Rheology of concentrated suspensions of viscoelastic particles

Colloids and Surfaces A: Physicochemical and Engineering Aspects 152 (1999) 79–88

Rheology of concentrated suspensions of viscoelastic particles P. Snabre a,*, P. Mills b a Institut de Science et de ge´nie des mate´riaux et proce´de´s, UPR 8521, BP 5, 66125 Font-Romeu, France b P.M.D. Universite´ de Mainne La Valle´e, 2 rue du promontoire, 93166 Noisy Le Grand, France Received 22 January 1998; accepted 9 June 1998

Abstract The viscoelastic properties of particles strongly influence the rheology of many concentrated industrial or biological dispersions (emulsions, paints or blood). The deformation-orientation of interacting particles under flow enhances the maximum packing fraction of particles and leads to a non-linear rheological behaviour. We first present a Kelvin Voigt model to describe the deformation and stable orientation of a viscoelastic particle in a simple shear field. We then use a viscosity law for concentrated suspensions of hard particles in purely hydrodynamic interactions and relate the maximum packing concentration to the component of the particle deformation tensor in the direction of the flow. In the range of small deformations and diluted suspensions, the model gives a viscosity law similar to the relation obtained by Lhuillier [Phenomenology of hydrodynamic interactions in suspensions of weakly deformable particles, J. de Phys. 48 (1987) 1887–1902]. In concentrated suspensions, a self-consistent theory based on an effective medium approximation yields a non-linear equation for the relative shear viscosity. The rheological law describes the viscosity of concentrated suspensions of viscoelastic particles such as red cells in saline solution and allows information about the non-linear viscoelastic properties of the cell membrane. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Rheology; Suspension; Viscoelastic particle red blood cell; Viscosity

1. Introduction The viscoelastic properties of particles strongly influence the rheology of concentrated industrial or biological dispersions (emulsions, paints, yoghurts or blood ). The deformation-orientation of interacting units induces a non-linear rheological behaviour [1]. The micro-rheological models based on understanding the relationships between bulk suspension properties and suspension microstructure are often limited to the diluted regime [2]. Our purpose is to propose a simple but general analysis of the non-Newtonian properties of con* Corresponding author. Fax: +33 04 68 30 29 40. E-mail address: [email protected] (P. Snabre)

centrated suspensions of viscoelastic particles such as liquid drops, macromolecules, microcapsules or biological cells. We consider particles larger than 1 mm in diameter so that the hydrodynamic effects dominate over Brownian motion. Reynolds numbers for flow around immersed particles are further assumed small enough compared to unity to neglect the inertial effects relative to viscous forces. The timedependent deformation of particles is relevant from either pure fluid mechanics, statistical physics or viscoelastic constitutive equations for composite materials [3–5]. However, the viscoelastic behaviour of deformable particles emphasizes the particular role of a common relaxation time related to the properties of the material (particle diameter,

0927-7757/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 9 2 7- 7 7 57 ( 9 8 ) 0 06 1 9 -0

80

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

interfacial tension, elastic modulus, internal viscosity, etc.) [6,7]. In a first part, we derive the first-order deformation of isolated liquid drops and complex bodies in a shear flow from a rotating Kelvin Voigt model [8] representative of linear viscoelastic behaviour. The non-linear rheological properties of concentrated suspensions mainly result from stresses exerted by the surrounding medium on deformable particles. We then propose a self-consistent model based on an effective medium approximation for estimating the shear viscosity of a concentrated suspension of viscoelastic particles. For this purpose, we use a reference rheological law for hard spheres in purely hydrodynamic interaction [9–11] and a first-order development of the maximum packing fraction as a function of the strain gradient in the direction of the flow. We finally analyse the viscoelastic properties of the red cell considered as a solid–liquid composite with shear-dependent relaxation times and deduce the shear viscosity of concentrated unaggregated red cell suspensions in saline solution.

