Light Scattering of PMMA Latex Particles in Benzene: Structural Effects

Transcription

Light Scattering of PMMA Latex Particles in Benzene:
Structural Effects
E. A. NIEUWENHUIS AND A. VRIJ
Van't H o f f Laboratory for Physical and Colloid Chemistry, Transitorium 3, Padualaan 8, 3584 CH Utrecht,
The Netherlands
Received February 19, 1979; accepted February 19, 1979
Intra- and interparticle structural effects were studied in polymethylmethacrylate (PMMA) latex
dispersions in a nonpolar solvent with the technique of light scattering. The required transparency of
the dispersions was attained by a close matching of the refractive index of PMMA and solvent, for
which benzene was chosen. Two latices were studied with divinylbenzene (DVB) and ethylene
glycol dimethacrylate (EGDM) as crosslinking agents. The latex particles were characterized by
electron microscopy in the dry state and by viscosity, sedimentation, light scattering, and laser light
scattering spectroscopy in dilute solutions. The molar mass of the particles is in the range of
5 × 109 g mole-L In benzene they are swollen and a hydrodynamic radius in the range of 200 nm was
measured. The light scattering intensity and its angular variation were found to be strongly temperature dependent, This could be explained by the difference in temperature coefficients of the refractive indexes of PMMA and benzene which is sufficient to change their matching appreciably. For
the latex containing DVB this phenomenon could be used to characterize the spatial distribution
of the DVB inside the particle. At higher concentrations (20-80 g dm 3) light scattering as a
function of scattering angle shows the emergence of a diffuse diffraction peak which points to interparticle structural effects. In one case the radial distribution function of particle centers was
determined by a Fourier transformation of the structure factor and is discussed in terms of some
theories of atomic liquids. The position of the diffraction peaks as a function of concentration is
also discussed in terms of an "expanded-lattice" structure, Results suggested that some structure
persisted down to concentrations of 2% (w/v) of latex. However, the calculated mean interpa~icle
distances seem much too large to be explained on a basis of interpenetration of peripheral chains
of the particles. Attempts were made to interpret the long range interaction by electrostatic repulsions. However, measurements of the electric conductivity of the dispersions do not corroborate
this view. Similar structural effects showed up with laser light scattering spectroscopy, where it
was found that the diffusion coefficient depended on the scattering angle. On prolonged standing
some of the systems showed sharp, bright diffraction peaks in visible light due to the formation
of "supramolecular crystals" as reported earlier in the literature.
few fundamental investigations have been
reported, the more recent ones being those
of Krieger et al. (2) and Hachisu et al. (3).
At higher concentrations these dispersions
show the typical iridescence of visible light
caused by interference of light waves scattered by the individual particles which are
packed in a regular fashion. A serious drawback in studying this iridescence is the very
large optical density of these dispersions.
This restricts the scope of these studies to
thin surface layers, the properties of which
1. INTRODUCTION
In 1958 Shashoua and Beaman (1) published the first systematic work on dispersions of polymer latex particles in organic
solvents. Already at that time they stressed
the importance of these systems for studying fundamental aspects of polymer behavior. The individual particles were called
microgets. The dispersions were characterized by ultracentrifugation, viscosity, and
electron microscopy. Since then relatively
321
Journal of Colloidand InterfaceScience, Vol. 72, No. 2, November1979
0021-9797/79/140321-21 $02.00/0
Copyright © 1979by AcademicPress, Inc.
All rightsof reproductionin a~lyform reserved,
322
NIEUWENHUIS AND VRIJ
TABLE I
PolymerizationData of Latices
Latex
Concn of MMA
(mole dm -3)
Concn of KPS
(mole dm -3)
Ratio crosslinking monomer
(vol%)
Reaction
time
(hr)
Reaction
temperature
(°C)
P10
P12
1.0
1.0
9.25 × 10-4
3.33 × 10-4
EGDM (5%)
DVB (5%)
7
8
70
70
may be influenced by the presence of the
wall of the vessel, it would therefore be
of interest if this large optical density
could be alleviated for in that case one could
also study the scattering properties of the
bulk of the dispersion, just like X-ray scattering in the case of molecular liquids, and
obtain information about the internal structure of the particle system.
In order to investigate such a possibility
we have chosen to study crosslinked PMMA
latex in benzene. In these dispersions the
refractive index difference between particles
and solvent is very small and the systems
are transparent.
In this paper we will present results on
the average light scattering intensity as a
function of scattering angle and dispersion
concentration as well as some results on the
time correlations in the scattering intensity
(quasi-elastic light scattering). The first
method gives information on the time average structure in space and the second method
on spatial structure changes with time. Both
kinds of information are essential ingredients in the formulation of better theories
on concentrated colloidal dispersions in
general.
In the near future one could hope to build
such theories on the foundations laid by the
theories of simple, molecular liquids, which
have improved so much in the last 10 to 15
years (4). The dispersed particles, acting as
"supramolecules," embedded in a "structureless" low-molecular-weight solvent,
would play the role of the molecules in a
simple, monoatomic liquid (5, 6).
Journal of Colloid and Interface Science, Vol. 72, No. 2, November 1979
2. LATEXPREI:;ARATION
Following Ono et al. (7) the latex dispersions were prepared by heterogeneous
polymerization in water without emulsifier
using different amounts of potassium
persulfate (KPS) as initiator. This method,
yielding so-called "soap-free" polymer
latex dispersions, was adapted by adding
the crosslinking agents divinylbenzene
(DVB) (3) or ethylene glycol dimethacrylate (EGDM) to the reaction mixture. The
latter (EGDM) has a refractive index nearly
equal to that of MMA, thus yielding an
optically more homogeneous particle than
DVB which has a higher refractive index.
The polymerization conditions of the two
latices are listed in Table I.
It turned out that without further purification the vacuum-dried latex did not disperse in benzene. Therefore the latex was
purified by dialysis (Union Carbide tube)
against deionized water for a few days to
remove residual MMA. During this period
the dialysate was changed continuously.
Excess initiator and other ionic impurities
were removed by ion-exchange (7-9), using
equal weights of Dowex 50W-X4 and Dowex
l-X4 resin. The purified latex was vacuum
dried at room temperature and was readily
dispersable in benzene under shaking in a
few hours. The solutions were almost
transparent when viewed under diffuse light.
Finally, excess free polymer was removed
by sedimentation of the latex particles and
redispersion in fresh solvent. Repeating this
procedure three times proved to be sufficient.
323
STRUCTURE IN LATEX DISPERSIONS
3. INSTRUMENTAL
The light scattering (LS) measurements
were performed with a F I C A 50 light scattering photometer. As a scattering standard
pure benzene was used (R90 = 15.8 × 10-6
cm -1 at )to = 546 nm andRa0 = 45.6 x 10-6
cm -1 at )to = 436 nm). The angular range
studied was 15° ~< 0 ~< 150 °. The incident
light beam was unpolarized. The solutions
were filtered through Millipore filters (1.2
and 3/xm) directly into the measuring cells.
Laser light scattering spectroscopy (LLSS)
was performed with a laboratory-built instrument containing a double-walled thermostated sample holder. The 514.5-nm line of
an argon ion laser (Spectra Physics Model
165) was used and the light was detected
with an E M T 9558 photomultiplier. The correlation functions of the fluctuations in the
scattered intensity were determined with a
S a i c o r - H o n e y w e l l 42 A photon correlator
with p h o t o n counting option. The correlation functions were numerically analyzed.
