The structure of sheared ordered latices

The Structure of Sheared Ordered Latices M. TOMITA AND T. G. M. VAN DE VEN Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Quebec H3A 2A 7, Canada Received July 29, 1983; accepted November 3, 1983 Shear-induced changes in the structure of ordered latices of moderate volume fractions have been observed by grating diffraction and right angle Bragg diffraction. The former method provides information about the two-dimensional lattice structure of the particle layers (net planes) parallel to the wall, while the latter gives information about the spacing between the layers. Both methods showed that the order was maintained when ordered latices were subjected to shear. Grating diffraction revealed that at moderate shear rates the two-dimensional lattice was compressed in the flow direction and expanded in the direction normal to the flow, and that at high shear rates the lattice was compressed in all directions. Bragg diffraction revealed that at high shear rates the spacing between the layers increased by 1,5 to 10%. These observations are qualitatively explained by a sliding layer model.

I. INTRODUCTION

In a previous paper (1) we investigated the optical properties of ordered latices under static conditions. The present paper deals with shear-induced changes in the structure of ordered latices studied by Bragg diffraction and the grating diffraction technique reported in the previous paper. Hoffman (2, 3) reported that concentrated ordered latices exhibit a discontinuous change in viscosity accompanied by a disappearance of a star-like diffraction pattern. He ascribed the viscosity change to a change in structure of the ordered latex. Clark et al. (4, 5) reported shear-induced melting of dilute ordered latices: Bragg diffraction spots diffused as the shear rate increased, and at high shear rates turned into diffraction rings characteristic of liquid structure. Our latices were at concentrations 0.05 to 0.45 in volume fraction intermediate between those of Clark et al. (4, 5) and Hoffman (2, 3), and did not exhibit the disappearance of order at any shear rate, but showed changes in the structure. In our optical studies distinct hexagonal patterns of diffraction spots and well-defined peaks in Bragg diffraction

curves were always recognized and displayed changes with increase of the rate of shear. It is well known that under suitable conditions latex particles assume a close-packed structure with the most close-packed planes with hexagonal symmetry oriented parallel to the wall of the container (1, 6-9). When ordered latex is placed between two parallel glass plates and subjected to shear, these parallel layers will slide over each other. It is known in crystallography that in atomic crystals sliding takes place along the most close-packed layer (10). Although it is reasonable to assume that atoms in a crystal pass over other atoms during the sliding process and that the distance between sliding layers increases, there has been no direct evidence to confirm this. It has been reported (11-13) that the macroscopic density of cold-worked metal is slightly smaller than that of annealed metal, an effect partly attributed to lattice expansion; but other factors, for example, introduction of more dislocations, could be responsible as well. If sliding takes place in ordered latices, then one can also expect an increase in spacing between sliding layers. However, latex cannot expand its volume, unlike molecular crystals, because

374 0021-9797/84 $3.00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science. Vol, 99, No. 2, June 1984

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SHEARED ORDERED LATICES

its total volume remains constant during flow. Therefore, one should expect a decrease of lattice constants in other directions to offset the increase in the spacing between sliding layers. As was mentioned in the previous paper (l), grating diffraction provides information about the structure of the two-dimensional layer parallel to the wall and a dark ring, present within a bright halo surrounding the point of illumination, provides information about the distance between parallel layers. However, shear-induced changes in the dark ring radius were too small to measure and the diffuseness of the ring prevented precise measurement. To measure the spacing between the parallel layers we employed the right angle Bragg diffraction method (14). It was found impossible to observe the hexagonal grating diffraction patterns and the right angle Bragg diffraction curves for the same latex sample because when the hexagonal pattern appears, the Bragg diffraction wavelength is outside the range of visible light. In spite of this disadvantage, one can obtain useful information by applying one of the two methods on various samples. We observed a small but significant expansion of the spacing between parallel layers by Bragg diffraction and compression of the hexagonal packing in the two-dimensional layers (at high shear rates) by grating diffraction. Before describing the experiments we will first propose a two-dimensional sliding layer model that can explain qualitatively the increase in spacing between parallel layers. II. TWO-DIMENSIONAL SLIDING LAYER MODEL

A particle in a sheared ordered latex undergoes a complex three-dimensional motion. To simplify the problem we propose a twodimensional analog that will provide some insight into the underlying physics. Figure 1 shows schematically the essential features of the two-dimensional sliding layer model, taking into account the condition of constant volume fraction. In Fig. 1, do, d~, and d2 are spacings between horizontal rows

117

ole o o

oo a0 Do (a)

oooo < >

dl ~I (b)

00%0 < >

d2 02 (C)

FIG. 1. Schematic representation showing the essential features of the two-dimensional sliding layer model. Do, D,, and/)2 are horizontal distances between particles, do, dl, and d2 are spacings between horizontal rows of particles. (a) Perfect close-packed configuration. As particles slide horizontally, the particles assume configurations (b) and (c) with do < dl < d2. From the condition of constant volume it follows that Do > D1 > /)2. The coordinate system 0(i, X2, )(3) is shown in each figure. The X, axis (not shown) is directed toward the viewer. Black circles denote the reference particle. The direction of sliding is shown on the left in the figure. The area of the dotted triangle is kept constant (condition of constant volume fraction) during the sliding process.