2. Deformation of a viscoelastic particle in a shear flow Solid spheres in a shear flow undergo a periodic rotary motion with a mean angular velocity v= c/2 where c denotes the local shear rate [1]. Liquid drops behave rather differently from a solid particle and undergo both deformation into a spheroid and steady state orientation in the shear field because of the transmission of velocity gradient across the interface and the flow of the internal fluid. Microcapsules and biological cells such as red cells represent an intermediate case between solid particles and fluid drops, behaving as a solidlike body in the low shear regime and attaining a steady state orientation with a tank-tread motion of the membrane only at high shear rates [12,13]. The first-order deformation-orientation of a viscoelastic sphere in a shear flow can be described by a rotating Kelvin Voigt element with a characteristic relaxation time h [8]. We consider an initially spherical particle of radius R centred at the origin O of a plane xy parallel to the fluid velocity vector V(cy, 0, 0) (Fig. 1). The position

Fig. 1. Deformation and orientation of a spherical viscoelastic particle in a simple shear flow.

vector r of an element of particle area rotates with an angular velocity v=c/2 and is assumed to behave as a Kelvin Voigt element with a characteristic relaxation time h. An element of the external surface bound to the end of the rotating vector r experiences a normal stress s =m c sin 2y involvn 0 ing the viscosity m of the suspending fluid and the 0 time varying angular position y(t). The time dependence of r=|r| then obeys a first-order differential equation [8]: dr

m r−R = 0 Rc sin[2y(t)]− 1.0 dt m h p ct t (1) with y(t)= v dt= +y 0 2 0 where m is an effective viscosity accounting for p the overall dissipative phenomena in the particle. We can write the above equation in the dimensionless form:

P

de dT

+e=

V L

sin(VT+2y ) 0

(2)

with t m , T= , L= p , V=ch R h m 0 The time-dependent deformation e(T ) then obeys the general solution: e=

r−R

e(T )=e cos(VT+2y −2y ) M 0 M =e cos(2y−2y ) M M

(3)

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

where e =(V/L)[1+V2]1/2 is the maximal relative M deformation and y =0.5/tan(1/V) the angle of M maximal deformation. In the low shear regime (c%1/h), the relative particle deformation e then scales as the shear stress m c and decreases with 0 the effective particle viscosity m . p We then deduce the components e =e(y=0) xx and e =e(y=p/4) of the shear strain tensor xy representative of the particle deformation gradient along the directions y=0 and y=p/4: V2

1.0 e =e(y=0)= xx L(1+V2) V e =e(y=p/4)= xy L(1+V2)

(4)

The steady state deformation of a liquid drop (viscosity m and interfacial tension s) in a i shear flow is reached with a relaxation time h#lC/c#m R/s depending on the viscosity ratio i l=m /m and the capillary number C=m cR/s [1]. i 0 0 When applied to the case of liquid drops with V= ch=19lC/20 and L=76l(l+1)/(95l+80), the Kelvin Voigt model gives exactly the steady state orientation y and the maximal relative drop M deformation e obtained by Cox for small deparM ture from the spherical shape (C%1) [7]. The frequency f of the periodic motion of the fluid particle further scales as the shear rate [14]. A Kelvin Voigt model thus gives a reasonable account of the drop motion in the range of small deformations.

3. Shear viscosity of suspensions of hard spheres The viscosity of a suspension of monodispersed non-Brownian hard spheres in purely hydrodynamic interactions depends on the pair distribution function and may exhibit non-Newtonian behaviour [15]. Neglecting normal stresses in a steady simple shear flow at low Reynolds number, the relationship between the viscous stress tensor and the rate of strain tensor involves a single viscosity coefficient: t=m(w, c)c

(5)

81

where m(w, c) is the apparent shear viscosity of the suspension, w the particle volume fraction and c the apparent macroscopic shear rate. For dilute suspensions of hard spheres in a viscous fluid (viscosity m ), Batchelor and Green [15] extended 0 the Einstein formula to second order in particle volume fraction: m(w)=m [1+2.5w+kw2] 0