For measurement of the specific refractive index increments a B r i c e - P h o e n i x
differential r e f r a c t o m e t e r was used. Transmission electron microscopy measurements
were performed with a Philips EM 301 apparatus. Carrier grids c o v e r e d with carboncoated Parlodion films were dipped in a
dilute dispersion and micrographs were
taken of the particles retained on the film.
Viscosity measurements were carried out
with Ubbelohde internal dilution viscosimeters at t = 25.0 _ 0.01°C. Sedimentation
velocity experiments were carried out in a
Beckman Spinco (Model E) analytical ultra-
FIG. 1. Electron micrograph of crosslinked PMMA
latex particles (sample P12) as taken from the aqueous
suspension after the preparation.
centrifuge using a 3-cm centerpiece and an
AN-J rotor at 25°C. Rotor speeds were
used between 2000 and 3000 rpm.
4. PHYSICAL CHARACTERIZATION OF LATEX
a. Electron Microscopy
Figure 1 shows latex particles as taken
from the aqueous phase. The particles are
almost perfect spheres with a standard
deviation in the radius, aEM, of about 10%.
The found absolute values of the average
radii, as given in Table II, should be considered with caution, however, because it
was observed that shrinkage and fusion of
the PMMA particles occurred at higher intensities of the electron beam. Similar
p h e n o m e n a were reported by Fitch (9) for
TABLE II
Physical Characterization of Latex Particles
Latex
aEM
(H20)
(nm)
OD
(H20)
(nm)
aG
(H20)
(nm)
Do
(benzene)
(10-Scm2sec i)
aB
(benzene)
(nm)
S0
(benzene)
(10 ~sec)
k~
(cmZg-I)
M
(10~g mole -1)
[7]
(g l c m z)
a~
(nm)
P10
PI2
103
115
130
161
129
158
1.88
1.38
177
240
1.03
1.18
54
33
4.6
7.1
8.8
9.4
210
265
Journal ofColloM and ~terface Science, V01.72, No. 2, November1979
324
PMMA, and by McDonald (!0) for PVC,
particles.
b. Laser Light-Scattering Spectroscopy
The correlation function of the fluctuations of the scattered light was measured
from dilute latex dipersions in water as
well as in benzene. In both solvents the
experimental data showed a single exponential decay. According to the theory, as
described in detail by Berne and Pecora
(11), the autocorrelation function, F(K,r),
for the fluctuations in the scattered intensity
of noninteracting particles is given by
F(K,.r) ~ exp(-2DoK2z).
[1]
Here Do is the diffusion coefficient, K
= (47rn/ho) sin (0/2) is the wave vector of
the fluctuation, with n the refractive index
of the medium and ~o the wavelength
of the light in vacuo, and z is the correlation time.
By plotting log F(K,.r) against z for a certain K a straight line was obtained. The
slopes were plotted against K 2, giving a
straight line the slope of which gives Do according to [1]. For dilute dispersions the
hydrodynamic radius of the particles, aD, is
related to Do by the Stokes-Einstein law,
Do = kT/(67r*/oaD),
[2]
is observed during the experiment. The
sedimentation coefficient s was plotted according to the relationship s -a = s~-l(1 + k~c).
Here So is the sedimentation coefficient at
infinite dilution and k~ is a proportionality
constant (12). From So, Do, and 1 - v~0,
the molar mass M of the particles was
calculated (see Table II). Here ~ is the partial specific volume of PMMA (13): ~ = 0.807
cm ~ g-l, and ~0= 0.874g cm -z is the
density of the solvent benzene at 25°C.
d. Viscosity
The relative viscosity */r was measured
and the specific viscosity ~/sp = (*/r - 1)/c
was then calculated for P10 and P12 in
benzene at 25,0°C. Most of the measurements were performed on samples that still
contained some free polymer. The measured
data were corrected afterward, simply by
considering the supernatant after centrifugation as the continuous phase instead of
pure benzene. Some checks on pure samples gave identical results.
Intrinsic viscosities [*/] = lim (c = 0)*/sp
were obtained by extrapolation of */sp to
zero concentration according to the equation (3),
*/sp = (5/2)q + Eq2c.
[3]
where k is Boltzmann's constant, T is the
absolute temperature, and */0 is the viscosity
of the medium. The so obtained diffusion
coefficient Do appears to be only slightly
dependent on the concentration of the latex
in dilute systems. Final results are sum.
rnarized in Table II.
Here q is a scaling coefficient to fit the term
linear in c to the Einstein equation: q
= (2/5)[,/]. It can be expressed as q
= o~b, where c~ is a volume expansion factor. A hydrodynamic radius an can be calculated from a n = a~aD(H20). Both [*/]
and an are given in Table II. The results
are comparable to those obtained earlier (3).
c. Sedimentation
e. Light Scattering
Because of the small difference in refractive index between particle and solvent
Schlieren optical measurements in benzene
could be done only above concentrations of
! g dm-L The Schlieren picture was a single
sharp peak; only a small peak broadening
We measured the angular dependence
of the light scattered from very dilute
aqueous dispersions. The results were
analysed in a so.called Guinier plot, in
which the logarithm of the scattered intensity (I) is plotted against the wave vector
Journal of Colloid and Interface Science, Vol, 72, No. 2~ November t979
325
S T R U C T U R E IN L A T E X D I S P E R S I O N S
(K) squared. According to Guinier (14) the
scattered intensity at low angles (in the
R a y l e i g h - G a n s - D e b y e approximation) falls
off with the wave vector as I ( K ) ~ exp
x (-K2a~/5) for small K, where aG is the
radius of the latex sphere (in water). The
radius of the latex particles was obtained
from the slope of the straight part of the
Guinier plot at small K and is given in Table
II. The agreement with radii obtained from
the diffusion experiments is v e r y good.
The p r o c e d u r e is based on the R a y l e i g h G a n s - D e b y e approximation which is valid
when m, the ratio of the refractive indices
of particles and solvent, is near unity. For
other values o f m the theory of Mie must be
used. H o w e v e r , according to K e r k e r (15)
the difference between both theories should
be less than 10% for our case where m = 1.12
and the sizes are still relatively small. So
we have used the much simpler RGD
approximation.
[ R~(K)/c]/cm 2 g-1
io I t-
I0 o
10
10
10-Jl
0
,
,
4
- - m ~ - - - -
~
8
K
12
cm
FIG. 2. Variation of light scattering intensity with
scattering angle (0) at different temperatures for
latex sample PI2 in benzene. K = (47m/h0)sin (0/2).
~0=546 nm; c ~ lg dm-3. Data are given in a
Guinier plot for 21°C (O), 17.5°C ([]), I0.5°C (©),
and 6.5°C (11).R*(K) = R(K)/(1 + cos2 0).
5. LIGHT S C A T T E R I N G OF D I L U T E
D I S P E R S I O N S IN B E N Z E N E ;
EFFECTS OF PARTICLE STRUCTURE
a. Light-Scattering Equations
Light scattering is one of the available
techniques to obtain structural and thermodynamic information on dispersions. F o r a
monodisperse system of spherically symmetric particles one may write for unpolarized light in the R a y l e i g h - G a n s D e b y e approximation (16),
R ( K ) = (1 + cos 20)Y{cMP(K),
[4]
with K = (47m/X0) sin (0/2).
