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Shear waves in colloidal crystals : II. Effects of finite height in cylindrical samples
M. Joanicot, M. Jorand, P. Piéranski, F. Rothen
To cite this version:
M. Joanicot, M. Jorand, P. Piéranski, F. Rothen. Shear waves in colloidal crystals : II. Ef- fects of finite height in cylindrical samples. Journal de Physique, 1984, 45 (9), pp.1413-1421.
�10.1051/jphys:019840045090141300�. �jpa-00209881�
Shear waves in colloidal crystals :
II. Effects of finite height in cylindrical samples
M. Joanicot, M. Jorand (*), P. Piéranski and F. Rothen (*)
Laboratoire de Physique des Solides, Bât. 510, Université Paris-Sud, 91405 Orsay, France
(*) Institut de Physique Expérimentale, Université de Lausanne, CH-1015 Lausanne-Dorigny, Switzerland
(Reçu le 9 novembre 1983, accepti le 22. mai 1984)
Résumé. 2014 On étudie les vibrations de symétrie axiale, excitées par des moyens mécaniques, dans un échantillon de cristal colloïdal ayant la forme d’un cylindre de rayon R et de hauteur H. On démontre que, à volume constant, le rapport 03B1
= R/H
= 0,5 minimise les fréquences des modes propres et optimise la précision des mesures du module élastique. Un nouveau montage expérimental nous a permis de comparer la théorie à l’expérience et d’en déduirele module élastique et la viscosité moyenne de plusieurs échantillons de cristal colloidal.
Abstract. 2014 The vibration spectra of a cylindrical sample of colloidal crystals are studied We indicate the proper choice of the aspect ratio 03B1 = R/H which optimizes the quality of resonances. A new experimental set-up allows
us to compare the theory with the experimental results and to deduce a precise value of the elastic modulus and of the viscosity for several latex suspensions.
Classification Physics Abstracts
47.00 - 36.20 - 62.20D
1. Introduction.
1.1 VISCOELASTICITY OF COLLOIDAL CRYSTALS. - One of the most striking properties of colloidal crystals,
that results when changing from an atomic to a colloi-
dal scale, is their very low elasticity; about ten orders
of magnitude smaller than in ordinary crystals [1].
Moreover, the viscous coupling between colloidal
particles, arranged in a crystal lattice, and the solvent in which they are immersed, results in a strong dissi- pation making the longitudinal modes purely diffu-
sive [2]. The transverse modes, where the crystal (of shear modulus E) and the solvent (of an affective viscosity 11) move together, propagate provided [2]
These unusual viscoelastic properties of colloidal
crystals required the development of special methods
to measure the relevant material constants such as
compressibility [11] or shear modulus [2-4]. In the latter
case, the methods, developed at Orsay [2] and at
UCLA [3], consisted in monitoring mechanical vibra- tions of polycrystalline samples contained in glass cylinders (such as the tube T in Fig. 2) of height H
and of radius R (Fig. 1).
Fig. 1. - Geometry : a cylindrical sample of a finite height
H is oscillating around the vertical axis-z. Boundary condi-
tions on S l’ St and Sb are discussed in section 2. 1.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045090141300
1414
Fig. 2. - Schema of the experimental set-up.
1.2 EFFECTS OF FINITE HEIGHT IN CYLINDRICAL SAMPLES. - In order to simplify theoretical cal-
culations, it has been assumed in the previous paper [2]
that
In this case, the boundary conditions on the base surfaces Stop and Sbottom of the cylindrical sample can
be neglected; the deformation is only a function of r and the mechanical resonances, given in section 3.4
by equation (22), are approximately
where p is an average density of colloidal crystal.
These resonances are sharp provided that their
frequencies are small compared to 1/i (condition (1));
for a given colloidal crystal, the larger the radius R, the sharper the resonances.
Unfortunately, the volume V = nR2 H of cylin-
drical samples increases with R 2 and becomes pro-
hibitively large when one wants to satisfy both condi-
tions (1) and (2).
The aim of the present paper is to treat the case of a
finite ratio HIR ( ~ 1), which optimizes the volume V under the initial condition (1). In section 3 we report
new calculations, more general than those of refe-
rence [2], of sample deformation, which depends both
on r and z and satisfies the boundary conditions both
on the sides and the ends of the cylinder. The theoretical
predictions are shown to agree with results of experi-
ments performed using a completely new experimental set-up, described in the next section.