(6)

where the k coefficient depends on the flow field and the two-particle hydrodynamic interactions through the pair distribution function [15]. For concentrated systems of hard spheres in a viscous fluid, the basic problem for predicting the suspension viscosity is how to relate the microstructure configuration and the many-body hydrodynamic interactions. A mean field theory based on the estimation of the viscous dissipation in the fluid volume leads to the expression of the relative shear viscosity m [9–11]: r m (w, w* )= r 0

m(w, w* ) 1−w 0 = m (1−w/w* )2 0 0

(7)

where w* is the critical volume fraction relative to 0 the transition threshold between the fluid state and the solid state. For an isotropic pair distribution function in the low shear regime, the suspension displays a Newtonian behaviour and the maximum packing volume fraction w* shows no dependence 0 upon the shear rate. The apparent shear viscosity of the suspension then mainly depends on the maximum packing fraction. Viscosity measurements for particles with bimodal size distributions indeed show a correlation between the maximum random packing of the dry solid and the maximum packing fraction w* [16 ]. 0 As shown in Fig. 2, Eq. (7) with the random isostatic packing fraction w* =4/7 for hard spheres i well describes experimental data around the zero shear rate limit from Maron and Levy-Pascal [17], Krieger [18], De Kruiff et al. [19] and GadalaMaria and Acrivos [20]. However, a deviation occurs in the high concentration regime (w>0.5, Fig. 2) which likely results from a change in the spatial distribution of particles and an increase in the maximum packing

82

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

transient anisotropic microstructures with a higher maximum packing volume fraction which results in a non-Newtonian behaviour. For weak particle deformations, we may estimate the shear rate dependence of the viscosity by expanding the maximum packing volume fraction w*(c) as a function of the shear strain gradient e (c) in the direction xx of the flow:

Fig. 2. Experimentally observed values of [(1−w)/m ]1/2 around r the zero shear rate limit for hard spheres in purely hydrodynamic interactions from Maron and Levy-Pascal (n) [17], Krieger (k) [18], De Kruiff et al. (&) [19], Gadala-Maria and Acrivos ($) [20]. The solid line is calculated from Eq. (7) with w* =4/7. 0

concentration. In the concentrated regime, shear forces make the pair distribution function anisotropic along the main strain axes [21]. Computer simulations from Brady and Bossis [22] based on Stokesian dynamics indeed show the formation of transient shear-induced microstructures in concentrated systems of hard spheres. The volume fraction dependence of the shear viscosity yet displays no slope discontinuity, since the lifetime of the finite transient structures remains short compared to the eddy diffusion time. The flow-induced anisotropic microstructures (deformation and orientation of transient clusters in a shear flow) and the maximum packing fraction then closely depend on the interparticle potential and the flow type [23]. In the following and for particle volume fraction not too close to the maximum packing concentration w* , Eq. (7) is therefore assumed to be the 0 reference rheological law for a random concentrated suspension of particles.

w*(c)=w* [1+be (c)+O(e2 )] (8) 0 xx xx where w* is the maximum packing concentration 0 around the zero shear rate limit and b a constant related to the flow type. The reference rheological law (7) together with (8) and the Kelvin Voigt model then gives the relative shear viscosity of a diluted suspension of weakly deformed viscoelastic particles: m (w, e )= r xx

1−w

A

1−

w w* (1+be ) 0 xx

B

2

(9)

with V2 e = and V=ch xx L(1+V2) The reference viscosity law assumes negligible viscous dissipation in the particulate phase of effective viscosity m . Considering the continuity p of the shear stress across the interface, the average shear rate c in the particle volume scales as i m c*/m #c*L−1 (c* is the average shear rate in the 0 p fluid phase). The condition for a dissipation energy d P per unit volume in the viscoelastic t phase much lower than the viscous dissipation d D in the fluid fraction (1−w) then takes the t form: d P#m (c*L−1)2w%d D#m c*2(1−w) t p t 0 We deduce the condition:

(10)

w

4. Shear viscosity of diluted suspensions of viscoelastic particles

L2&

The orientation of deformable particles along the main strain axes induces the formation of

For a viscosity ratio L=m /m &1 or particle p 0 volume fraction not too close to the maximum