Here R ( K ) is the reduced scattering intensity (Rayleigh ratio) of the dispersion
o v e r that of the solvent, 0 is the scattering
angle, M the particle molar mass, P ( K )
= Po is the particle scattering factor, and
c = o M / N A is the weight concentration
with p the particle n u m b e r density.
Further,
if{ = 2 7 r 2 n 2 ( d n / d c ) 2 ( X ~ N h )
-1.
[5]
Here n is the refractive index of the dispersion, X0 the wavelength of the light in
vacuo, and NA Avogadro's number.
b. Experiments
Because the particle mass is v e r y large,
a solvent (benzene) was chosen in which
the refractive index increment dn/dc of the
particles is very small. At 20.0°C we found
for P10: d n / d c - ~ 0.0075 (X0 = 546 rim);
0.0009 (X0 -- 436 nm) and for P 1 2 : 0 . 0 1 1 0
(X0 = 546 nm); 0.0048 (X0 = 436 rim), all
positive, and in units cm~/g. The accuracy
is low because of the limited sensitivity of
the refractometer. This implies that it is not
feasible to obtain a good absolute value
of M from light scattering intensity. Note
that dn/dc of P12, containing DVB as a
326
NIEUWENHUIS
AND VRIJ
tRIKI/cl/J g-1
I01F
100
I(]I
16 2
1(]~IO
i
~
t
j
8
i
i
12
I
I
I
16
2
r
K
cm,
FIG. 3. As Fig. 2 but for k0 = 436 nm.
cross-linker, is larger than that of P10,
which contains E G D M .
It turned out that the light scattering was
rather sensitive to variations in temperature. Results for X0 = 546 and 436 nm for
P12 are shown in Figs. 2 and 3. One observes that R ( K ) at small angles varies with
a factor of about 3 for h0 = 546 nm and with
a factor of about 50 for h0 = 436 nm. Also
the slopes are temperature dependent, in
particular for )~0 = 436 nm. It will be clear
that the main cause of this abnormal behavior must be searched in (small) changes
of dn/dc with T. In most cases these small
changes are not observable. In our systems,
however, the absolute values of dn/dc are
so small that these changes may b e c o m e
of comparable magnitude. This would still
have no consequence on the shape of the
light scattering curves if the particle compositions were completely homogeneous.
It will be apparent, h o w e v e r , that any deviation of homogeneity, e.g., because of the
presence of a small n u m b e r of crosslinks
with a rather different dn/dc, might also
change the shape of the curves. In turn,
these changes in shape will then contain
information about the inhomogeneity. This
will now be investigated in more detail.
c. Scattering o f Particles with Several
Scattering C o m p o n e n t s
We assume that the swollen P M M A latex
particles contain two scattering components,
i.e., P M M A chains with a refractive index
increment dn/dc = vB and a relatively small
n u m b e r of crosslinks, e.g., DVB, with a
dn/dc = va. It will be apparent that such a
particle is in fact a degenerate case of a
c o p o l y m e r molecule with two types of
chains, the scattering behavior of which is
known (17). The swollen latex particles,
however, have some features often not encountered in the more familiar copolymers.
The particles are practically spherically
STRUCTURE
-Y2
s y m m e t r i c and h o m o g e n e o u s in size ( s o m e
d i s p e r s i o n s o f these particles f o r m supram o l e c u l a r crystals; see S e c t i o n 7). T h e centers o f m a s s o f b o t h units A and B will
n e a r l y coincide.
Light scattering near 0 = 0. F o r 0 --~ 0,
P ( K ) in [4] b e c o m e s u n i t y and for particles
p o l y d i s p e r s e in size and c o m p o s i t i o n , vZM
m u s t be r e p l a c e d b y (17)
v2M ~
~ TiMiu ~ .
327
IN LATEX DISPERSIONS
3
[6]
i
H e r e v = the m e a s u r e d dn/dc o f the particles; y~ = q~ ~ c , M~ is the m o l a r m a s s o f
species i and
5
10
15
20
t:/%
l) i -~- W i l ) A "~- ( 1
-- Wi)IIB,
[7]
where w~ is the weight fraction of the crosslinker (A) in species i. F u r t h e r [6] m a y be
written as
F r o m Eq. [7] one has with wi = w
v2M = v~Mw + 2PV(VA + VB)
+ Q(vA
FIG. 4. The square root of the scattering at zero
angle divided by the concentration as a function of
the temperature for sample PI2 in benzene. (3,
X0 = 546 nm; ~, X0 = 436 nm. R*(0) = R(0)/2.
-
v~) ~, [8]
d v M T = wdvAMT + (1 - w)dvBMT.
[10]
F o r the s e p a r a t e c o m p o n e n t s we take (17)
where
P = ~ yiMi(w~ - (v),
re= be(he-no),
i
Q = ~ yiMi(w~ - vb)z,
[9]
( ( = A , B),
[11]
w h e r e be is the partial specific v o l u m e and
i
and M is the a p p a r e n t and Mw the weight
a v e r a g e m o l a r m a s s o f the particles.
B e c a u s e it c a n be e x p e c t e d that v is a
linear f u n c t i o n o f t e m p e r a t u r e w e h a v e
p l o t t e d [VzR(O)/c] 1/z as a f u n c t i o n o f T as
s h o w n in Fig. 4. T h e plots are linear, so
a p p a r e n t l y the terms c o n t a i n i n g P and Q do
n o t c o n t r i b u t e p e r c e p t i b l y . F r o m Fig. 4, w e
calculated v as a f u n c t i o n o f T as s h o w n in
Fig. 5. We u s e d M = 6.0 x 10~g m o l e -1
to obtain a g o o d fit with the t w o values o f
v m e a s u r e d with the r e f r a c t o m e t e r . T h e M
obtained
from
sedimentation-diffusion
m e a s u r e m e n t s is 7.1 x 109g m o l e -~ (see
Table II) thus in r e a s o n a b l e a g r e e m e n t .
L e t us n o w investigate h o w far the t r e n d
with t e m p e r a t u r e c a n be explained on the
basis o f separate variations in vA and vB.
9/10 -3
c m 3 g-I
12
/
J
J
y
J
J
0
5
1~0
1~5
210
~/°c
Fio. 5. Specific refractive index increment, v, as a
function of the temperature, t, as calculated from the
light scattering results assuming M = 6.0 × 109g
mole -~ for sample P12 in benzene. (3, h0 = 546 rim;
A, h0 = 436 rim. Also indicated are two measured
values of v (11).
328
[ rg 2//3] /I011 cm 2
6
2
0
t
0
5
I
10
15
,
20
~
25
-1
( t - t ' ] -1/10 -2 K
FIG. 6. Dependence of the apparent radius of gyration, rg, on temperature for sample P12 in
benezene at X0 = 546 nm (O) and )to = 436 nm ([~). Note that T* (T* = T at which dv/dt = 0) is
different for both wavelengths (see text).
ne is the refractive index of c o m p o n e n t
and no the refractive index of the solvent.
This equation reflects the influence of the
solvent on re. Using this equation for a
single p o l y m e r in different solvents it was
found (18) that the variation of v~ with no
is indeed linear and that ne is v e r y near to
the refractive index of the bulk p o l y m e r .
