On the anomalous viscosity of monodisperse latex in the disordered state

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On the anomalous viscosity of monodisperse latex in the disordered state

K. Okano, S. Mitaku

To cite this version:

K. Okano, S. Mitaku. On the anomalous viscosity of monodisperse latex in the disordered state.

Journal de Physique, 1980, 41 (6), pp.585-589. �10.1051/jphys:01980004106058500�. �jpa-00209283�

On the anomalous viscosity ^of monodisperse ^latex in the disordered state

K. Okano and S. Mitaku

Department ^of Applied Physics, Faculty ^of Engineering, University ^of Tokyo, Tokyo 113, Japan (Reçu ^{le 8 octobre} 1979, révisé le 12 février, åccepté ^{le 21} fevrier 1980)

Résumé.

On trouve que la viscosité du latex monodispersé ^de polystyrène croit anormalement au voisinage

du point ^de transition ordre-désordre lorsqu’on approche ^{dans la} phase desordonnee. On analyse ^le comportement anormal en utilisaut la théorie de Cohen-Turnbull pour la diffusion moléculaire du liquide.

Abstract.

It is found that the viscosity ^{of a} monodisperse polystyrene latex in the disordered state increases

anomalously ^{when the} disorder-order transition point ^is approached. This anomalous behaviour is analysed

based on the Cohen-Turnbull theory of molecular diffusion in a liquid.

Classification

Physics ^Abstracts

46.60

51.20

1. Introduction.

Monodisperse polymer ^latexes

can exist in ordered lattice structure and%or ^disor-

dered structure depending ^{on the} concentration of added salt as well as on the volume fraction of latex

particles. ^{In the} previous ^{papers [1, 2] we have} reported

the dynamic mechanical measurements of charged polystyrene ^{latexes over the different states from the}

ordered state to the disordered state through ^the phase transition region by changing ^{the amqunt of}

added salt. We have found that a latex in the ordered state is nothing ^{but a real} crystal having ^{a finite shear} rigidity ^{as well as a well defined} yield ^{stress for a} steady ^{shear flow.}

In the disordered state the latex showed no yield

stress and the flow behaviour was essentially ^Newto- nian, ^{but the} steady ^state viscosity ^increases sharply

in the vicinity of the transition point ^{from the disor-}

dered to the ordered state when the concentration of added salt is decreased [2]. Upon further reduction of the concentration of salt there _appears eventually

a finite yield ^stress and the latex transforms into the ordered state. In the previous paper ^we have analysed

the anomalous viscosity ^{of the} disordered latex by

use of Brinkman’s equation ^{on the} assumption ^that

the hydrodynamic ^{volume of a} sphere ^is equal ^to

the effective thermodynamic ^volume including ^the

effect of the electrostatic repulsive potential [2].

However, ^this assumption ^{seems to be} logically improper because the solvent can flow within the electric double layer. ^{In the} present work, ^{we have} carefully measured the anomalous salt concentration

dependence ^{of the} viscosity ^of monodisperse ^{latex in}

the disordered state and will present ^an alternative

explanation ^{for this} anomaly ^{based on} the free volume

theory ^{of molecular} transport ⁱⁿ liquids ^{due to Cohen}

and Turnbull [3].

2. Experimental.

Monodisperse polystyrene spheres ^were polymerized by ^{the emulsion} polyme-

rization and deionized by ^ion exchange resin. The diameter of polystyrene spheres ^{was 125 nm with}

the dispersion ^{of 5 nm as measured} by ^electron microscopy. ^{The surface} charge density ^{was deter-}

mined by the conductometric titration with NaOH

as 1240 elementary charges per ^a sphere. ^{The deio-}

nized monodisperse ^{latex of 11.7} % ^{showed a distinct}

iridescence which indicates the ordered lattice struc- ture of polystyrene spheres.

The measurement of the steady flow behaviour

was performed by ^a concentric cylinder ^viscometer

at 30 °C. In this apparatus ^{a constant} torque ^was

applied ^{to the rotor} by ^the eddy ^{current induced in}

an aluminum plummet ^{in the rotor} by ^the rotating magnetic field. The rotor was suspended ^{in a} sample liquid by ^the buoyancy ^{as well as} the surface tension,

and the rate of shear was determined from the angular speed ^{of the rotor. The shear stress was scanned}

from 10-3 to 2 dyn./cm2, ^{and the shear rate as low}

as 5 x 10-5 s-1 was detected with the aid of a radiate pattern ^{in the rotor} [4].

When KCI was added to the suspension ^{of 11.7} %

in the volume fraction, the iridescence associated with the ordered structure disappeared ^{at the salt}

concentration of 160 gM. Figure ¹ shows the flow

curves of the polystyrene latex in the disordered state above 160 yM ^{KCI. In order to} analyse quantitati-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004106058500

586

Fig. ^1.