of particles, and Do, D1, and/)2 are horizontal distances between particles. The direction of sliding is shown on the left of Fig. 1 and the Cartesian coordinate system is shown in each Figs. 1a-c. The black circles in Figs. 1a-c denote the reference particle. Its coordinates are expressed relative to another reference particle placed at the origin of the coordinate system. Figure I a represents the configuration of close packing. If we assume that the relative motion of each particle is identical, the particles will take consecutively the configurations shown in Fig. 1. In going from configuration (a) to (c) in Fig. 1, the distance d increases because of the increased repulsion between horizontal rows of particles, while D decreases because of the requirement that the volume fraction must remain constant. Decreasing D results in increased repulsion between vertical rows of particles opposing the decrease in D and thus also the increase in d. It follows that the trajectory of a particle will be wavy and that both d and D will be oscillating between their maximum and minimum values. In calculating the motion of a reference particle one has to consider the hydrodynamic Journal of Colloid and Interface Science, Vol. 99, No, 2, June 1984

376

TOMITA

A N D V A N DE V E N

and colloidal forces acting on it. In order to do so we will make the following assumptions and approximations: (i) The total volume remains constant. (ii) The relative motion of each particle is identical, i.e., we can concentrate on the motion of a reference particle. Assuming an initially symmetrical configuration of particles, the configuration always remains symmetrical. This assumption also implies that the flow field is linear. (iii) The forces acting on each particle consist of colloidal and hydrodynamic forces. Under our experimental conditions discussed below, van der Waals interactions can be neglected and the pair potential for colloidal interaction is assumed to be given by 4~rEa~2 V(~) = - exp(-Ka~)

~+2

[ 1l

When the position (x2, x3) of the reference particle and the distance D (see Fig. 1) is specified, the position of all particles surrounding the reference particle is known since the particle configuration is assumed to be symmetrical (assumption (ii)). Hence the distance li between the reference particle and a surrounding particle i depends on x2, x3, and D, i.e. li = li(x2, x3, D). From assumptions (iii) and (iv), the potential energy Vt of the reference particle can be calculated from Vt(x2, x3, D) = ~ Vi[li(x2, x3, D)].

[3]

i

In a typical latex K-~ may be several times as large as a, in which case several shells of particles around the reference sphere must be included in the calculation of Vt. The potential energy given by Eq. [3] is a function of x2, x3, and D. However, assumption (i), i.e., the assumption of constant volume requires the following relation:

where ~ = ho/a; ho is the gap width between the two spheres, and a is the particle radius, is the permittivity of the suspending medium, Dx2 = S, [41 the surface potential of the particles, and K the Debye-Hfickel parameter. The hydrodynamic resistance forcej~ is as- where S is a constant which equals the area of the triangle formed by the centers of three sumed to be equal to particles (dotted triangles in Fig. 1). Therefore, fh = -6zc nav, [2] S is a function of volume fraction and can be determined from the configuration of closewhere ~ is the viscosity of the suspending mepacking, in which case the triangle is equidium and v the velocity of the particle. In lateral (Fig. la). From Eqs. [3] and [4] it folprinciple, because of hydrodynamic particle lows that Vt can be expressed as a function of interactions, Eq. [2] must be corrected for by x2 and x3. a factor f ( x z , x3) which depends on the poAt a given x3, the vertical components (in sition of the reference particle relative to its the x2 direction) of colloidal and hydrodyneighbors. As yet no theory exists to calculate namic forces, f~2, fh2, should balance: f(x2, x3) and, for the sake of simplicity and because of various other simplifying assumpfh2 q - f e 2 ~---O. [51 tions a more rigorous analysis is not warranted, According to assumption (iii), Yh2 is given by we will assume that f(x2, X3) = 1. (iv) The potential energy of particle interfh2 = --6~r~aV2 [61 actions is additive, i.e., the potential a reference particle feels can be calculated by summing where v2 is the relative vertical velocity of parthe pair potentials between the reference par- tides, while fc2 is given by ticle and the surrounding particles. a V~(x2,x3) (v) In the calculation of motion of a reffc2 -[71 erence particle, inertia is neglected. Ox2 Journal of Colloid and Interface Science, Vol. 99, No. 2, June 1984

SHEARED ORDERED LATICES Substituting Eqs. [6] and [7] into Eq. [5], the velocity o f a reference particle in the vertical direction can be written as

a

377

a= 249nm ~b: 50mY

b

a= 249nm ~: 100 mV

I

1.005

0=~25 / ]

dx2 _ v 2 - dt

1 0 V t ( x z , x3) 6zc~a Ox2

[8]

F r o m assumption (ii) the relative horizontal velocity v3 is given by

ax3

v3 -

dt

1

-

OVt(x2, X3)

67r~lGado

@

Ox2

[to]

"..~G , ; 5 x104s_1" . G=lxl04S 1 ;G=5xl03S-1 2

1.005

}: 50 mV O:cxu

r

d

a= 64nm q~=lOOmV

[9]

The trajectory o f a reference particle is obtained by integrating Eq. [10] numerically. Examples are shown in Fig. 2. The initial position o f the reference particle was taken as x2 = do, x3 = 1)o/2 (close-packed configuration). As can be seen f r o m Fig. 2, the trajectories b e c o m e fairly s m o o t h as the shear rate exceeds G -~ 10 4 s e c -~. In right angle Bragg diffraction the time average, d, o f the oscillating x2 value is measured.

I

o!c o:o4om

~:0,05~0

- Gx2 ~- Gdo

where G is the shear rate a n d do the x2-value at close packing. T h e a p p r o x i m a t i o n introduced in Eq. [9] is justified because o f the fact that the horizontal velocity does not change significantly at a given shear rate G since changes in x2 are small. Dividing Eq. [8] by Eq. [9] results in

dx2 dx3

o, : J o . J

1.000

x3/%

5

FIG. 2. Calculated trajectories of a reference particle at various shear rates. The curves are calculated for the conditions a = 249 nm, q~= 0.35, ~ = 50 mV, and K-1 = 0.33 um. The x2 and x3 values are nondimensionalized by do, the spacing between horizontal rows of particles and Do, the horizontal distance between particles at close packing, respectively, xz/do = 1 corresponds to the close-packed configuration.'When the shear rate exceeds G --- 104 secthe curves become fairly smooth.