Then one m a y write
dvjdT
P(K) = exp[-K%~/3].
[12]
F o r ve and d ( n , - no)/dT we will take
values for bulk p o l y m e r and solvent (17, 19).
T h e n one finds for A (crosslinker): 104 dvA/
d T = +4.8 c m 3 g-1 K-1 and for B (PMMA):
104 dvBMT = +2.8 c m 3 g-1 K -1, independent of wavelength. Thus 104 dv/dT = 4.8w
+ 2.8(1 - w) = 2.8 + 2.0w. This is rather
independent of the (small) value of w. The
values found f r o m the light scattering exp e r i m e n t s in Fig. 5 are 104 d v M T = 3.3 cm 3
g-1 K -~ for h, = 546 nm and 3.1 cm 3 g-~
K -~ for X0 = 436 nm. The wavelength
i n d e p e n d e n c y is in a c c o r d a n c e with theory.
The calculated values for dv/dT h o w e v e r
are s o m e w h a t small.
Journal of Colloid and Interface Science, Vol. 72, N o . 2, N o v e m b e r 1979
[13]
F o r a c o p o l y m e r the radius of gyration, rg,
takes the f o r m (17)
r2g = r2gB + y(r~A-- r2gB) + y ( I -- y ) F ,
= f~d(n e - no)/dT
+ v~ldb~/dT.
Radius o f gyration. L e t us now investigate the light scattering b e h a v i o r at finite
scattering angles as o b s e r v e d in Figs. 2 and
3. F o r the linear parts of the plots one has
the Guinier a p p r o x i m a t i o n (14) for P ( K ) :
[14]
with y = WVA/V.
H e r e rg2A and r~B are (average) square
radii of gyration of the c o m p o n e n t s A and B
and F is the (average) square distance bet w e e n the centers of A and B. N o t e that
according to Eq. [14] rg depends on v and is
often called a p p a r e n t radius of gyration.
In our case y takes the form,
y(T) = WVA[(dv/dT)(T - T*)] -1,
[15]
where T* is the (extrapolated) t e m p e r a t u r e
where v ~ 0.
F r o m Fig. 5 one obtains T'a6 = 259.3°K
and T*36 = 275.3°K. Equation [15] suggests to plot the slope of the straight part
of the natural logarithm of P ( K ) versus K 2
as a function of ( T - T*) -1. The result for
329
R~K) ~10-2 cm -1
6.4
4.8
3.2
1.6
I
~
2
i~
4
6
8
,,,
,±
10
K2 /
I
12
1010 cm-2
FIG. 7. R e d u c e d light scattering intensity, R * ( K ) , v e r s u s K z for latex sample P12 in b e n z e n e at
h0 = 546 n m , as m e a s u r e d at a t e m p e r a t u r e of 6.5°C. T h e concentration (c/10 -2 g cm 3) data are
as follows: (11), 1.36; (D), 2.26; (A), 3.77; (O), 5.39. R * ( K ) = R ( K ) / ( 1 + cos 2 0).
P12 is shown in Fig. 6. The values for X0
= 436 and 546 nm both fall on the same
straight line. Apparently the term in Eq.
[14] causing nonlinearity is negligible in our
case. The extrapolated value at (T - T*) -~ -->
0 gives rgB = 145 nm.
Because vA and d v / d T are positive the
negative slope implies that rg2A< r~B. Indeed, one would e x p e c t the spatial distribution of the crosslinker (DVB) to be less extended than that of the PMMA. Unfortunately the values of w and VA are not well
known. Because the initially used w in the
reaction was - 5 % , its actual value in the
particle cannot be larger. The estimated
value of uA was ~0.11 cm3/g. The product
w vA can be estimated from the difference
v(P12) - v(P10) ~ 0.0035 for X0 = 546 nm
and 0.0039 for X0 = 436 nm (see [7]),
Taking w u A = 0.0037 as a c o m p r o m i s e one
finds from the intercept on the abscissa of
Fig. 6: rgA/rg B --~ 0.8 or rgA ~ 115 nm.
Further one finds w -~ 0.034 which seems
reasonable,
6. L I G H T S C A T T E R I N G A T H I G H E R
CONCENTRATION; 1NTERPARTICLE
STRUCTURAL EFFECTS
The relatively weak scattering power of
the particles makes it possible to investigate
these systems also at higher particle concentrations, Experiments were performed
both with P10 and P12 at several temperatures. As an example, scattering curves o f
the sample P12 at k0 = 546 nm are shown
in Fig. 7 for T = 6.5°C where the overall
scattering power is lowest (see Fig. 2). One
observes that with increasing concentration
well-defined peaks develop superimposed
on the scattering o f the individual particles
indicating the emergence of some order in
the system. Similar curves were found for
the sample P10. This behavior which is well
known from X-ray scattering of simple
liquids was observed and discussed earlier
for small-angle X-ray scattering o f biological
macromolecules by Riley and Oster (16, 20).
It was also found recently in light scattering
o f very dilute dispersions of latex particles
Journal of ColloM and lnterfoee Science, Vol. 72, No. 2, November 1979
330
in deionized water (21, 22) where it is caused smaller magnitude and a smaller angular deby extraordinary long-range interactions of pendence in the lower K-range. We surmise
ionic double layers.
• that with increasing concentration of the
To treat the ordering effects quantita- latex particles their peripheric segment
tively, Eq. [4] must be supplemented clouds are compressed and/or deformed
with a structure factor S(K). For spherically when they interpenetrate each other. So the
symmetric particles one finds (16):
growing number of particles results in two
R(K) = (1 + cos ~ 0)Y{cMP(K)S(K) [16] effects. First, the angular dependence will
be less pronounced since the particles are
with
optically smaller. Second, the average segment density distribution between the
S(K) = 1 + 47rp
particles is more uniform which results in a
higher average refractive index of the back× I~ r2h(r)[sin (Kr)](Kr)-ldr [17] ground and thus in a lower scattering
power of the particles. These two effects
and h(r) = g(r) - 1. Here g(r) is the radial complicate the extraction of S(K) from the
distribution function which measures the experiments. To overcome this difficulty we
probability to find a particle center in a will introduce an apparent scattering factor
volume element separated at a distance r for particles in the concentrated solution
from the center of a given particle. The denoted by Pc(K). This Pc(K) was obtained
function h(r) is often called total correlation by drawing a smooth curve between the
function. Further p = cNAM -1 is the oscillations of the measured scattering
particle number density.
curve, under the following restrictions: (a)
Because the integral in Eq. [17] is a the Pc(K) coincides with the experimental
Fourier transform, it is, in principle, possible curve for large K, because S ( K ) ~ 1 for
to obtain h(r) from the reverse transform,
large K; (b) Pc(K) is a straight line for
small K in a semilogarithmical plot against
h(r) = (2~'2p)-1
K 2, according to Guinier's approximation;
× I~ Kz[S(K) - 1](Kr)-i[sin Kr]dK. [18] and (c) the resulting structure function
S(K) has to satisfy the condition (23, 24)
The function h(r) is determined by the interaction forces between the particles and depends further on p and T. In the next section we will consider the evaluation of S(K)
and h(r) in more detail and also discuss how
these functions may be related to an interaction potential with the help of some
models taken from liquid state theory.
a. Evaluation of S(K) and h(r)
In principle, S(K) can be obtained by
simply dividing the scattering results by
P(K) according to Eq. [16]. Doing so,
however, completely unrealistic results are
obtained. It appeared that a somewhat
different P(K) had to be chosen with a
I f K2[S(K) - 1]dK = -2zr2p,
[19]
which follows from Eq. [18] by taking the
limit for small r, where h(r = 0) = 1. By
estimation of Pc(K) from the first two
requirements (a and b), S(K) is calculated
and checked by performing the integration.