2. Experimental
2.1 APPARATUS. - Although the principle of mea-
surements is the same as that of reference [2], the new experimental set-up contains many important modi-
fications which we mention below :
- Container : in order to make the mechanical
resonances narrower than previously and so improve
the accuracy of determination of E, the radius R
of the cylindrical container (tube T in Fig. 2) was chosen
to be 1.75 cm more than twice larger than previously [2]. As the boundary conditions on the ends of the
cylindrical sample had to be taken into account, the bottom of the container was made from an optically
flat glass plate P epoxied to the tube T, perpendicular
to its axis; a non-slipping boundary condition was
assumed in calculations. The upper surface of the
sample was left free but the vertical position of the
tube T was carefully adjusted in order to make this
upper surface St parallel to Sb. Theoretically, in the
calculation of section 3, a zero-stress boundary condi-
tion was assumed at this upper surface.
- Excitation : in order to induce only the vibra-
tions of cylindrical symmetry, the sample should have
only one degree of freedom of rotation around its z-axis. This requirement was achieved by mounting
the tube T on two ball bearings (B) using two conical joints (Cj) and a cylindrical metallic part (C).
As previously, oscillations of the sample were induced by a loudspeaker LS, coupled to the cylinder C by means of a rigid rod (R) (Fig. 4).
Particular care was taken in order to detect in situ the vibration of the container : a laser beam (LB)
was reflected from a mirror (M) onto a four-quadrant position sensing photodiode Dl. This photodiode
delivered a voltage Vout proportional to the angle 0
of rotation around the vertical axis z.
Let V;n( f ) be a voltage of frequency f applied to
the speaker. In figure 3a we show a plot V 0 t(f),
for the excitation voltage V;n( f ) (constant Oy
is the result of squaring Vout using an analog multi- plier and subsequent time-averaging by a low-pass filter). Clearly, due to inertia of moving parts and
elasticity of the speaker suspension, the vibrations
of the container, detected in situ, show a reso-
nance centered at fr = 20 Hz. In order to suppress this undesirable effect and keep the amplitude of
oscillations 4>0 at a constant level throughout all the
Fig. 3. - Plots of oscillation of the tube T, detected in situ, as a function of frequency : a) without regulation, the system of excitation shows a resonance at fr = 20 Hz; b) with a proper adjustment of the feedback loop, shown in figure 5, the
resonance is suppressed.
Fig. 4. - Schema of the excitation system with the feed- back loop.
frequency range a feedback loop, was introduced, as
shown in figure 4, through a PID amplifier and an analog multiplier.
With proper adjustments of this feedback loop,
the excitation system worked correctly and the
resonance was suppressed as shown in figure 3b.
- Detection of vibrations of the C. C. sample [5] :
the vibrations induced in the colloidal crystal sample, by the axial oscillations of its container, can be detect- ed either by optical [2] or by mechanical [3] methods.
For the purpose of the present study, where we
are also interested in the z- and r- dependence of a
rather complicated deformation of cylindrical samples,
the use of light as a probe has two main advantages;
this method is local and non-perturbative.
In our new apparatus, we detected the motion of Kossel lines [6] by a four-quadrant position sensing photodiode which can be moved around the glass sphere S by means of two rotation stages. During
measurements, this diode is located at the edge of the oscillating Kossel line.
2.2 PREPARATION OF SAMPLES. - Our samples of
colloidal crystals were prepared by dilution of a polystyrene latex synthesized by a standard emulsion
polymerization method (Ref 16], Fig. 7). The diameter of particles as determined by electron microscopy was
0.14 um. The crystallized suspension was poured
in the tube T through a long funnel terminated by a
sintered glass filter. In spite of very careful cleaning
of all glass surfaces, the suspension, once introduced
in the tube T, sometimes melted In those cases, the
suspension was drawn up back to the funnel (con- taining ionic exchange resin) several times until crys- tallization occurred
Once prepared and crystallized, our samples were
characterized in situ using their Kossel diagrams.
All samples of concentrations between n = 0.6 x
1014 p/CM3 and n = 1013 p/CM3 had Kossel diagrams, corresponding to the b.c.c. structure. From the angu- lar diameter of Kossel lines with known Miller indices,
the concentration n was determined accurately (An/n =
± 5 %). The diameter 2 R = 3.5 cm was constant and the height H was measured using a millimetric gradua-
tion painted on the tube T.