1.00 2

103

10a

lO2

lO3

lO"

G(~-I) FIG. 3. Calculated time averaged spacing, a~,for various volume fractions as a function of shear rate. d is nondimensionalized by dL, the value of d as G --~ 0. The conditions are a = 249 nm, q~ = 0.25-0.35, ~b = 50 mV in (a) and ff = 100 mV in (b), and a = 64 nm, q~ = 0.050.15, ff = 50 mV in (c) and ~b = 100 mV in (d). Curves are fitted through calculated points. The trajectories in Fig. 2 show that dincreases as the shear rate increases up to a certain value. Figures 3 a - d show plots o f d vs the shear rate G, calculated for various conditions. In Figs. 3a-d, d i s nondimensionalized by dL, the value o f d as G ~ 0. F o r ordered latex, assuming a perfect close-packed configuration in the absence o f shear, the distance between horizontal layers is do. For the conditions o f Figs. 3a-d, do is 4.6% smaller than dL. The model predicts, therefore, that when ordered latex is sheared, a j u m p will occur in the average spacing o f magnitude dL -- do. It can be seen f r o m Figs. 3 a - d that d/dL starts to increase at G - 103l 0 4 s e c -1 depending on the surface potential o f the particles. In all cases the higher the surface potential the higher the shear rate at which d starts to increase. The reason for this is that the higher the surface potential, the stronger the colloidal force, hence the higher the shear rate required to affect the trajectory o f a reference particle. It can also be seen f r o m Fig. 3 that the h i g h e r the v o l u m e fraction, the higher the shear rate at which d/dL starts to increase, thus the smaller the increase o f d/dL o f a given shear rate. Again this is because the higher the volume fraction the stronger Journal of Colloid and Interface Science, Vol. 99, No. 2, June 1984

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TOMITA AND VAN DE VEN

the colloidal force, hence the higher the shear rate required to affect the trajectory of a reference particle. c The above calculations are based on a two,/ dimensional sliding layer model. Although the I................... I actual system is three-dimensional, the twodimensional model explains qualitatively some of the experimental observations. B The main shortcoming of a two-dimensional model is probably that distortion in the two-dimensional lattice of the layers cannot be taken into account. When hexagonally FIG. 4. Experimental system for right angle Bragg difpacked layers are sliding over each other, col- fraction experiments. LX, latex sample; MC, Microlisions occur between particles in neighboring Couette; B, nonslip belt; P, right angle prism; BS, beam splitter cube; PH, pin hole; PMP, photomultiplier; C, collayers. Their impact will be diminished when limator; MO, monochromator; SM, scanning motor; PM, the spacing between layers is expanded, but potentiometer; LS, light source; LM, light meter; SA, signal also by a suitable distortion of the two-di- averager; PG, pulse generator; WVT, wavelength-voltage mensional lattice of the layers. Which of these transformer; XY, X-Y recorder. two phenomena occurs is a matter of competition between the forces between particles mator, MO. The wavelength was scanned at inside a layer and in different layers. Our ex- constant speed (25 nm/min) by a scanning periments show that two-dimensional lattice motor, SM. The axis of the scanning motor distortion occurs at low shear rates and an was connected to the axis of a potentiometer, increase in spacing between layers at high shear PM, to transform the wavelength to a voltage rates. As a result the j u m p from do to dE at using a wavelength-voltage transformer, WVT. low shear rates does not occur. The predicted The light leaving the monochromator was increase in spacing is found to occur when collimated by a collimator, C, then directed the model predicts a rather pronounced in- to the latex surface through a pin hole, PH (1 crease (see Fig. 3). m m dia.), a right angle prism, P, a beam splitter cube, BS, and the upper glass plate of the IIl. EXPERIMENTAL TECHNIQUES micro-Couette. The prism, beam splitter cube, AND PROCEDURES and the glass plate were stuck together with immersion oil to avoid unwanted reflections a. Right Angle Bragg Diffraction from their interfaces. The diffracted light from Figure 4 shows the experimental set-up for the latex sample was reflected in the beam right angle Bragg diffraction. A latex was splitter cube toward a photomultiplier, PMP. placed between the two parallel glass plates of The output signal from light meter LM was the micro-Couette, MC. The diameters of the connected to a signal averager, SA. The waveplates were 200 ram, the thickness of each length-voltage transformer was connected to plate was 6.00 mm, and the gap between the a pulse generator, PG, which generated a single two plates was variable between 0.10 and 0.30 pulse when receiving a certain voltage to trigmm. The lower plate was driven by a nonslip ger sweeping of the signal averager. The signal belt, B, while the upper plate was held fixed. accumulated in the signal averager was after The shear rate was calculated from the rota- a sufficient number of experimental runs, read tional speed of the lower plate and the gap out onto an X-Y recorder, XY. The wavewidth. length of the recorded signal was calibrated A light beam from the light source, LS (100 by comparison with the reading of the waveW tungsten lamp), entered the monochro- length dial of the monochromator. Journal of Colloid and Interface Science, Vol.99, No. 2, June 1984

SHEARED ORDERED LATICES

379

The peak in the reflectivity vs k curve occurs when the condition

m k = 2hid

[ 11 l

is satisfied. Here X is the wavelength of the light in the air, nl the mean refractive index of the latex, d the spacing between layers, and m is an integer. Equation [ 11 ] is valid for fight angle Bragg diffraction at low volume fractions (15). If the change in d is small, ni can be considered constant for the following reason. The relation between the spacing d and the volume fraction ~bat close-packing is given by

(1) (16~r~1/3 a d = 1,91/~]

~bl/3.