A trial and error procedure is used until
Eq. [19] is satisfactory fulfilled. In Fig. 8
some examples are given together with the
calculated values of the integral. These can
be directly compared to the value of the
right-hand side of Eq. [19] obtained from
the concentration and the particle mass
from sedimentation measurements. Clearly,
331
.¢
Iog(R(K) /g-1 cm2)
C
/
3 2
-
1.(
~*'"-..
i
0
"
J
2
FIG. 8. Guinier plot of the scattering
sample P12 in benzene at 6.5°C (h0 =
cm -z and the drawn lines are different
Pc(K). Values of the integration in Eq.
all in 1014 c m - L F r o m the fight-hand
= R(K)/(1 + cos 2 0).
-
4
,
K2 //1010cm-2
6
intensity divided by concentration, R*(K)/c, versus K 2 for
546 n m ) . •
are experimental data for c = 5.39 × 10-2g
estimated curves for the single particle scattering function
[19] are: curve 1, +0.97; curve 2, -0.92; curve 3, -1.55,
side of Eq. [19] one obtains - 0 . 8 0 × 10TM cm -3. R*(K)
the integral condition is a rigorous one,
and some selection of the S(K) can be made.
In Fig. 9 we show for P12 one calculated
S(K). One of the characteristic features of
this curve is the sharp peak, implying that
considerable spatial ordering of the particles
has occurred. Further the position of the
peak, denoted by Km shifts to higher values
14
S(KJ
1.2
1.0
015
i
i
K/10 5 crn
i
2.0 . /
08
0
.
4
~
FIG. 9. Best fit for the structure function S(K) versus K for latex sample P12 in benzene at
c = 5.39 × 10-2 g cm -3, as measured at 6.5°C and at h0 = 546 nm.
332
when the particle concentration increases,
as is summarized for P10 and P12 in Table
III. It appeared that the values of cKm 3 are
nearly constant. This will be discussed further in the next section.
In a further analysis we have determined
the total correlation function, h(r), and also
the so-called direct correlation function,
c(r), which is defined by Fourier transformation of the structure function S(K) (23),
h(r) = (27r2p)-a I f K2[S(K) - 1][sin Kr](Kr)-~dK
[201
c(r) = (27r2p)-1 Io K 2 [ S ( K ) - 1]S(K)-l[sinKr](Kr-a)dK"
[21]
Equations [20] and [21] require knowledge
of S ( K ) from K = 0
to K = ~ .
Below
K = 0 . 5 × 105cm - ~ ( f o r 0 = 15° a t h 0 = 546
nm) S ( K ) has to be extrapolated to K = 0.
Because in all cases we found that S ( K )
equals unity for K I> 3.0 × l0 s cm -a integration is performed only up to this point.
h(r) and c(r), as obtained in this way, for
example P12 is shown in Fig. 10. The
spurious peak in h(r) at small r is often found
in these transforms [see discussion by
Karnicky et al. (23)] and probably must be
attributed to truncation errors or inaccuracies in the S(K). It will further be discarded.
TABLE III
Position of the MainDiffractionPeak in Concentrated
Latex Dispersions
c
Km
cK~ a
d~
(10-2 g cm-3)
(105 cm-')
(10-'7 g)
(nm)
L a t e x P10 at 23°C (h0 = 546 nm)
7.7
1.60
5.8
1.45
4.0
1.28
2.3
1.07
1.88
1.90
1.89
1.86
480
530
600
720
L a t e x P10 at 23°C (h0 = 436 nm)
7.7
1.58
5.8
1.47
1.95
1.84
490
520
L a t e x P12 at 6.5°C (h0 = 546 nm)
5.4
1.22
3.8
1.08
2.3
0.92
3.0
3.0
2.9
630
710
840
Journal o f Colloid and Interface Science,
Vol. 72, No. 2, November 1979
b. Interaction Potential
The spatial ordering of the particles in
the dispersion, as described in the radial
distribution function g(r), is due to the
interacting forces between the individual
particles. The problem of relating the pair
potential for interacting particles with the
pair distribution function g(r), is known
from the theory of simple liquids and dense
gases. Starting from a given effective pair
potential ~(r), Monte Carlo and Molecular
Dynamical methods can be used to calculate
numerically the distribution functions. Besides a number of analytical equations are
derived by several authors using different
types of approximations. For recent reviews
see, e.g., Refs. (25) and (26). All these
equations are not exact over the whole range
of the particle number density as can be
tested by comparison with the computer
experiments. The Percus-Yevick (PY) and
Hypernetted Chain (HNC) theories have
been proved to work reasonably well in
the investigated cases. So for the inverse
route, to determine the effective pair
potential ~(r) from the distribution functions we used the PY and HNC equations
with/3 = (kT)-l:
/3cbpy(r) = In [1 - c(r)(1 + h(r)) -a]
[22]
/3qbHNc(r) = h(r) - c(r) - In [1 + h(r)].[23]
Equations [22] and [23] are exact in the lowdensity limit but are only approximate to
STRUCTURE
333
IN LATEX DISPERSIONS
0./~
h(r)
0
,
,
r AO 2
nm
I-0.4
-0.8
FIG. 10. T o t a l c o r r e l a t i o n f u n c t i o n h(r) for l a t e x p a r t i c l e s P12 in b e n z e n e at c = 5.39 x 10 -2 g c m -3.
the behavior of denser systems. In Fig. 11
The small minimum of the order of a few
we show one of the calculated pair po- tenths of k T is somewhat puzzling. In our
tentials. The £b(r) found from HNC and PY systems we expect a purely repulsive pair
are nearly the same. The exact shape of potential because benzene is a good solvent
the resulting qb(r) must be considered with for PMMA. It seems that the application
some caution, however, because of the of the PY and HNC approximations is not
uncertainties along the step by step proce- reliable in this range of r. This is cordure starting from the experimental scatter- roborated by the fact that the potential of
ing data. Nevertheless dp(r) shows a steep mean force, V ( r ) - - k T In g(r), which is
increase below a certain interparticle center- the potential (free) energy of a pair in the
to-center distance pointing to a strong presence of the other particles, is nearly
repulsion between the hard particle cores at the same as the above calculated qb(r) (see
r = 450 rim.
Fig. 11). Because the above analysis could
We note here that also some uncertainty only be accomplished with the highest
can be produced by an unknown amount
2
of multiple scattering.
In the Appendix a rough estimation is
'\
made of the double scattering contribu- ~ ( r ) / k T I
tion to the measured intensity at 90° scattering angle. When applied to our geometry
the double scattering is less than 10% of the
single scattering. Calculation of this effect
at other angles will be much more difficult
due to the asymmetric geometry of the ex\\
perimental arrangement. Moreover the
influence of multiple scattering on the
i
I
I
I
I
I
structure function will be reduced in divid4
6
8
ing the measured intensity data by the semir / 1 0 2 nm
empirical curve for the apparent particle
FIG. 11. E f f e c t i v e p a i r p o t e n t i a l th(r ) (
) a n d poscattering function. So we expect only small
t e n t i a l o f m e a n f o r c e V(r) = - k T In g(r) in u n i t s k T
deviations in the foregoing calculations due ( - - - ) , for l a t e x p a r t i c l e s , P12 in b e n z e n e at c = 5.39
X 10 -2 g c m -3.
to more than once scattered light.