In hgure 5a we show a series of experimental spec- tra V;,y(f) obtained with a typical sample of height
H = 3.0 + 0.1 cm of concentration n = 4.2 x 1013
1416
Fig. 5. - Spectra of a sample _with R/H = 0.58.
a) Experimental plots of Vx( f ) measured at different height z.
b) Theoretical plots, calculated numerically, using the form (27) (continuous line). Dotted line : plots corresponding
to the case of infinite height H.
part/cm’. Each plot corresponds to a different height
z of the incident laser beam.
Obviously, these typical spectra are more complex
than those of reference [2]; the spacings between
resonances are not uniform and their heights do not
decrease monotonically. On the other hand, they are
very reproducible and show a progressive transfor-
mation as z changes. We will show in the following
that these differences, with our previous results [2],
are all due to the finite height H.
3. Theoretical.
3.1 I EQUATION OF MOTION. - Usually, the samples of
colloidal crystals that we studied were polycrystalline.
As the dimensions of crystallites ( ~ 0.1 ’--. 1 mm) are
much smaller than H and R, this polycrystalline infra-
structure can be neglected and the samples can be
considered as made of an isotropic, homogeneous, incompressible material.
Let S(r, z, t) be the local displacement in the crystal.
The equation of motion for S, in the use of an incom- pressible deformation of low frequency, has been
shown (Rcf [2]) to have the following form :
where p is an average density of the suspension,
E is a shear elastic modulus (supposed to be frequency, independent),
and il is an effective shear viscosity (frequency inde- pendent).
In the following we will discuss the solution of (4)
which will satisfy the boundary conditions imposed by the walls of the container.
3.2 NORMAL MODES. - Let us first suppose that the container T (in Fig. 2) is at rest The slipping
conditions at the bottom and on the lateral walls of the container are :
The upper surface (free) must be stress-free :
The normal modes which satisfy equation (4) under
the above boundary conditions are :
where yj = 3.83; 7.01; 10.17 ; 13.22 ; are zero’s of the first order Bessel function J1 and n = 0, 1, 2, 3, ...
The frequencies of resonances of these modes are
Fig. 6. - Frequencies G
normal modes plotted as a function of the aspect ratio (HjR) = a for the sample volume V = Cte.
where a = R/H is the « aspect » ratio of the cylindri-
cal sample.
In equation (8), each resonance is a-dependent.
For a given value of j, the « principal >> resonance
progressively splits as a increases, into different « sub »
resonances (J)jn«(X) as shown in figure 7. The initial
order, depending upon the values of j in the limit H -+ oo, is progressively lost under subsequent overlapping of subresonances; but, finally, for very
large values of a tlre resonances with the same number
n tend to
1418
Fig. 7. - Determination of the elastic modulus using equation (7).
Obviously, in experiments where, for example,
a = 1, a confusion between different modes is pos- sible unless one can measure the local deformation
S(r, z) which differs from one mode to another.
3.3 CALCULATION OF THE DISPLACEMENT S. - Let these axial oscillations of the container be
so(r, z) has to satisfy the following boundary condi-
tions :
1 ) Sø(R, z) = A, where A = ROO is the amplitude
of motion of side walls of the tube T; it corresponds
to the non-slipping condition at this glass surface.
2) Se(r, o) = A
2013,
R corresponds to the non-slipping boundary condition at the bottom surface (section 2.1).3) S,(r, H) = f(r), where J’(r) is an unknown func- tion of r which has to be determined. On the other hand, as the upper surface is free is must be stress free which means that :
Introducing S = Se e irot in equation (4) and using cylindrical coordinates one gets :
where
Let us suppose now that Se(r, z) can be represented
as a sum of three functions :
satisfying the following boundary conditions, equiva-
lent together to the above three conditions :
where a(r) and B(r) are to be determined from conti-
nuity of Si(r, z) (i = 1, 2, 3).
S 1, in equation (14), represents the deformation of the cylinder, compatible with the motion of the lateral surface of the container and independent of the condi- tions on the two ends. In other words, S1 is the solu- tion for the previously discussed case of the cylinder
of infinite height (as 0) [2].
Let us remember that, when Se is z-independent,
the equation (13) is the Bessel equation of first order,
and one gets :
The two other terms, S2 and S3, must be z-dependent; their role is to match the total function So to condi-
tions 2) and 3) which, once a(r) and p(r) are known, write :
Let us suppose that
Fig. 8. - Dependence of the spectra on the viscosity q.