1−w

83

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

packing fraction, we may neglect the viscous dissipation in the particle phase. In the range of small deformations and dilute suspensions (w%w*), then Eq. (9) yields a viscosity law similar to the relation obtained by Lhuillier [2]: 1−w m (w, e )= r xx (1−w/w* )2 0

A

1−

2w/w* 0 be 1−w/w* xx 0

B (11)

Considering a maximum packing concentration at zero shear rate close to the random isostatic packing fraction w* =4/7 of solid spheres, a firsti order expansion of Eq. (11) with w* =4/7 further 0 leads to the rheological law obtained by BarthesBiesel and Chim [24]: 7 bL−1V2 m (w, C )=1+2.5w− w r 2 1+V2

(12)

5. Shear viscosity of concentrated suspensions of viscoelastic particles In concentrated suspensions, hydrodynamic interactions enhance shear gradients and particle deformation but the flow-induced anisotropic microstructures lead to a decrease in viscous dissipation and suspension apparent viscosity. Therefore, the non-Newtonian behaviour of deformable particles is essentially due to the nonlinear relation which exists between the suspension viscosity and the particle deformation. Some works introduce a phenomenological expression of the structure parameter w*(c) for describing the nonlinear rheological properties of concentrated suspensions [25]. Here, we propose a self-consistent theory based on an effective medium approximation in agreement with data from rheo-optical experiments [11,26 ]. We consider that viscoelastic units behave like isolated particles in a fluid of viscosity equal to the viscosity m(w, c) of the suspension and experience an effective shear stress t=m(w, c)c. From Eq. (4), it follows that the relative deformation gradient e of particles in a concentrated xx

Fig. 3. Theoretical variations of the relative shear viscosity m (w, c) of concentrated suspensions of viscoelastic spheres with r the Deborah number V=ch and the particle volume fraction w. Solid curves are obtained from Eq. (14) with L/b=10 and w* =4/7 (# w=0.2, % w=0.3, $ w=0.4, & w=0.5). 0

suspension is given by: V2 e =m (w, e ) xx r xx L(1+V2)

(13)

By introducing the shear rate dependence (8) of the maximum packing concentration in the reference rheological law (7), a non-linear equation for the relative shear viscosity m (w, V) is obtained: r 1−w m (w, V)= r w bL−1V2 −1 2 1− 1+m (w, V) r w* 1+V2 0 (14)

C

A

B D

Taking b/L=0.1 and w* =4/7, the viscosity was 0 determined numerically as a function of the dimensionless shear rate V. Fig. 3 clearly shows the shear thinning behaviour of the suspension associated with the shear-induced deformation of viscoelastic spheres. At high shear rates (V&1), the relative viscosity reaches a constant value m (w, w* , b/L) r2 0 which decreases with the maximal particle elongation e (V2)=1/L in the flow direction. xx 6. Viscoelastic properties of red cells Red cells or erythrocytes are deformable discshaped cells of 8 mm diameter and 2 mm thickness which consist of an external biological membrane enclosing a Newtonian fluid of viscosity

84

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

m #6×10−3 Pa s containing haemoglobin and i salts. Microscopic observations and mechanical experiments demonstrate that the red cell membrane is easily deformed at constant area but greatly resists area dilation because of the strong hydrophobic interactions of the membrane lipids and the adjacent aqueous media [27,28]. Mechanical extension of the red cell membrane at constant area in a micropipette yields an elastic shear modulus G#6×10−6 N/m without dependence on the degree of cell deformation [29]. The swollen network of spectrin molecules weakly bonded to the internal lipid bilayer mainly accounts for the elasticity of the membrane since the lipid bilayer is in a fluid state. Experimental results from Stokke et al. [30] indicate that spectrin molecules free in solution or cross-linked into three-dimensional networks behave essentially like purely entropic springs and constitute a reversible ionic gel with non-linear viscoelastic properties [31]. A simple elastomer theory yields an elastic modulus G#3×10−6 N/m for the human erythrocyte spectrin network [31] in good agreement with experimental values. In a shear flow, isolated red cells rotate like a rigid ellipsoid at low shear stresses and behave in a manner similar to suspended fluid drops when subjected to shear stresses t>0.1 N/m2 [32–34]. Cells deformed into ellipsoids then align at a constant angle to the direction of the flow with a tank-treading motion of the membrane about the cell interior scaling as the shear rate (tank-tread frequency f#c/20 [35]). Red cell deformation further increases with the shear stress and reaches a maximum elongation at high shear rates [36 ]. The backscattering diagram of sheared red cell suspensions probed by a laser light indeed displays an angular asymmetry indicating a cell orientation at angles 20–35° with respect to the flow direction [37]. For hardened red cells, the scattering diagram remains symmetric because of the flip-flop motion and the random orientation state of rigid particles in a shear flow [37]. A Kelvin Voigt model may thus be used for describing the deformation-orientation of deformable red cells in a shear flow when suspended in a