334
0.2
log
( K m a x / 1 0 5 cm-1}
0.1
- 0:1
I
i
'
0.2
~_
016
'08
I
log (c/lO-2gcm 3 )
FIG. 12. Double logarithmic plots of apparent Bragg spacing against weight concentration for
the latex dispersions in benzene. Q, sample P10 at 23°C; O, sample P12 at 6.5°C.
concentrations, it is also of interest and useful to discus.s the position, Km, of the
maxima in S(K) with a "quasi-lattice"
approach.
c. Quasi-lattice Approach
On the basis of a consideration of
scattering data in liquid mercury, Oster and
Riley (20, 27) came to the practical conclusion that for solid spherical particles the
center-to-center distance of separation between neighboring particles in "expanded"
liquid-type configuration is given approximately by the apparent Bragg spacing
Am of the maximum in S(K). It is therefore
of interest to use their analysis for our
systems. The apparent spacing, Am, is derived from the angular position of the maximum by the straight forward application of
Bragg's law, Am = k[2 sin (0/2)] -1 = 27rKm1.
The values of Am at different stages of
dilution are plotted versus the concentration (w/v) on a double logarithmic scale
in Fig. 12.
The linearity of the plot is a measure of
Journal of Colloid and Interface Science,
Vol. 72, No. 2, November 1979
the"degree of crystallinity" in the solutions
due to quasi "long-range" order, while the
slope = +0.33, indicating the lattice expansion on dilution is in three dimensions. The
relation between the Bragg spacing, Am, of
the principal peak and the mean interparticle distance, do, is given for a closest
cubic packing (fcc) by (20):
de = (3/2)1/2Am.
[24]
Numbers for dc are shown in Table III.
From the number of lattice points per
unit cell (=4), M follows from M = (3/4)
X 31/2NACA3m,and was found to be 3.7 x l09 g
mole -1 for sample Pl0 and 5.8 × 109g
mole -1 for sample P12. Both values are in
reasonable accordance with those obtained
from sedimentation plus diffusion.
d. Effective Diffusion Coefficient
Structure formation not only shows up in
the light scattering intensity as a function of
0 but in the quasi-elastic light scattering as
well. In Fig. 13, it is shown how Deff as a
function of K is also structured at higher
3.0
1.5
D o / D e f f (K)
S(K)
1.0
335
2.0
1
----,-.--
0.5
K/lO cr(
1.0
FIG. 13. Structure function S(K) and relative reciprocal effective diffusion coefficient Do/Dell(K)
(Q) v e r s u s K , for P10 latex particles in b e n z e n e at c = 7.7 × 10-2 g c m 3. Do is the " i n d e p e n d e n t
particle" diffusion coefficient for c --~ 0.
concentrations. Recent theoretical (11)
and experimental (21, 22) work has shown
that S ( K ) can be inferred indirectly through
the dynamics of concentration fluctuations.
In Fig. 13 one observes that the extrema
in D~-f} coincide with those of S(K). This
result is in good agreement with those of
others (21, 22) for latex dispersions in
extremely dilute electrolyte solutions. We
think that in both cases the interaction must
be characterized as soft, long-range repulsions. The discrepancy between the static
structure factor and D~-f}in our systems has
to be attributed to an additional term to the
instantaneous velocity of a particle in an
interacting dispersion: the hydrodynamic
interaction. The theories on this point
are not yet worked out in detail.
7. C R Y S T A L S
In this section we will describe the
crystal-like structure which we observed in
a test tube with a dispersion of sample P12
after standing for several months. When a
moderately concentrated latex dispersion
was set apart, the particles settled to the
bottom of the container by gravity, leaving
an almost clear solution in the upper part
of the tube. The sediment consisted of
ordered arrays of particles. This crystal-
like structure occupied a relative large
volume and no more changes take place
after a few months. Under more detailed
investigation the ordered sediment appeared to be very transparent and, when
placed in a beam of white light, very
strong and sharp diffraction colors could
be observed at different angles (see Fig.
14). Between the "particle-free" supernatant and the concentrated crystal-like sediment we observed a small region that
showed the behavior of a highly concentrated (liquid-like) dispersion. To determine the distances between the net planes
(d), we placed the sample in a beam of
monochromatic light and measured the
angle (0) of the first diffraction peak. According to the Bragg relation for the first
order diffraction, 2d sin (0/2) = X, the distance d can be calculated from 0, and the
wavelength of the light in the medium
X (=X0/n). In Table IV the results are given
for three different wavelengths. Assuming
an fcc lattice the observed diffraction
corresponds to the reflection from the (111)
plane; reflections from (100) and (110) are
destroyed by geometric structure effects.
In this case the center to center distance,
dcr, of the lattice points can be calculated
from the distance between the net planes:
dcr = (3/2)~/2d, and becomes d e r = 418 nm.
Journal of ColloM and Interface Science, Vol. 72, No. 2, November 1979
336
FIG. 14. Diffraction of white light by a crystal-like latex sediment of sample Pl2. The test tube is
placed in flask with toluene to avoid reflection of light by the glass wall. The incident light beam
passes through the test tube from right to left. The transmitted beam gives an oval white spot on the
glass wall of the flask. A: blue; B: green; C: yellow; D: red.
TABLE IV
Crystal Diffraction Data for Latex
Sediment of Sample P12a
8. FINAL DISCUSSION
h0
(rim)
n
sin 0
d
(nm)
436
546
633
1.521
1.503
1.495
0.418
0.535
0.621
343
339
341
h0, Wavelength of incident light in vacuo; n, refractive index of benzene; 0, angle of diffraction; and d
distance of net planes.
A n o v e r a l l v i e w o f the r e s u l t s o b t a i n e d
in the p r e v i o u s s e c t i o n s will b e g i v e n in
t e r m s o f the d i f f e r e n t sizes a n d d i s t a n c e s o f
i n t e r a c t i o n o f the p a r t i c l e s as s u m m a r i z e d
in Fig. 15.
W h e r e a s a d e t a i l e d p i c t u r e o f the i n t e r n a l
p a r t i c l e s t r u c t u r e is n o t k n o w n , all the
m e a s u r e d p a r t i c l e sizes are e x p r e s s e d as
diameters of apparent, homogeneous spheres
(i.e., G u i n i e r light s c a t t e r i n g , q u a s i - e l a s t i c
P12
rim
9oo -
800
P10
nm
c/gcm-3
%1
0.023 ~
0.09
Ool/dc
700
.
337
0.20
c/gcm-3
~ ~I
0.023~
008
!!!!yd c o,lz.