I. . . .

:':> . . . .

////

/#// ///(/

'

<1

FiG. 5. Experimental set-up for the grating diffraction experiment. LX, latex sample; MC, micro-Couette; S, cone-shaped paper screen; CL, cone-shapedplastic lens; M1, mirror; M2, mirror with hole at center; L, lens; LA, He-Ne laser; CA, camera.

[12]

If it is assumed that expansion occurs in all directions, a 5% increase in d corresponds to a decrease in the volume fraction q$ of about 16% which, because of the relation (1, 8) nl = 1.32(1 - q$) + 1.58~b

cA

[13]

corresponds for ~b = 0.1 and 0.3, to changes in nl of about 0.3 and 0.9%, respectively. In reality, expansion occurs only in one direction and is accompanied by compression in other directions. Hence, changes in nl are negligible and changes in d can be estimated from the shifts in peak wavelengths by means of Eq. [111.

b. Grating Diffraction Figure 5 shows the experimental set-up for two-dimensional grating diffraction experiments. A latex sample, LX, was placed between the two glass plates of the microCouette, MC, similar to the case of the right angle Bragg diffraction experiments. A laser beam from a H e - N e laser wavelength (633 nm), LA, was directed perpendicular to the latex surface by a mirror, M1, then it was collimated by a lens, L. The collimated beam illuminated the latex through the top area of a truncated cone-shaped plastic lens, CL, in contact with the glass plate with immersion oil. A paper screen, S, is glued on its cone surface with immersion oil. The lens avoids

total reflection of the diffracted light at the glass-air interface allowing the diffraction spots to appear on the screen. The diameter of the plastic lens was 40 m m and the cone angle was 142 ° . Both its top and bottom surfaces were polished. The diffraction spots were photographed by a 35 m m camera, CA, through a mirror M2, which was fixed at 45 ° relative to the incident beam direction. All photographs thus taken in each series of experiments were printed at the same calibrated magnification. The distances between diffraction spots were measured on the prints. Figure 6a shows a top view of a hexagonal lattice while the diffraction spots produced by the lattice are shown in Fig. 6b. The three sets of parallel lines in Fig. 6a with spacings DA, DB, and Dc are responsible for the diffraction spots, A, B, and C in Fig. 6b, respectively. Therefore, the distance between spots O and A is related to DA and, similarly, distances O-B and O - C are related to DB and Dc, respectively (1). Figure 7 shows the geometry of the system. 0=l is the diffraction angle in the latex, and 0sg, the corresponding angle in the glass plate and lens (the refractive indices of glass and plastic were taken equal), and the radial distance r~ of a diffraction spot from the position of the incident beam. The two-dimensional grating equation is given by (1) Journal o f Colloid and Interface Science, VoL 99, No. 2, June 1984

380

TOMITA AND VAN DE VEN / //~,'/ / /

A

©

//

0c

BI

oe C°

©

\

where Sh0 and do are the undistorted area and spacing, respectively. F r o m Eq. [17], and readily derived relations between Sh, DA, and DB, it can be shown that

d = [4(DA/Do) 2 - (DB/Do) 2]1/2

e~

do

[ 18]

f3(DA/Do)2(DB/Do)

where Do is the undistorted distance between rows of particles in a two-dimensional layer. It should be noted that right angle Bragg diffraction does not detect the distortion of the two-dimensional lattice. FIG. 6. Schematicrepresentationof (a) two-dimensional hexagonallypacked latex particles, and (b) corresponding hexagonal pattern consisting of grating diffraction spots. c. Latex Samples and Note that the orientation of the hexagon in (a) is different Experimental Conditions from the one in (b). The distances between diffraction spots O-A, O-B, and O-C are related to the spacingsDA, The latex suspensions used were monodisDB, and Dc, respectively. perse polystyrene latices consisting of particles of diameter 497 and 128 nm, determined by X =nlD sin 0,1 [ 14] electromicroscopy. The standard deviations were 11 and 2.5 nm, respectively. These samwhere D is the spacing between rows of latex ples were kindly supplied by Professor S. particles. 0sl is related to 0sg by Snell's law: Hachisu and were prepared by the method nl sin 0sl = ng sin 0sg. [15] described in ref. (6). T h e range of volume fraction studied was 0.05-0.45. The volume Therefore, Eq. [ 14] can be rewritten as fraction was determined from the dried weight = ngD sin 0~g. [ 16] of the latex. In preparing samples of higher Since 0~g can be calculated from the value of volume fractions, the suspensions were conr, and the geometry of the cone-shaped lens, centrated by ultrafiltration in a pressurized the spacing D can be calculated from the ex- dialysis tube. Depending on the final volume perimental value of r~. F r o m DA, DB, and Dc, obtained from the r~ values in all three direcincident beam --diffraction spots tions, we can calculate the distortion of the two-dimensional lattice. As mentioned earlier, grating diffraction and right angle Bragg diffraction cannot be applied on the same sample. However, we can estimate the change in spacing d between sliding layers by assuming that the distortion (and expansion) of particle packing is the same in FIG. 7. Schematic representation of the diffracted light all layers. When DA =/=DB = Dc, the hexagon and diffractionspots in gratingdiffraction experiment. 0~, made of latex particles (see Fig. 6a) is distorted. diffraction angle in the latex; 0~, diffraction angle in the Let the area of the distorted hexagon be Sh glassplate and cone-shapedplastic lens; r,, radial distance and the spacing between layers d. Then, be- of the diffraction spots appearing on the cone-shaped screen, Sincethe refractiveindices of glass and plastic are cause of the condition of constant volume we very similar, no refraction is assumed at the glass-plastic have interface. Figure is not to scale; the spacing between the first latex layer and the glass is very much smaller than Shd = Shod0 [ 17] the thickness of the plate.

