Stress chains formation under shear of concentrated suspension 

Didier Lootensb, Nicos S. Martysa, William Georgea, Steve Satterfielda, Pascal Hébraudc

a NIST, Gaithersburg, MD, USA. b

Sika Technology AG, CH-8048 Zürich, Switzerland 

c

IPCMS, UMR 7504, F-67034 Strasbourg, France

Results comparing experiments on a model system of mono-disperse silica-particles with the numerical simulation of a 

highly concentrated suspension of spherical particles subject to a constant rate of strain are presented. Giant fluctuations 

of the shear stress and the first and second normal force difference are studied. Stress chain formation and evolution 

under shear are visualized in order to make the relation between the stress fluctuations and the suspension microstructure. 

Keywords: Concentrated suspension, shear thickening, jamming, chain force, normal force.

PACS: 82.70.Dd, 8360.Rs, 83.80.Hj

INTRODUCTION 

 When submitted to a low stress, concentrated suspensions show an elastic response. By increasing the imposed 

stress, suspensions begin to flow with a plastic deformation. At higher shear stress they can shear thicken or even 

jam and stop flowing. Although these phenomena are of great importance for engineering applications such as 

extrusion or pumping, the basic understanding is very weak [1]. The simplest model to explain shear-thickening has 

been proposed by Boersma et al [2]: shear thickening appears when the hydrodynamic compressive forces are of the 

same order of magnitude as repulsive - either Brownian, steric or electrostatic - forces. The jamming transition can 

be continuous or discontinuous. In the latter case, when the suspension is forced to flow, we observe a fluctuation of 

the applied stress above a base line [3]. Two flow states are present: one plastic, which is the same as that in the 

plastic zone and the second flow state, where the stress is higher. The organization of the particles can be followed 

by experimental techniques such as confocal microscopy or neutron scattering [3-5]. While these techniques are 

used to follow the global organization of the particles under shear, they can not give information on the stress 

distribution in the suspension. Therefore, we turn to numerical simulation, using a dissipative particle dynamics 

(DPD) model [6], to study the organization of colloidal suspension under shear. The fluctuations of the tangential 

and normal stress are correlated to experimental data. The contacts between the particles are used to build a map of 

the stress occurring during the shear process. We can observe the formation of stress chains when highly 

concentrated suspensions are sheared. 

JAMMING: EXPERIMENTAL VIEW 

 We use concentrated suspensions of silica particles of controlled roughness. These are prepared according to a 

Stöber-like synthesis [7] with an average size, measured by dynamic light scattering, of 1 μm. Rheological 

measurements are made with a con/plate (Rheometrics* RFSII, 25 mm diameter). Confocal microscopy images are 

made with a Leica (TCS-4D, oil immersion 100 objective, equipped with an Ar-Kr laser), where the sample is 

sheared with a homemade cone-plate shear cell (radius 50 mm). 

 A concentrated suspension is sheared in a cone/plate geometry in order to measure the normal stress applied by the 

sheared suspension. We can then measure indirectly the anisotropy and spatial organization of the particles during 

the flow. This measurement is, however, difficult to perform: the jamming transition appears either at a high shear 

rate where the suspension is ejected from the gap or at a lower shear rate when the suspension is concentrated then a 

rapid drying occurs. We use rough silica particles, where the jamming transition occurs at a lower volume fraction 

and lower shear rate. The normal force fluctuates in phase with the tangential stress. The normal force fluctuations 

are negative at low volume fraction and low shear rate. At higher volume fraction, these are positive (see Figure 1). 
0 10 20 30 40 -100

-50

0

50

100

150

100

150

200

250

γ (s-1

)

σ, N1 (Pa)

TIME (s)

42.5%

0 10 20 30 40 -100

-50

0

50

100

100

150

200

250

43%

TIME (s)

σ, N1 (Pa) 

γ (s-1

)

0 10 20 30 40

0

100

200

300

400

6

8

10

12

14

16

TIME (s)

σ, N1 (Pa)

γ (s-1

)

44%

FIGURE 1. Evolution of the tangential stress and normal stress as a function of the imposed shear rate and for different volume 

fractions (from left to right 42.5 %, 43 % and 44 % w/w). Data from reference 5. 