5oo -
0,20
VISCOSITY
500
DIFFUSION [C6H5}
~/kT: 1
-
400 -
-
CRYSTAL
GUiNIER
/
3oc
0.27
- -
VISCOSITY
- - -
DIFFUSION (C6H6}
OUINIER
PMMA
DIFFUSION (H201
OUINIER OVB
- -
DIFFUSION (H20}
EM
2oc
-
-
EM
lOC
Fro. 15. Diameters and distances in PMMA dispersions. EM: 2aEM(H20); Guinier DVB:
2(5/3)l/2rgA; diffusion (H20): 2aD(H20); Guinier PMMA: 2(5/3)lmrgB; crystal: dcr; q~ = lkT from
Fig. 11; diffusion (C6H6):2aD = kT(37r'oD)-~; viscosity: 2an; d c = 27r(3/2)11ZKml, all in nm.
light scattering, intrinsic viscosity). From
the numbers found in Fig. 15 it seems
reasonable to conclude that the particles
have a rather open crosslinked kernel with a
much less dense periphery. The periphery
does not contribute much to the light
scattering but still strongly obstructs the
motion of solvent. At higher particle conc e n t r a t i o n s - e v e n near the closest packing
volume fraction: cbn = (4/3)~-a~--the dispersions still showed a relatively low viscosity as compared with high-molecular-weight
polymer solutions of the same weight concentration from which we conclude that the
particles still behave as separate entities
and that interchain entanglements between
overlapping peripheries are absent.
From Figs. 10 and 11 we conclude that
interparticle repulsions of - lkT arise when
the interparticle distance reaches 450 nm
(for P12), i.e., somewhat below the hydrodynamic diameter (2an and 2a9), which
seems acceptable. The crystal diffraction
data of P12 show an even smaller inter°
particle distance of 420 nm (with ~b -1.5kT
from Fig. 11). This implies a marked overlap
of the peripheries after settling of the dispersion in the terrestrial gravitational field.
Let us turn the attention now to the
spatial ordering of the concentrated (liquid)
dispersions. The relevant distances de as
obtained from the quasi-lattice approach
(see Section 6) are given in Table III and also
in Fig. 15. Here (see [24]) dc = (3/2)1/2Am
= 7.695 Km1, and Kin is the position of
the maximum in S(K). One may doubt
whether dc has any significance beyond the
concept of a "lattice." It can be argued,
however, that d~ must be very close to
the position, rm, of the principal maximum
in the radial distribution function g(r).
For many liquids--rather independent of
density--it is found (empirically) that Kmr m
= 7 to 8, e.g., liquid argon (28): 7.1; dilute
338
N I E U W E N H U I S A N D V R IJ
aqueous latex (29): 7.4; our system (see
Figs. 9 and 10) = 7.6; computer simulation
with a purely repulsive pair potential (30):
7.40; the so-called Ehrenfest rule (the position of the minimum in (Kr) -1 sin (Kr)): 7.73.
The large variation of dc is remarkable,
for both P10 and P12, with concentration
(see Figs. 12 and 15), cd~ being approximately constant and with very large values
of de, particularly at the smaller concentrations. This implies that ordering takes place
at distances much larger than the hydrodynamic diameters. Both the proportionality
of d~ 3 with c and the very large values of
de were also reported for charge-stabilized
latex in water (29, 30). This was explained
by the very extended electrical double
layers in these systems, which contained
a very small amount of electrolyte. This
suggests that ordering by electrical charge
effects also occurs in our systems. Others
(31, 32) have described the occurrence of
charge effects in nonpolar media especially
in relation to inorganic colloids. It is unquestionable that these effects, when
present, give rise to forces of a very long
range (say - 1 / ~ m ) , because the number of
charges can only be very small in solvents
with a low dielectric constant.
We investigated the presence of free
charges with some simple electrical conductivity measurements. Conductivities
down to about 10-14 12-1 cm -1 were measured with an accuracy of about 5% by a
simple dc method using a Jones-type cell
with a cell constant of 16 cm (33, 34), a
Fluke 412 B power supply, and a Keithley
602 electrometer. For benzene (Baker
Analyzed) without further purification we
found a specific conductivity ~Cs~c~-8
× 10-14 12-1 cm -1 and for P12 latex dispersions Kso~c= 16 × 10-14 and 30 x 10-14
12-1 cm -1 for c = 0.006 and 0.010 g cm -3,
respectively. Whereas nothing is known
about the distribution of charges over
particles and counterions let us assume that
all the charges are present on the latex
particles alone, which will give an overestimation of the number of charges per
particle. Using the equation Ks~c = F2z2D
× ( R T ) - l c M -a (see e.g., (35)), where z is the
average number of elementary charges per
particle and F is Faraday constant one finds
from the values of D and M determined
above: z2 1011Kspeec-1.
This gives a z value of the order of unity.
The presence of (much smaller) counterions
would even give a much smaller number.
We assess that, e.g., the well in V(r) given
in Fig. 11 would require a z value of 5 or
10. As a consequence we conclude that free
electrical charges cannot be important in our
systems and that electrostatic repulsion
seems no plausible explanation for the longrange ordering in the dispersions. Further
research is required to elucidate the nature
of the interactions at intermediate concentrations. In the meantime we have started
static and dynamic light scattering experiments with somewhat smaller microgel
particles (36).
=
APPENDIX: E V A L U A T I O N OF
DOUBLE SCATTERING
Because no appropriate treatment was
found in the literature for the description
of multiple scattering in our experimental
geometry, we will make a rough estimate of
the contribution of the double scattered
intensity. The experimental set-up is schematically represented in Fig. 16. Calculations are made only for the scattering
angle 90° . In this arrangement, single scattering originates only from the scattering
volume in the center of the cell. Double
scattered radiation can reach the photomultiplier from a second scatterer situated
in the detection path of the photomultiplier.
The incident light on this scatterer comes
only from a first scatterer somewhere in
the incident beam. The dimensions of the
incident beam and the detected beam are
taken equal. In the figure, dl represents the
width of the beams in the plane of the
339
?
?2
/
\
I
dl
t
FIG. 16. Experimental setup in the light scattering apparatus. The picture is given for an observation angle of 90°. The Roman numerals refer to the four quadrants. Also given is a possible path for
double scattered light. For explanation of symbols see text.
figure and d2 the height in the perpendicular
direction. The first scattered intensity at an
angle 02, with the direction of the incident
beam by a volume element d~dsdx in the
incident beam at a distance r from the
volume element, can be expressed as:
dI1 = Io•r-Sp
X e x p [ - h s sins (Oi/2)]dxdsdx.
[A1]
Here Io is the intensity of the incident
beam, Y[ is an optical constant (see Eq.
[5]), p is the particle number density, and
the exponential term is the Gulnier approximation for the particle scattering function,
with h 2 = (4zrna~t-1)s/5 (see Eq. [13]).
If this light is scattered again by particles
in the detection path the intensity is given as
d[2 =
dI~ Y[R-Sp
× e x p [ - h s sins (02/2)]d~d2dy.
the first and second scattered radiation.
Note that we have neglected the pathlength
of the light in the cell with respect to the
distance R.
Let us first determine the contribution of
the scattering center to the double scattering. With the help of calculations performed by Blech (37) for cylindrical samples
it is easily shown that the secondary scattering in the scattering volume is less than 2%
of the single scattering when applied to our
data. So we will calculate double scattering from the parts of the beams outside the
center. By substituting Eq. [A1] in Eq. [A2],
and some algebra, the following relation is
obtained
dis = Io( ~{d~d2p)2R-2r -s exp(-h s)
× exp[hS(cos 01 + cos 02)/2]dxdy
[A2]
Here R is the distance of the scattering cell
to the detector, and 02 is the angle between
[A3]
which must be integrated over both axes.