(a)

A

(b)

Z,c ,ens

Osl

Journal of Colloid and Interface Science, Vol. 99, No. 2, June 1984

Osg ~latei

381

SHEARED O R D E R E D LATICES

agonally close-packed (atomic) crystals (10). Therefore this observation strongly suggests that hexagonally close-packed layers in ordered latex slide over each other during shear, maintaining the two-dimensional order.

fraction of the latex suspensions, it took 3 to 7 days to concentrate the suspensions. Before use, the latices were subjected to deionization with a mono-bed ion-exchange resin which was subsequently removed by filtration. Small aggregates in the latex could not be removed in this way. These aggregates, as well as the occasional presence of air bubbles in the sample, seemed to affect the diffraction intensity, which was found to be rather irreproducible. However the reproducibility of the shift in peak wavelength and of the distortion of the hexagonal pattern was found to be satisfactory.

b. Shear-Induced Distortion of Hexagonal Patterns For the latex with a = 249 nm, 4) = 0.2610.355, hexagonal patterns were observed at all shear rates. Their intensity, size, and distortion depended on the rate of shear. As the shear rate increased from zero to intermediate values, the hexagonal patterns became less clear and subsequently became clear again as the shear rate increased further. When the hexagonal pattern was not clear, the halo (1) was seen to be bright. It was also found that even at zero shear rate the hexagonal pattern was usually distorted, but the distortion was rem o v e d by moving the lower glass plate of the micro-Couette apparatus. Figure 9 shows changes in DA, DR (Dc = DB within experimental error), and d (calculated from DA and Ds) as a function of shear rate. It can be seen from Fig. 9 that DA increases by about 1.2% at intermediate shear rates, then decreases at high shear rates, while DB decreases monotonically as the shear rate in-

IV. RESULTS AND DISCUSSION

a. Orientation of Latex Structure Figure 8a is a photograph of a hexagonal pattern taken for the conditions a = 249 nm, ~b = 0.289, G = 5840 sec -1. The flow direction is indicated by the arrow. Figure 8b shows schematically the orientation of the crystal structure deduced from the diffraction pattern. Sets of white circles and black circles represent two neighboring layers. As can be seen, the flow direction is one of the three directions (dotted lines) along which latex particles line up most densely. This is in accordance with the most c o m m o n sliding direction in hex-

---0-

.....

.......

•

.....

O- .....

.......

0

....

0-

-0- ..... ....

(3- ........

--0 .....

0

......

0

....

- - - 0 . . . . -O . . . . . • . . . . . • . . . . . .

iI D A v~ "

---0

'-

.....

-0

-Q .....

.....

O-

.....

<3 .....

-0 .......

...... • ..... O---O ..... • .....

A

X3

1

y r

a

b

FIG. 8. (a) Photograph of observed hexagonal pattern under shear. The flow direction is indicated by the arrow. Experimental conditions: a = 249 rim, ~b = 0.261, G = 5840 sec -~, shutter speed 0.5 see. (b) Schematic representation of deduced orientation of crystal structure. Two hexagonally packed layers of particles are shown by open and closed circles, respectively, to indicate the arrangement of particles in neighboring layers. Dotted lines represent the most densely packed lines, which are along the flow direction indicated by the arrow. Journal of Colloid and Interface Science, Vol. 99, No. 2, June 1984

TOMITA AND VAN DE VEN

382

-/J ,

curves obtained for a latex with a = 64 nm, q~ = 0.064 at shear rates ranging from 35 to 2250 sec -1. Each curve shown was obtained in one single experiment, i.e., no averaging procedures were followed. In general the noise level, as well as the height of the curve, as v ~ 1.10 mentioned earlier, were rather irreproducible, o a~'60C i o i ~ - especially at intermediate shear rates. When the noise was high, 2 to 15 experiments were 580~ -r~ G 1.00 required to obtain a smooth curve. Figure 10 shows that clear Bragg diffraction signals can be observed even at high shear rates. Although 2 x 10| 102 103 104 hexagonal patterns cannot be observed under o(~-1 ) these conditions, the existence of hexagonally FIG. 9. DA and DB values calculated from the size of packed two-dimensional layers (distorted or the hexagonal pattern observedin grating diffraction experiments, and changes in d calculated from DA and DB undistorted) is likely. In Fig. 11 the peak using Eq. [18] as a function of shear rate. The latex used wavelength and peak height measured for the was a = 249 nm, 4~ = 0.261. Curves are fitted through same latex sample (cf. Fig. 10) are shown as experimental points. Typical error bars are indicated in a function of shear rate. As can be seen from the upper right corner. this figure the peak wavelength starts to increase around G - 500 sec -1, while the height creases. Thus the hexagon made up of latex of the diffraction curves appears to be miniparticles is almost equilateral at low shear m u m at the same shear rate. Although, owing rates, it is compressed in the flow direction to the irreproducibility in the height of the and expanded in the direction normal to the diffraction curves, quantitative comparisons flow at intermediate shear rates, and at high are doubtful, the m i n i m u m in Fig. 11 seems shear rates it is again almost equilateral but to correspond to the least clear hexagonal patsmaller, i.e., the two-dimensional structure is terns observed at intermediate shear rates. more densely packed. As remarked upon earlier, the sliding layer model is unable to explain d. Spacing between Sliding Layers from the distortion of the two-dimensional lattice. Grating Diffraction and Right Angle F r o m our experiments it follows that at low Bragg Diffraction and intermediate shear rates two-dimensional As was seen above (Figs. 9-11), the spacing lattice distortion is preferred above an increase in spacing between layers, while the opposite between sliding layers can be estimated either from grating diffraction or from right angle occurs at high shear rates. Bragg diffraction. In Fig. 12 spacings measured by right angle Bragg diffraction (open squares) c. Peak Shift in Right Angle Bragg and obtained from grating diffraction (other Diffraction Experiments symbols) are plotted as a function of shear For latex samples with a = 249 nm, the rate for latices with a = 249 nm, ~b -- 0.261grating diffraction method can be applied for 0.445. Since experimental points for ~b = 0.320 ~b = 0.261-0.355 and right angle Bragg dif- and 0.355 are almost indistinguishable, one fraction for ~b = 0.449. For latex samples with curve was drawn through the experimental a = 64 nm, right angle Bragg diffraction ex- points for both volume fractions. In the right periments can be performed for ~b = 0.055- angle Bragg diffraction experiments the spac0.154, but no grating diffraction was possible. ing at zero shear rate, do, could not be deterFigure 10 shows examples of Bragg diffraction mined uniquely (see Section IV, f). However, Journal of Colloid and Interface Science,