^{Flow curves} of the disordered latex of 11.7 % ^{at various}

salt concentration (o : ¹⁶⁰ yM ; ^{0 : 180} IlM; . : ²⁰⁰ IlM; ð. : 300 pM; ^{Q : 500} pM; ^{0 : 1} mM; ^{0 : 2} mM ; ^broken line, water).

vely the anomalous behaviour of the steady ^state viscosity, ^{we have} changed the salt concentration with the step ^{of 20} yM KCI in the vicinity ^{of the}

transition point. ^{The flow curves indicate that the}

Fig. ^2.

^The dependence ^{of the} steady ^state viscosity ^{in the}

disordered latex

the concentration of added salt (KCI). ^The

volume fraction of latex is 11.7 %.

disordered latex is a Newtonian fluid whose viscosity considerably depends ^on the salt concentration. On the other hand, ^the ordered latex showed the well defined yield stress, which ^is reported ^{in a} separate paper [5].

The steady ^state viscosity calculated from the slope

of the flow curves in figure ^{1 are shown in} figure ²

as a function of the salt concentration. The viscosity

is 1.3 cP at sufficiently high salt concentration above 1 mM and increases with the decrease of the salt concentration. Divergent increase of the viscosity ^is

observed in the vicinity of the transition point ^at

160 yM. Figure 2 appears ^to show the details of the salt concentration dependence ^{of the} viscosity ^more clearly ^{than the} previous ^result [2].

3. Theoretical.

3. 1 EFFECTIVE VOLUME FRACTION.

-

In the previous ^{paper [2] we have} analysed ^this

anomalous increase of the steady ^state viscosity ^of

the disordered latex near the transition point ^{in terms}

of Brinkman’s equation [6] for the concentrated sus-

pension ^of rigid spheres ^{with the} modification that the effective volume fraction T* ^{defined below is}

substituted for the volume fraction in the original equation. ^Modified Brinkman’s equation ^reads

where

In the above equation, ^{ns is the} viscosity ^{of solvent,}

qJ is the true volume fraction of suspended spheres, K-1 is the Debye screening length of the added salt and a is a constant of the order of unity. ^In figure ³

the steady ^state viscosity ^is replotted against ^{the effec-}

tive volume fraction T* ^{with a}

1.71 ± 0.07 toge- ther with the theoretical curve of eq. (1).

The idea of regarding ^{a latex as the} suspension ^of

effective rigid spheres ^each consisting ^{of the latex} particle surrounded by ^its electrical double layer,

which adds an amount a/K ^{to the} particle ^{radius, has}

been quite successful in explaining ^the phase diagram

of the order-disorder transition of latexes [7-9] except for the low salt concentration region ^{where the} repul-

sive potential energy between particles ^{is rather}

soft [10, 11]. However, ^{on the} logical ground ^{we are}

not satisfied by eq. (1) ⁱⁿ conjunction ^with eq. (2)

for the anomalous viscosity of disordered latex, ^even though eq. (1) reproduces rather well the observed behaviour of the viscosity ^{as shown in} figure ^{3. The}

volume of the spheres contained in Brinkman’s

equation ^{is, as} in the Einstein formula, ^the hydro- dynamic volume that is the excluded volume for the solvent, whereas the effective volume fraction defined

by eq. (2) is related to the thermodynamic ^excluded

volume between spheres.

Therefore, ^we would like to present ^an alternative

Fig. ^3.

Viscosity of disordered latex is plotted

function of effective volume fraction (p*. ^Full

^shows

modified Brink- man’s formula. The diameter of

polystyrene sphere ^{is 125}

and the constant

in _eq. (2) ^{is taken as 1.71} ± 0.07 which gives

the best fit of the phase diagram ^{in the} high’ salt region.

explanation of the anomalous viscosity of the disor- dered latex based on the free volume theory ^{of mole-}

cular transport ⁱⁿ liquids by Cohen and Turnbull [3].

The present theory is still tentative but, ^we believe, ^it

is physically ^{more sound.}

3.2 FREE VOLUME THEORY.

Let us review briefly

the Cohen-Turnbull theory ^of transport process in a

liquid ^{of hard} spheres [3]. Consider the system ^of N rigid spheres of radius a contained in the vessel of volume V. Let us define the _average free volume _per

particule u f as,

where ₉

N/ Y.4 na3/3 is the volume fraction of

spheres. If the volume fraction is ^not small, particles

are most of time confined to a cage ^or cell surrounded

by their immediate neighbours. Occasionally ^{there is}

a fluctuation in density which opens up ^a hole within the cell large enough ^to permit ^a considerable dis-

placement ^{of the} particle ⁱⁿ the cell. Such a displace-

ment followed by ^the jump of another particle ^into

the hole gives ^{rise to the diffusive} motion. Thus the diffusion constant is considered to be proportional directly ^{to the} probability ^of finding a hole of volume vt just large enough ^to permit ^another particle jump

into the hole after the displacement ^{of the centre}

particles. The diffusion is considered ^{to occur as a} result of the statistical redistribution of free volume within the system.