Figure 16 is divided into its four quadrants.
Because of the angular dependency of the
340
scattering of the particles the contribution
of the double scattering across each
quadrant is different. By using simple
geometrical properties and changing from
rectangular to polar coordinates one obtains the following equation, where d12 2/2 is
the smallest distance between incident and
detection beam and 122/2 is the largest distance in the cell.
12 --= Io( ff~dld2p)2R -2
× e x p ( - h 2) In (21d~2)Q,
[A4]
where Q = Q I "-[- QII + QIII + Qlv, as contributions from each of the quadrants. In
our case only Qin is of importance,
ACKNOWLEDGMENTS
The authors wish to acknowledge Dr. W. van der
Drift for his advice and assistance in preparing the
latex particles. We also thank Mr. J. Pieters of the
Center of Electronmicroscopy of the Department of
Biology for making the electron micrographs. We
are indebted to Dr. C. Pathmarnanoharan for helpful
discussions on multiple scattering. Grateful thanks
are extended to Professor J. Th. G. Overbeek for
helpful suggestions on the electric conductivity measurements. Dr. H. Fijnaut and Mr. H. Mos are gratefully thanked for making the diffusion experiments
and Mr. J. Suurmond for making the ultracentrifuge measurements. Finally we thank Miss H. Miltenburg and Mrs. M. uit de Bulten for typing the
manuscript and Mr. W. den Hartog for drawing the
illustrations.
REFERENCES
Qm =
If
exp[h2(cos 01 + sin 00/2]d01. [A51
Because no analytical solution is known for
this type of integrals, and the sum cos 01
+ sin 02 changes only slightly in the interval
of 0, we took an average value for this
sum of 1.2. Then one obtains
QIII -~ (~r/2) exp(3h2/5).
[A6]
For our samples where h 2 ~ 12 the sum of
the other Q's is less than 10%. The total
single scattering by the center of the cell is
given by
11 =
IoY{d~d2R-2p exp[-h2/2]
[A7]
and the ratio of double to single scattering
becomes
12/12 = (~r/2)Rod2 In (21/dl)
x exp(h2/10)
[A8]
where Ro = 3fp is the Rayleigh ratio at zero
scattering angle. For our geometries l = 1
era; d2 = 0.16 cm one finds 12/12 = 2.1
× (R0/cm-2). In our experiments R0 = 0.03
cm -1 for the highest concentrations (see
Fig. 7). So we conclude that the multiple
scattering is, say, less than 10% of the total
scattering intensity (at 90 ° scattering angle).
1. Shashoua, V., and Beaman, R., J. Polymer Sci.
33, 101 (1958).
2. Hiltner, P. A., and Krieger, I. M., d. Phys. Chem.
73, 2386 (1969).
3. Kose, A., and Hachisu, S., J. Colloid Interface
Sei. 46, 460 (1974).
4. Hansen, J. P., and McDonald, I. R., "Theory of
Simple Liquids." Academic Press, London,
1976.
5. Vrij, A., Pure Appl. Chem. 48, 471 (1976).
6. Vrij, A., Nieuwenhuis, E. A., Fijnaut, H. M.,
and Agterof, W. G. M., Disc. Faraday Soc.
65, 101 (1978).
7. Ono, H., and Saeki, H., Colloid Polym. Sci.
253, 744 (1975).
8. Hul, H. Van Den, and Vanderhoff, J., J. Colloid
Interface Sci. 28, 336 (1968).
9. Fitch, R., and Tsai, C., in "Polymer Colloids"
(R. Fitch, Ed.), Vol. III. Plenum Press, New
York, 1971.
10. McDonald, S., Daniels, C., and Davidson, J.,
J. Colloid Interface Sci. 59, 342 (1977).
11. Berne, B. J., and Pecora, R., "Dynamic Light
Scattering." Wiley, New York, 1976.
12. Tanford, C., "Physical Chemistry of Macromolecules," Chap. 22. Wiley, New York, 1963.
13. Schulz, G. V., and Hoffmann, M., Makromol.
Chem. 23, 220 (1957).
14. Guinier, A., and Fournet, G., "Small Angle
Scattering of X-rays," p. 126. Wiley, New
York, 1955.
15. Kerker, M., "The Scattering of Light and Other
Electromagnetic Radiation," p. 427. Academic
Press, New York, 1969.
16. Riley, D. P., and Oster, G., Disc. Faraday Soc.
11, 107 (1951).
17. Benoit, H., and Froelich, D., in "Light Scattering
from Polymer Solutions" (M. B. Huglin, Ed.),
Chap. 11. Academic Press, London, 1972.
18. Cordier, P., J. Chim. Phys. 64, 133 (1967).
19. Brandrup, J., and Immergut, E. H., "Polymer
Handbook," 2nd ed. Wiley, New York, 1975.
20. Oster, G., Receuil trav chim. 68, 1123 (1949).
21. Brown, J., Pusey, P., Goodwin, J., and Ottewill,
R., J. Phys. A: Math. Gen. 8, 664 (1975).
22. Schaefer, D., J. Chem. Phys. 66, 3980 (1977).
23. Karnicky, J., Hollis Reamer, H., and Pings, C.,
J. Chem. Phys. 64, 4592 (1976).
24. Krogh-Moe, J., Acta Crystallogr. 9, 951 (1956).
25. Barker, J. A., and Henderson, D., Rev. Mod.
Phys. 48, 587 (1976).
26. Andersen, H. C., Chandler, D., and Weeks, J. D.,
Adv. Chem. Phys. 34, 105 (1976).
27. Oster, G., and Riley, D., Acta Crystallogr. 5,
1 (1952).
28. Mikolaj, P., and Pings, C., J. Chem. Phys. 46,
1412 (1967).
29. Brown, J., Goodwin, J., Ottewill, R., and Pusey,
30.
31.
32.
33.
34.
35.
36.
37.
341
P., "Colloid Interface Sci.," Vol. IV (Hydrosols and Rheology). Academic Press, New
York, 1976.
Megen, W. Van, and Snook, J., J. Chem. Phys.
66, 813 (1977).
Osmond, D. W. J., and Waite, F. A., "Dispersion Polymerization in Organic Media" (K. E. J.
Barrett, Ed.), Chap. 2. Wiley, London, 1975.
Lyklema, J., Advan. Colloid Interface Sci.
2, 65 (1968).
Koelmans, H., and Overbeek, J. Th. G., Disc.
Faraday Soc. 18, 52 (1954).
Albers, W., and Overbeek, J. Th. G., J. Colloid
Sci. 14, 501 (1959).
Macinnes, D. A., "The Principles of Electrochemistry," Chap. 3. Dover Publications, New
York, 1961.
Fijnaut, H. M., Pathmamanoharan, C., Nieuwenhuis, E. A., and Vrij, A., Chem. Phys. Lett.
59, 351 (1978).
Blech, I. A., and Averbach, B. L., Phys. Rev.
137, A 1113 (1965).

Light Scattering of PMMA Latex Particles in Benzene: Structural Effects

Transcription

Similar documents

15-322 Retail Light-Fit - UIS

GEL-TEK

Achievements and Future Carear Plans:

Coulomb Scattering

neopreno TRAD AN

SAXS/SANS data processing and overall

How to Tie a Woggle

lnstallation - ConservationMart.com