Separating variables in equation (13) one gets the following two equations for R2 and Z2 :
The solutions of the two equations above, satisfying the required boundary conditions are :
and
J
The unknown coefficients B, as determined from the boundary condition
are : O O
The same method can be applied to find S3.
SO is then :
where is given by (22) and k2 = pw2/(E + iw il).
In the present case, where we are only interested in the relative measurements of ou it is enough to
say that for any position of the diode, the voltage Vout is proportional to ðýJ = OSIOR (6§ « 1).
1420
The derivative ðSlðr, as calculated from (23) is :
4. Discussion.
Among the parameters H, R, z, E and tj of the above
equation, H, R and z are just the geometrical factors
known directly from experiments. The two material
constants E and tj are to be determined from the fit with experimental spectra.
4.1 DETERMINATION OF E. - When the resonances are as sharp as those in figure 6a, the elastic modulus E
can be determined simply using the equation (8).
First, each resonance, present on the spectra of
figure 6a was identified as a particular normal mode and its frequency OJj,n was plotted in figure 9 as a
function of the dimensionless quantity Wj,,. =
J PI + (n + 1/2)2 n2 a2. Knowing R and p, the
elastic modulus E was calculated from the slope El 1
of the straight line which best fits the experi-
p R
mental points. The accuracy AE in the particular case
of the figure 9 is about AE/E = 0.05. This accuracy
depends mostly on the proper identification of the
indices j and n corresponding to each of the reso-
nances present in the spectra. The correct identifica- tion should be confirmed by the numerical plots
of the spectra using equation (27).
4.2 NUMERICAL PLOTS, CHOICE OF tl. - The use of the equation (27) for numerical plots is simple enough
in cases when one can neglect the higher order terms
of the Bessel series. The convergence of this series
depends on all parameters H, R, z, E, q and to. For
the particular plots shown of figure 6a, it is sufficient to consider only the first term since the factor
COSJk2 - kf (H - z)/cosJk2 - kf H decreases ra-
pidly with j.
In figure 6b we show the theoretical spectra plotted by computer for E = 40 dyn/cm2 and il = 0.08 (a = 0.58). In the same figure, we show (dotted line) the
theoretical spectra corresponding to the case of infinite
height (a = 0). The differences between the plots
for a = 0 and a = 0.58, very striking for small
values of z, are due to the proximity of the bottom with non-slipping boundary conditions.
The relative height and width of the resonance depend mainly on the choice of the viscosity n. In figure 10
we show how the form of the spectra is modified by varying n, all other parameters being kept constant
The choice of q can be made numerically by looking
for the best (mean-square-error) fits of the whole spectra. However, this procedure requires digitizing
of the experimental data and is rather time consuming.
Our choice of il for the plots in figure 6b was based
on a criterion such as the half width of resonances. For
increasing values of q the resonances with lower values of indices n and j are progressively swept
out The accuracy of such a subjective method can
be estimated to be of the order of å"l" = 10 %.
5. Conclusion.
In the present paper we have pointed out that, for a given volume V of cylindrical samples of colloidal
crystals, the lower frequencies of the normal modes
occur when the aspect ratio R/H = a is of the order of aoptimum = 0.5. This case of finite cylinder height is
therefore important when one wants to achieve the best determination of the elastic modulus under the condition of the smallest possible volume V.
We have solved theoretically the problem of vibra-
tion in such cylindrical samples of finite height. Our
theoretical plots are in agreement with the experimen-
tal spectra obtained using a completely new experi-
mental set-up.
The method described in the present paper is
actually applied to the determination of the elastic moduli of different types of Colloidal Crystals [8].
Acknowledgments.
We are greatful to E. Dubois-Violette and to J. No- gues for numerous discussions and for help in nume-
rical calculations.
References
[1] CRANDALL, R. S. and WILLIAMS, R., Science 198 (1977)
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[2] DUBOIS-VIOLETTE, E., PIERANSKI, P., ROTHEN, F., STRZELECKI, L., J. Physique 41 (1980) 369.
[3] LINDSAY, H. M. and CHAIKIN, P. M., J. Chem. Phys.
76 (1982) 3774.
[4] BENZING, D. W. and RussEL, W. B., J. Coll. Int. Sc. 83 (1981) 178.
[5] PIERANSKI, P., DUBOIS-VIOLETTE, E., ROTHEN, F. and STRZELECKI, L., J. Physique 42 (1981) 53.
[6] PIERANSKI, P., Contemporary Physics 24 (1983) 25.
[7] BUDAK SAMARSKI, Collection of Problems in mathematical
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