Vol.99,No.2,June1984

SHEARED ORDERED LATICES

383

10

1.1o ~,~a

G=35 s-1

~

"

~ G = 4 4 9 0

~" 5

f

.~.1.05 "O

s-1 G=2250 s~1

,7' A oo

1.oo ~

-

o

i

102

600

620 640 ~, (nm)

FIG. 10. Bragg diffraction curves at various shear rates observed in right angle Bragg diffraction experiments. I is the intensity of the diffracted light and Xthe wavelength. Latex used: a = 64 nm, ~b = 0.064. A pronounced shift in peak wavelength can be observed. s i n c e for t h e t h r e e c u r v e s i n F i g . 12 t h e s p a c i n g is c o n s t a n t a t l o w s h e a r r a t e s , d w a s n o n d i m e n s i o n a l i z e d b y t h i s v a l u e (dL). I n Fig. 13 s p a c i n g s m e a s u r e d b y r i g h t a n g l e B r a g g diff r a c t i o n a r e p l o t t e d as a f u n c t i o n o f s h e a r r a t e f o r t h e l a t i c e s w i t h a = 6 4 n m , ~b = 0 . 0 5 5 0 . 1 5 4 . T h e v o l u m e f r a c t i o n d e p e n d e n c e is n o t v e r y p r o n o u n c e d b u t t w o g r o u p s , i.e., o n e i n t h e r a n g e ~b = 0 . 0 5 5 - 0 . 0 6 4 a n d t h e o t h e r i n t h e r a n g e 4~ = 0 . 1 4 8 - 0 . 1 5 4 , s e e m t o b e s l i g h t l y d i f f e r e n t . T h e t w o c u r v e s i n Fig. 13 a r e f o r these two ranges in ~ respectively. It can be s e e n f r o m Figs. 12 a n d 13 t h a t (i) t h e l o w e r the volume fraction, the higher the increase 630

,~=0.261

.

•

~ •

~:0.320-0.351 0:0.44.

i

i

1031

104

O(~-)

FIG. 12. Estimated change in d values from fight angle Bragg diffraction (open squares) and grating diffraction experiments (other symbols) as a function of shear rate. Since the value of do cannot be determined uniquely in right angle Bragg diffraction experiments, d values are nondimensionalized by dL, the value of d as G ---, 0. In order to plot d values obtained in grating diffraction experiments on the same graph, the d values are also nondimensionalized by dr. Curves drawn are for 4~ = 0.261, ~b = 0.320-0.355, and 4~ = 0.449. Since experimental points of ¢ = 0.320 and ¢ = 0.355 are almost indistinguishable, one curve was drawn through the experimental points for both volume fractions. Latex used: a = 249 nm. Typical error bars are indicated in upper left corner. i n d, (ii) t h e h i g h e r t h e v o l u m e f r a c t i o n , t h e higher the shear rate at which d starts to increase, a n d (iii) t h e s h e a r r a t e a t w h i c h d s t a r t s t o i n c r e a s e is a b o u t G ~ 1 0 0 0 - 2 0 0 0 sec -1 i n Fig. 12 a n d G ~ 7 0 0 - 1 0 0 0 sec -1 i n Fig. 13. All t h e s e t h r e e f e a t u r e s a r e i n f a i r a g r e e m e n t with our two-dimensional sliding layer model (see Figs. 3 a - d ) . H o w e v e r , q u a n t i t a t i v e l y , t h e

10 1.02

{

(5=0,055-0.064

.,~ 620

5 ~o v

610

o o ~ o 'Is ~ m ~ . ~ o 1.00 ,~ ~ o - - ~ - - -qj - ~ . aIA ~ A• • ~ o t~?

