, 201 (1945); Ind. Eng. Chem. 41, 2259 (1949); JELLINEK, M. H., SOLOMON,E., AND FANKUCtIEN,I., Ind. Eng. Chem., Anal. Ed. 18, 172 (1946); BARTON H. M., AND BRILL, R., d. Applied Phys. 21, 783 (1950). 6. DuiVIoND~J. W. live., AND 1VJ[ARLOW,D., Rev. Sci. Instruments 8, 112 (1937).