viscous fluid. However, the discoid shape and the viscoelastic properties of the cell membrane result in differences between the periodic motion of red cells and liquid droplets in a shear flow [36 ]. The non-linear viscoelastic properties of the spectrin network influences the relaxation time of the membrane. The deformation of a red cell aspired into a micropipette indeed exhibits a timedependent behaviour [38]. The relaxation time ranging from 0.1 to 0.01 s decreases when increasing the deformation rate, which indicates a change in membrane surface viscosity m and a shear m thinning behaviour of the spectrin network during rapid deformation. Since the tank-tread motion frequency of the membrane scales as the shear rate [35], then the relaxation time h(c) of the red cell in a shear flow may be written as:

AB

AB

c n c n m (c) 0 with m =m 0 h(c)= m =h 0 m c G c c

(15)

where h =m /G is a characteristic relaxation time 0 c for the spectrin network, m the characteristic surc face viscosity of the cell membrane close to the resting configuration, and n a scaling exponent. The relaxation time h further scales as the charac0 teristic shear rate c . The shape recovery of a red 0 cell expelled from a micropipette exhibits a single relaxation time h #h #0.1 s since the deformation r 0 rate is slow and the membrane structure remains close to the resting configuration of characteristic surface viscosity m #h G#6×10−7 N s/m (with c 0 G#6×10−6 N/m) [29,39]. In the following we take n=3/4 since the shear rate dependence of the viscosity obeys a power law with 0.7
85

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

flow then scales as m c0.5, in good agreement with 0 data from rheo-optical experiments [41]. When plotted as a function of the variable m c0.5, changes 0 in diffuse reflectivity resulting from red cell deformation-orientation in a shear flow indeed lie on a single curve without significant dependence upon the viscosity of the suspending fluid [41]. A semi-empirical law proposed by Quemada [42] and often used to describe the blood rheology further involves a structure parameter depending on the variable (c/c )0.5, in good agreement with a c scaling exponent p=0.5 (c is a characteristic shear c rate related to shear-induced changes in the microstructure of the suspension). The parameter L involves an effective viscosity m accounting for the overall viscous dissipation p in the sheared red cell. Since the viscous dissipation rate in the cell interior dominates over the viscous dissipation rate in the cell membrane [43], the effective viscosity m is close to the viscosity m of p i the intracellular fluid and we may consider the approximation L#l=m /m . i 0

7. Viscosity of unaggregated red cell suspensions Eqs. (14)–(16) together with the condition L= l give a non-linear expression for the relative shear

viscosity of deformable red cell suspensions: m (w, V)= r

1−w

C

1−

w w* 0

A

B D

bl−1(h c)p −1 2 0 1+m (w, V) r 1+(h c)p 0 (17)

The relative viscosity m (w, w* , b/L) of red cell r2 0 suspensions at high shear rates reduces to: m

r2

=

1−w

C

1−

w

D

2

(18)

(1+m bl−1)−1 r2 w* 0 We have measured the apparent shear viscosity of unaggregated normal red cell suspensions in a Couette geometry (Contraves low shear 30 viscometer, controlled temperature 20±0.2°C ). The whole blood of healthy donors was centrifuged three times (10 min at 3000 rpm) and then washed each time in saline solution (ionic strength 150 mM, pH#7). Hardened red cells were prepared by suspending one volume of washed cells in 10 volumes of 2% glutaraldehyde saline solution for 60 min at room temperature. Finally, normal or hardened red cells are suspended in saline solutions. The particle volume fraction ranging from 0.15 to 0.55 was adjusted by microhematocrit technique. The calibration of the rheometer was performed

Fig. 4. Normal red cells in physiological saline solution. Resting shape ( left picture). Deformation-orientation in shear flow (right picture).