0 10

t02 G(, -1) 103

FIG. 11. Peak wavelength and intensity of observed Bragg diffraction curves as a function of shear rate. I is the intensity of the diffraction at the peak wavelength and 2, the peak wavelength of the diffraction curves. Latex used: a = 64 nm, 4~= 0.064. Typical error bars for intensity measurement and peak wavelength measurement are indicated in lower right corner.

i

lo

1°2 G

(,-I) 1°3

--0.154 -

~ .....

i

1°4

FIG. 13. d/dL values estimated from right angle Bragg diffraction as a function of shear rate. The latices used were a = 64 nm, ~b = 0.055-0.154. A, ~b = 0.064; , , ~b = 0.062; O, q~ = 0.055; O, q~ = 0.154; D, q~ = 0.148. The two curves are for two groups of sarnples having ~b= 0.0550.064 and 4~= 0.148-0.154. A typical error bar is indicated in upper left corner. Journal of Colloid and Interface Science, Vol. 99, N o . 2, J u n e 1984

384

TOMITA AND VAN DE VEN

m a x i m u m increase in d is about 10% in Fig. 12 and 1.5% in Fig. 13, while the two-dimensional sliding layer model predicts only 0.5%. However, if one assumes that dE = do and the increase in spacing starts to occur at higher shear rates (because at small shear rates DA rather than d increases), our model predicts an increase of about 5% (see Fig. 2). The j u m p in spacing from do to dE, predicted by the model, should have been observed in grating diffraction experiments, where both do and dE values can be measured, if it existed. However, as can be seen from Figs. 9 and 12, no evidence of such a j u m p exists, suggesting that dE = do. An increase in spacing must be accompanied by a decrease in the number of horizontal layers, because the sample thickness was kept constant during the experiments. A simple calculation shows that the latex with a = 249 nm, ~b = 0.261 consisted of about 350 layers at low shear rates, which n u m b e r was reduced by 35 layers when the spacing was increased by 10% at high shear rates. Similarly the latex sample with a = 64 nm, ~b = 0.055-0.064 consisted of 780 layers at low shear rates and 770 layers when the spacing was increased by 1.5% at high shear rates. The mechanism of rearrangement of latex particles is at present unknown.

e. Linearity of Velocity Field If the velocity field in the latex suspension had not been linear as assumed in our model, the shear rate would not be constant throughout the sample, and a discussion of experimental results in terms of a single shear rate would become inaccurate. I f the velocity field is nonlinear, the profile will be a function of the sample thickness between the two glass plates of the micro-Couette. I f it is linear, the rotational speed of the glass plates to produce a certain shear rate is inversely proportional to the sample thickness. Hence the linearity of the velocity profile can be tested by doing experiments with different sample thicknesses. In Fig. 14, peak wavelengthsobtained from right angle Bragg diffraction are plotted for Journal of Colloid and Interface Science,

Vol. 99, No. 2, J u n e 1984

.°S

670

T ° o

•

660

10

i

,

i

102

103

14

o (~-l)

FIG, 14. Peak wavelength observed for various sample thicknesses in right angle Bragg diffraction experiment. The shear rate G was calculated by assuming a linear velocity profile, e, 0.10 mm; O, 0.20 ram; [3, 0.30 ram. The curve is fitted through all experimental points. A typical error bar is indicated in upper right corner.

the same sample, but at various sample thicknesses as a function of shear rate. The latex used was a = 64 nm, q~ = 0.055. The sample thickness was changed from 0.10 to 0.30 ram. The shear rate G was calculated from the rotational speed of the glass plate and the sample thickness, assuming a linear velocity profile. As can be seen from Fig. 14, there is no significant difference in the shear rate dependence of samples of different thickness. Because no light can pass through our samples, one cannot be certain that the observed p h e n o m e n a occur throughout the whole sample and not just in the neighborhood of the plates. The fact that the velocity profile is approximately linear seems to indicate that the structure extends through the entire sample. For chaotic particle distributions plug flow should occur at the volume fraction used. It is likely that the flow is linear because of the symmetry in the system.

f Distortion at Zero Shear As mentioned in Section IV, d, the value of do could not b e determined uniquely in right angle Bragg diffraction using the microCouette cell. Table I shows the m e a n peak wavelength together with the standard deviation observed in the micro-Couette and a con-