A simple statistical consideration [3] ^{shows that}

the diffusion constant D of a rigid sphere liquid ^is given by

or

where

Do ⁱⁿ eqs. (4) ^and (5) is the diffusion constant of the centre particle ^{within a} cell, ^and y is a constant between 1/2 ^{and 1} introduced to correct for overlap

of the free volume. Note that w that _appears in _eq. (5)

is related ^to the thermodynamic excluded volume between rigid spheres.

We consider that the Cohen-Tumbull theory,

eq. (5), would describe the transport process in ^a concentrated suspension ^of spherical particles, ^because

a concentrated suspension with random distribution of spheres may be regarded ^{as a} liquid ^as the ordered

suspension ^{is a real} crystal [5]. ^In particular, ^{the dif-}

fusion constant of a charged ^latex particle ^{will be} given by eq. (5), if w ^is replaced by ^{the effective volume}

fraction (p* ^defined by eq. (2).

As for ut we may put

where

because in a closed packed regular ^{structure of} spheres

or radius a* the centre particle ^can just-escape ^{the cell} [ 12] ^if

Thus we have for the diffusion ^constant of a particle

in a latex suspension ^the following equation

where Do is the diffusion constant of a suspended particle ^{in a} cell. In the first approximation, Do ^is given by

where k is the Boltzmann constant and T is the abso-

lute temperature.

588

The viscosity ^{of a latex} suspension ^is given by

In eq. (12), L ^{is the} lifetime of a cell or the relaxation time of the short range order in a disordered latex and is given by

where d is a length of the order of the interparticle

distance and D is given by eq. (10). ^G in eq. (12) ^is

the high frequency (glass like) shear modulus of disordered latex which would be observable for a

measurement at frequencies higher ^than 1/r. ^{In a}

time interval much shorter than 1:, the distribution of

particles ^{is frozen and the} suspension will behave as a glass with the shear rigidity ^G. On the other hand, particles ^{can move} according ^to the external stress at frequencies ^much lower than 1/r, giving ^{a Newto-}

nian liquid ^whose viscosity ^is expressed by eq. (12).

Thus, ^{we have}

At present ^{we do not know how to calculate} precisely

the shear modulus of frozen disordered latex. However,

we have already measured the high frequency ^shear

modulus of various disordered latexes and the values from 0 to 102 dyn./cm2 have been obtained [1, 2, 4].

Therefore, ^{we assume} here the shear modulus as an

adjustable parameter ^and compare it to the above values after fitting eq. (14) ^to experimental ^results

in figure ^2.

The experimental ^{results are} replotted ⁱⁿ figure ⁴

in which the ordinate is ln n and the abscissa is

cp*(1 - cp*) - 1. ^{The full line} represents ^the equation

or

which fits fairly ^{well the} experimental values. These

equations ^are quite consistent with the theoretical

formula, eq. (14). The factor 1.87 in the exponential

of _eq. (15’) corresponds to y

0.66, which is in the range between 1/2 ^{and 1.} Furthermore, the preexpo- nential factor of 7.82 ^x 10-3 together with d 2-- 230 nanometer, ^{2 a}

^{125 nanometer} and _qs

0.008 poise give the value of about 1 dyn./cm2 ^{as the} glass-like rigidity ^{of the} disordered latex. This value of the

rigidity is consistent with the experimental ^values

between almost zero and 102 dyn./cm2 ^{at 40 kHz,} although ^the experimental ^{error is} fairly large [1,2,4].

At low values of cp*/(1 - 9*) ^the experimental points

Fig. ^4.

^A replot ^of figure 3; In q ^is plotted against qJ* /0 - (p*).

Full line represents the eq. (15) ^{in the text.} Open ^circles

^the previous ^results [2] ^and square corresponds ^{to water.}

seem to become apart from the theoretical curve.

However, since the concept of the diffusion due ^to.

the redistribution of the free volume is considered to be applicable only ^to the concentrated particle

systems, ^this discrepancy ^{at low effective volume}

fraction is reasonable.

4. Conclusion.

The anomalous increase of the

steady ^flow viscosity ^{of a} disordered monodisperse polystyrene latex in the vicinity of the order-disorder transition point ^{has been} analysed successfully by invoking the Cohen-Tumbull theory of molecular transport ⁱⁿ liquids. ^We may conclude that the _pro-

cess of flow (or diffusion) ^{in a} concentrated latex

dispersion ^is quite ^{the same as that of a molecular} liquid in which the free volume plays ^{an essential}

role in the elementary process of transport. ^{If we} could increase T* ^without occurrence of crystalline lattice, ^we would have a glass transition in a latex

suspension ^{as the} Cohen-Turnbull theory predicts.

In order to substantiate this conclusion we are now

planning ^to measure more extensively ^the steady

state viscosity ^of latexes of different amount of sur-

face charges together ^{with the} direct measurement of the high frequency glass like shear modulus of disor- dered latexes.

Acknowledgment.

^We would like to thank Drs.

Y. Wada and T. Ohtsuki for their interest and dis- cussion. This work was partly supported by ^the

Grant-in-Aid for Scientific Research, ^446046.

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