86

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

Fig. 5. Maximum packing volume fraction w* as a function of 0 particle volume fraction for normal ($) and hardened red cells (#). The structure parameter w* is derived from viscosity meas0 urements at low shear rates and the reference viscosity law (7).

with Newtonian silicone oils. The sample was stirred at a high shear rate (130 s−1) before each measurement and the shear was further reduced to the value at which the measurement was performed. A new equilibrium was reached after some time and the viscosity was calculated from the steady state torque readings. In repetitive experiments with a fresh sample, the procedure gives reproducible data with an error of about 4% for the relative shear viscosity. The deformation-orientation of deformable cells at high shear rates (c>10 s−1) induces a shear thinning rheological behaviour [44] ( Fig. 4). The maximum packing concentration w* (w) is esti0 mated from the reference Eq. (7) and the viscosity data in the low shear regime. At low particle volume fraction, hardened or deformable red cells are randomly distributed and then the maximum packing fraction is close to the random isostatic packing w* #4/7 ( Fig. 5). Above a critical volume 0 fraction w #0.2, the virtual spheres encircling the c random collection of non-spherical particles interpenetrate, thus inducing a structuration of the suspension and an increase of the maximum packing fraction w* (w #w* 3V/4pR3#0.2 where 0 c 0 V#95 mm3 is the red cell volume and 2R#8 mm the maximum dimension of particles) (Fig. 5). For particle volume fraction above the random dense packing w* #0.6, hardened red cell suspend sions no longer flow. Suspensions of normal red

Fig. 6. Relative shear viscosity m (c, w)=m(c, w)/m against the r 0 shear rate for normal red cells suspended in saline solution ($) ( m =0.9 cP) with a particle volume fraction w=0.15, 0.25, 0.35, 0 0.45 or w=0.55. The solid curves are calculated from the relation (17) with p=1/2, h =5×10−2 s, b=0.3, l=m /m and 0 i 0 m =6 cP. The maximum packing fraction w* (w) is derived from i 0 the shear viscosity of unaggregated particles in the low shear regime (Fig. 5).

cells remain fluid at high particle concentration since cell deformability allows membrane bending and motion of near contact particles (Fig. 5). Cell deformability favours the structuration of concentrated suspensions, resulting in an increase of the maximum packing fraction in the low shear regime. Fig. 6 shows the shear rate dependence of the relative viscosity for unaggregated red cells in saline solution. The rheological Eq. (17) with n= 3/4 ( p=2−2n=1/2), b=0.3 and h =0.05 s pre0 dicts the observed shear thinning behaviour. Shearinduced deformation and orientation of red cells in shear flow result in a viscosity decrease. The inflection point occurs for shear rates c #20 s−1 0 in relation to the characteristic relaxation time h #1/c #0.05 s of the spectrin network. 0 0 The shear viscosity reaches an asymptotic value m at high shear rates as predicted by the rheologir2 cal model. For a red cell volume fraction w=0.45, the non-linear Eq. (18) with w* #0.65, b=0.3 and 0 l=m /m shows a considerable decrease of the high i 0 shear relative viscosity when increasing the viscosity m of the suspending fluid, in very good 0 agreement with the data of Chien [45] (Fig. 7).

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88

Fig. 7. Relative shear viscosity m ( m ) in the high shear rate r2 0 regime of unaggregated red cell suspensions (particle volume fraction w=0.45) versus the viscosity m of the suspending fluid. 0 Experimental results ($) from Chien [45] are compared to the theoretical predictions given by Eq. (18) with w* =0.605, 0 b=0.3, l=m /m and m =6 cP. i 0 i

8. Conclusion The motion of deformable particles in a shear flow was described by a Kelvin Voigt model. The shear viscosity of a collection of deformable particles was derived from the concept that the deformation-orientation of particles enhances the maximum packing fraction of particles. A selfconsistent theory using a reference rheological law and an effective medium approximation then predicts the viscosity of concentrated suspensions of viscoelastic particles such as red cells in saline solution and allows information about the nonlinear viscoelastic properties of the particles.