SHEARED

ORDERED

LATICES

385

their three-dimensional structure under shear, Peak Wavelengths (in nm) Obtained from Bragg although slightly distorted, (ii) the orientation DiffractionExperimentsat Zero or Near Zero ShearRates of the structure in sheared latices coincides (a = 64 nm) with what is expected from crystallographic considerations, (iii) latex under shear consists x (G = 0) of a stack of sliding layers, (iv) the two-diX(G ~ 0 ) Sample q~ Micro-Couette Rectangular cell Micro-Couette mensional hexagonal packing in a layer is distorted by shear, (v) the spacing between the A 0.057 656 + 6 660 + 2 6 5 6 -+ 2 layers increased under high shear, (vi) the B 0.136 5 1 2 -+ 5 514 + 2 5 1 2 __+ 2 lower the volume fraction, the larger is the increase in spacing, and (vii) the higher the ventional rectangular cell (1 × 1 X 3 cm). For volume fraction, the higher is the shear rate comparison, peak wavelengths measured for at which d increases. G --~ 0 are included as well. The experimental The star-like patterns of Hoffman (2, 3) and set-up used in the rectangular cell experiment our diffraction patterns are of the same origin, was identical to the fight angle Bragg diffrac- but the spots of our patterns were clearly seption experiment (Fig. 4) except for the way of arated, allowing a precise quantitative analysis. handling the sample and filling the cell. The Furthermore, his latex was illuminated at the latices used were a = 64 nm, q~ = 0.057 and center of the glass plate, where the shear rate 0.136. For both samples A and B in Table I, was essentially zero. As explained in Section for do values measured in the micro-Couette, III, b, the diffraction pattern reflects the structhe spread about the mean is about three times ture at the illuminated site, not of the site of larger than for samples in the conventional the pattern (1). Therefore, his observation that cell, and much larger than the experimental the diffraction pattern disappeared at a certain error. This suggests that at zero shear rate, the shear rate does not necessarily imply the disstructure of the latex can be distorted to var- appearance of order everywhere in the sample. ious extents. The same conclusion follows We observed less clear hexagonal patterns at from the observation of hexagonal patterns medium shear rates. This might correspond (see Section IV, b). Therefore it can be con- to the disappearance of his diffraction patterns. cluded that the structure of ordered latex at We also did some viscosity measurements on rest is not always perfectly close packed, but our samples which showed no discontinuity some distortion remains. On the other hand, in viscosity in contrast to Hoffman's samples. the do values measured in the rectangular cell The discontinuity was ascribed (2, 3) to the have a narrow distribution. In introducing the rotational motion of aggregates which perturb latex sample into the rectangular cell, applying the order. An alternative explanation is that some form of shear is unavoidable. Therefore the latex undergoes a change in order at a some distortion in structure should be ob- given shear rate by reducing its number of served in the rectangular cell, but far less than layers. Shear-induced melting (4) was not obin the micro-Couette, because in the rectan- served in our systems. It is interesting to note gular cell the direction of shear is not unique that at high shear rates, very dilute samples and hence distortion can be assumed to occur (Clark's samples) showed melting; moderately in more or less random directions and mag- concentrated samples (our samples) showed nitudes and thus to average itself out. A study clear hexagonal patterns and well-defined of the annealing process of the distortion might Bragg diffraction curves; and very concenbe of interest. trated samples (Hoffman's samples) showed a discontinuity in viscosity. V. C O N C L U D I N G REMARKS Some o f our observations can be qualitaThe results of our experiments can be sum- tively explained by a two-dimensional sliding marized as follows: (i) ordered latices maintain layer model in which the motion of a reference TABLE I

Journal of Colloid and Interface Science, Vol. 99, No. 2, June 1984

386

TOMITA AND VAN DE VEN

particle can be calculated from a force balance equation. According to the model the trajectory of a reference particle is wavy, but the waviness decreases with increasing shear rate. The time averaged steady-state spacing, assumed to correspond to the experimentally observed spacing, starts to increase at a certain shear rate. Observations were in qualitative agreement with the model, except that for low shear rates no increase in spacing was observed. Instead the two-dimensional lattice of each layer was found to be distorted. The reasons for this discrepancy must be found in the simplifying assumptions made in the theoretical model. The weak points of the model are probably that it is two-dimensional and that the relative motion of each particle is assumed to be identical (assumption (ii)). A consequence of these assumptions is that the spacing between layers is predicted to increase and decrease periodically (see Fig. 2). Because the sample thickness is constant during an experiment, this implies that the number of layers in the sample must also vary periodically. Hence, a rearrangement of particles should occur, in contradiction to assumption (ii). In a real system, wavy particle motion is not only allowed in the X2X3-plane but also in the X1X3-plane (see Fig. 1). This will reduce the wavy motion in the X2X3-plane because it does not require changes in the number of layers and is probably energetically more favorable at low shear rates. As a consequence, it can be expected that DA values will be larger, DB values smaller, and the increase in d values less than predicted by our model. Grating diffraction experiments shown in Fig. 9 support this speculation. Our calculations suggest that at high shear rates it is energetically favorable for ordered latices to

Journal of CoUoid and Interface Science, Vol. 99~ No. 2, June 1984

have fewer (more densely packed) layers with larger spacings than at lower shear rates. Extension of the model to three-dimensions might remove some of its weaknesses; nevertheless the present model provides, in a qualitative way, some explanations for our observations. ACKNOWLEDGMENTS The authors wish to thank Professor S. G. Mason for valuable discussions and encouragement. They are also indebted to Professor S. Hachisu for kindly supplying latex samples. REFERENCES 1. Tomita, M., Takano, K., and van de Ven, T. G. M., J. Colloid Interface Sci. 92, 367 (1983). 2. Hoffman, R. L., Trans. Soc. Rheol. 16, 155 (1972). 3. Hoffman, R. L., J. Colloid Interface Sci. 46, 491 (1974). 4. Clark, N. A., and Hurd, A. J., Nature (London) 281 57 (1979). 5. Ackerson, B. J., and Clark, N. A., Phys. Rev. Lett. 46, 123 (1981). 6. Kose, A., Ozaki, M., Takano, K., Kobayashi, Y., and Hachisu, S., J. Colloidlnterface Sci. 44, 330 (1973). 7. Krieger, I. M., and O'Neill, F. M., J. Amer. Chem. Soc. 90, 3114 (1968). 8. Takano, K., and Hachisu, S., Sci. Light (Tokyo) 25, 19 (1976). 9. Goodwin, J. W., Ottewill, R. H., and Parentich, A., J. Phys. Chem. 84, 1580 (1980). 10. Cottrell, A. H., "Dislocations and Plastic Flows in Crystals," p. 70. Oxford University Press, London, 1965. 11. Zener, C., Trans. A.LM.E. 147, 361 (1942). 12. Clarebrough, L. M., Hargreaves, M. E., and West, G. W., Phil. Mag. (Set. B) 1, 528 (1956). 13. Clarebrough, L. M., Hargreaves, M. E., and West, G. W., Acta Met. 5, 738 (1959). 14. Takano, K., and Hachisu, S., J. Colloidlnterface Sci. 66, 124 (1978). 15. Tomita, M., and van de Ven, T. G. M., J. Colloid Interface Sci. (in press).