Two images of the suspension, at rest, where the particles exhibit a liquid-like order, and after shear, above the

jamming transition, where the particles are more heterogeneous, are given in Figure 2, as well as the Fourier

transform of the centers of the particles. The volume fraction of the particles is 50 %. In the shear regime, where

crystallites are observed, the suspension is highly shear thickening and becomes dilatant. However, as we are in the 

high concentration regime, we are not able to characterize the orientation of particle contacts. nm nm

A B

FIGURE 2. Confocal images of suspensions at rest (A) and just after shear above the jamming transition (B). The Fourier 

transforms of the centers of the particles are on the left. Particles are in a mixture of water/glycerol at 23/77 w/w. 

NEAR-JAMMING: SIMULATION RESULTS 

The simulation approach used in this study is based on a hybrid dissipative particle dynamics approach (DPD) [6].

While interstitial fluid is modeled using DPD particles, the model explicitly incorporates lubrication forces when the 

particles are very close. For this paper, we consider a mono-size sphere suspension with solid volume fraction equal

to 51 %. In the infinite Peclet Number regime, such systems have been shown to jam at small strains. The simulation

approach used here can produce a jamming transition. However, as a consequence of modeling an interstitial fluid,

with the compressibility of water, the usual jamming of a dense hard spheres suspension can be suppressed at lower 

shear rates. This effect may be controlled, allowing us to probe flow regimes very close to the jamming transition. 

Similar fluctuations are observed with simulation as represented in Figure 3. Here giant fluctuations are found in

the shear stress and first and secondary normal stress differences [1]. The first and secondary normal stress 

differences are generally positive, however, they occasionally showed negative fluctuations (not shown). 

FIGURE 3. Shear stress (green), and first (red) and second (blue) normal stress differences. Results shown are normalized to the 

solvent fluid viscosity times shear rate. 

60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

nm ujam zoomée (sur 900 particules) 500 450 400 350 300 250 200 150 100 50 0 0 50 100 150 200 250 300 350 400 450 500 60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

nm jam zoomée (sur 980 particules) 500 450 400 350 300 250 200 150 100 50 0 0 50 100 150 200 250 300 350 400 450 500
Figure 4. shows evidence of the giant stress fluctuation in a highly correlated formation of stress chains 

FIGURE 4. Stress chains visualization. Here, the suspension is straining to the right at the top and to the left on the

bottom. The images are thresholded to only show the largest stresses with yellow indicate of the highest values. Note the lines 

are directed along the compression quadrant (oriented towards the bottom right). The image on the right is taken shortly after

that on the left, indicating a minor change in microstructural organization at the onset of the stress jump. 

In summary, we have shown, by computer simulation, evidence of transient stress chains that are oriented in the 

compression flow quadrant and rapidly reorganize themselves. These chains are responsible for the very large stress

fluctuations. As found in experiment, these giant stress fluctuations appear near the jamming transition.

ACKNOWLEDGMENTS 

We would like to gratefully acknowledge support from the Virtual Cement and Concrete Testing Laboratory

consortium (VCCTL). The flow simulations were performed under award SMD-05-A-0129, “Modeling the 

Rheological Properties of Suspensions: Application to Cement Based Materials,” for NASA’s National Leadership

Computing System Initiative on the “Columbia” supercomputer at the NASA Ames Research Center. 

REFERENCES 

1. H. A. Barnes, Shear-thickening (dilatancy) in suspensions of non-aggregating solid particles dispersed in Newtonian fluids, 

Journal of Rheology 33, 1989 pp. 329-366. 

2. W. H. Boersma, J. Laven and H. N. Stein, Shear thickening (dilatancy) in concentrated dispersions, AIChE 36, 1990, pp.

321-332. 

3. D. Lootens, H. Van Damme and P. Hébraud, Giant stress fluctuations near the jamming transition, Phys. Rev. Lett. 90(17), 

2003, 178301. 

4. B.J. Maranzano and N.J. Wagner. Flow-small angle neutron scattering measurements of colloidal dispersion microstructure 

evolution through the shear thickening transition. Journal of Chemical Physics, 117, 22, 2002. pp. 10291-10302. 

5. D. Lootens, H. van Damme, Y. Hémar and P. Hébraud. Dilatant flow of concentrated suspensions of rough particles. Physical 

Review Letters, 95, 268302, 2005. 

6. N. S. Martys, J. Rheol, 49(2), 401-424 (2005). 

7. W. Stöber and A. Fink, J. Colloid Interface Sci. 26, 62-69.(1968).

* Certain commercial products are identified in the paper to specify the materials used and procedures employed. In no case 

does such identification imply endorsement by the National Institute of Standards of Technology, nor does it indicate that the 

products are necessarily the best available for the purpose.