References [1] T.G.M. van de Ven, Colloidal Hydrodynamics, Academic Press, New York, 1989. [2] D. Lhuillier, Phenomenology of hydrodynamic interactions in suspensions of weakly deformable particles, J. Phys. 48 (1987) 1887–1902. [3] J.M. Rallison, A. Acrivos, A numerical study of the deformation and burst of a viscous drop in an extensional flow, J. Fluid Mech. 89 (1978) 191–200. [4] R. Roscoe, On the rheology of a suspension of viscoelastic spheres in a viscous fluid, J. Fluid Mech. 28 (1967) 273–293. [5] D. Barthes-Biesel, Motion of a spherical microcapsule

87

freely suspended in a linear shear flow, J. Fluid Mech. 100 (1980) 831–853. [6 ] R.G. Cox, The deformation of a drop in a general timedependent fluid flow, J. Fluid Mech. 37 (1969) 601–623. [7] D. Barthes-Biesel, J.M. Rallison, The time dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251–267. [8] M. Bitbol, P. Mills, The deformation of a viscoelastic sphere in a time dependent shear flow, J. Phys. Lett. 45 (1984) L775–780. [9] P. Snabre, Rhe´ologie des suspensions concentre´es et agre´ge´es de particules de´formables. Application a` la suspension sanguine. The`se d’e´tat, Universite´ Paris VII, 1988. [10] P. Mills, P. Snabre, The fractal concept in the rheology of concentrated suspensions, Rheol. Acta 26 (1988) 105–108. [11] P. Snabre, P. Mills, Rheology of weakly flocculated suspensions of rigid particles, J. Phys. III 6 (1996) 1811–1834. [12] R.L. Powell, External and internal streamlines and deformation of drops in linear two-dimensional flows, J. Colloid Interface Sci. 95 (1983) 148–162. [13] D. Barthes-Biesel, H. Sgaier, Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flows, J. Fluid Mech. 160 (1985) 119–135. [14] S. Torza, R.G. Cox, S.G. Mason, Rotation of viscoelastic particles in a linear shear flow, J. Colloid Interface Sci. 38 (1972) 395–415. [15] G.K. Batchelor, J.T. Green, The determination of the bulk stress in a suspension of spherical particles to order c2, J. Fluid Mech. 56 (1972) 401–427. [16 ] A.P. Shapiro, R.F. Probstein, Random packing of spheres and fluidity of monodisperse and bidisperse suspensions, Phys. Rev. Lett. 68 (9) (1992) 1422–1425. [17] S.H. Maron, A. Levy-Pascal, Rheology of synthetic latex, J. Colloid Sci. 10 (1955) 494–503. [18] I.M. Krieger, Rheology of monodisperse lattices, Adv. Colloid Interface Sci. 3 (1972) 111–136. [19] C.G. De Kruiff, E.M.F. van Iersel, A. Vrij, Hard sphere colloidal dispersions: Viscosity as a function of shear rate and volume fraction, J. Chem. Phys. 83 (9) (1985) 4717–4725. [20] F. Gadala-Maria, A. Acrivos, Shear-induced structure in a concentrated suspension of spheres, J. Rheol. 24 (1980) 799–814. [21] F. Parsi, F. Gadala-Maria, Fore-and-aft asymmetry in a concentrated suspension of spheres, J. Rheol. 31 (1987) 725–732. [22] J.F. Brady, G. Bossis, Stokesian dynamics, Ann. Rev. Fluid Mech. 20 (1988) 111–157. [23] L. Petit, Ecoulement et mise en structure de suspensions macroscopiques, Ann. Phys. Fr. 16 (1991) 155–191. [24] D. Barthes-Biesel, V. Chim, Constitutive equation of a dilute suspension of spherical microcapsules, Int J. Multiphase Flow 7 (1981) 473–493. [25] D. Quemada, Phenomenological rheology of concentrated dispersions: I. Clustering effect and the structure-depen-

88

[26 ] [27] [28] [29] [30]

[31]

[32]

[33]

[34] [35]

[36 ]

P. Snabre, P. Mills / Colloids Surfaces A: Physicochem. Eng. Aspects 152 (1999) 79–88 dent packing fraction, J. Theor. Appl. Mech. Nspec. (1986) 267–288. P. Snabre, M. Bitbol, P. Mills, Cell disaggregation behavior in shear flow, Biophys. J. 51 (1987) 795–807. E.A. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton, FL, 1980. E.A. Evans, R.E. Waugh, Erythrocyte Mechanics and Blood Flow, Alan R. Liss, New York, 1980, pp. 31–56. R. Waugh, E.A. Evans, Thermoelasticity of red blood cell membrane, Biophys. J. 26 (1979) 115–132. B.T. Stokke, A. Mikkelsen, A. Elgsaeter, Some viscoelastic properties of human erythrocyte spectrin networks endlinked in vitro, Biochim. Biophys. Acta 816 (1985) 111–121. B.T. Stokke, A. Mikkelsen, A. Elgsaeter, The human erythrocyte membrane skeleton may be an ionic gel. I. Membrane mechanochemical properties, Eur. Biophys. J. 13 (1986) 203–218. H.L. Goldsmith, J. Marlow, Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions, Proc. Roy. Soc. London, Ser. B 181 (1972) 351–384. T.M. Fisher, H. Shmid-Schonbeim, Tank treading motion of red cell membrane in viscometric flows. Behavior of intra cellular and extra cellular markers (with film), Blood Cells 3 (1977) 351–365. M. Bitbol, Red cell orientation in orbit C=0, Biophys. J. 49 (1986) 1055–1068. T.M. Fisher, M. Stohr-Liesen, H. Schmid-Schonbeim, The red cell as a fluid droplet. Tank tread like motion of the human erythrocyte membrane, Science 202 (1978) 894. T.M. Fisher, M. Stohr, H. Schmid-Schonbeim, R.B.C.

[37]

[38]

[39]

[40]

[41]

[42]

[43] [44]

[45]

microrheology: comparison of the behaviour of single R.B.C. and liquid droplet in shear flow, AI.Ch.E. Symp. Ser. 182 (1978) 38–45. A.H. Gandjbakhche, P. Mills, P. Snabre, Laser light technique for the study of orientation and deformation of viscoelastic particles in a Couette flow, Appl. Opt. 33 (1994) 1070–1079. S. Chien, K.L.P. Sung, R. Skalak, S. Usami, A. Tozeren, Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane, Biophys. J. 24 (1978) 463–487. R.M. Hochmuth, P.R. Worthy, E.A. Evans, Red cell relaxation and the determination of membrane viscosity, Biophys. J. 26 (1979) 101–114. E. Puchelle, J.M. Zahm, D. Quemada, Rheological properties controlling mucociliary frequency and respiratory mucus transport, Biorheology 24 (1987) 557–563. A.H. Gandjbakhche, P. Mills, P. Snabre, J. Dufaux, Aggregation and deformation of red blood cells as probed by a laser light technique in a concentrated suspension, Soc. Photo-opt. Inst. Eng. 2136 (1994) 90–103. D. Quemada, An overview of recent results on rheology of concentrated colloidal dispersions, Progr. Colloid Polym. Sci. 79 (1989) 112–119. T.M. Fisher, On the energy dissipation in a tank-treading human red blood cell, Biophys. J. 32 (1980) 863–868. A.L. Copley, C.R. Huang, R.G. King, Rheogoniometric studies of whole human blood at shear rates from 1000 to 0.0009 s−1, Biorheology 10 (1973) 23–28. S. Chien, Present state of blood rheology, in: Hemodilution. Theoretical Basis and Clinical Application, Int. Symp. Rottach, Egern, 1971.