A generalized frictional and hydrodynamic model of the dynamics and structure of dense colloidal suspensions ~o M. Maiac) Arman Boromand,a) Safa Jamali,b) Brandy Grove, and Joa Department of Macromolecular Science and Engineering, Case Western Reserve University, 2100 Adelbert Road, Cleveland, Ohio 44106-7202 (Received 28 September 2017; final revision received 6 April 2018; published 1 June 2018)
Abstract Controlling the structure and the rheological properties of colloidal suspension is essential in numerous applications to control the phenomenon known as shear-thickening. Here, we report on the nontrivial interplay between hydrodynamic and frictional interactions using mesoscopic characterization of semidense, u ¼ 0.48, and dense, u ¼ 0.58, colloidal suspensions. Monitoring computationally both rheology and microstructure of these complex fluids under an external deformation, we show that in the semidense regime the interactions in colloidal suspensions are dominated by hydrodynamics while the fraction of frictional bonds remains negligible and consequently the size of frictional clusters remain small. For these systems, the normal stresses remain negative and large. For dense suspensions, frictional forces are necessary to capture discontinuous shear-thickening (DST); however, the microstructure and rheology are sensitive to the level of roughness of colloidal particles. Furthermore, we show that the frictional bonds in the dense and semidense regime follow the same statistics as random networks introduced by Erd}os–Renyi where the presence of frictional bonds in dense suspensions promotes formation of a Giant percolated cluster. We show that for both semidense and dense regimes hydroclusters initially form, within which the frictional contacts nucleate. In the case of dense suspensions these nuclei grow and percolate and form a frictional network. We show that the presence of such a percolated C 2018 The Society of Rheology. https://doi.org/10.1122/1.5006937 cluster is also necessary for DST to occur. V
I. INTRODUCTION Colloidal suspensions are ubiquitous in many industrial applications, displaying a wide range of non-Newtonian behaviors. These are known to be determined through the interplay between Brownian (thermodynamic) forces and viscous (hydrodynamic) interactions, as well as interparticle interaction potentials [1,2]. At low shear rates, the main contribution of the stress tensor comes from Brownian forces that are proportional to the N-body probability distribution of particles. Near equilibrium, the microstructure of colloidal particles evolves linearly with shear rate which results in pseudo-Newtonian behavior, g0(u), which diverges at the volume fraction u ¼ ug 0.58 [2–4]. Upon increasing shear rates, Brownian forces contribution drops and the microstructure cannot relax to the isotropic state. At the intermediate rates, the anisotropic structure and sublinear growth of the Brownian stress with respect to the external driving force result in the emergence of the shear-thinning in suspensions. At a particular shear rate (stress), however, shear-thinning behavior stops and the suspensions become shear-thickening [1,2,5]. Since the rate-dependency of the colloidal suspensions is determined through competition between Brownian stresses and the stresses generated by the flow (hydrodynamic
a)
Present address: Department of Mechanical Engineering and Material Science, Yale University, 9 Hillhouse, Mason Lab, New Haven, CT 06511. b) Present address: Department of Mechanical and Industrial Engineering, Northeastern University, 360 Huntington Ave., Boston, MA 02115. c) Author to whom correspondence should be addressed; electronic mail:
[email protected]
stresses), the nondimensional Peclet number is used to describe the flow behavior of suspensions defined as: Pe ¼ rh =rB ¼ ð6pg0 c_ Þ=ðkB T=a3 Þ. In a shear-thickening regime, there is a long-standing model based on the hydrodynamic interactions that predicts shear-thickening through the formation of hydro-clusters. In this model, the increase in the viscosity of suspensions is related to the presence of lubrication forces. At high volume fractions and high driving forces (Pe), the particles are forced together in the compressive direction under shear deformation. A pair of colloidal particles in the close proximity experience significant squeezing and tangential lubrication forces which are dissipative in nature (retard the relative motion of the pair). Consequently, this gives rise to the formation of long-lived clusters of particles under the flow (i.e., hydroclusters). These reversible mesoscopic structures can interact with the flow and increase the rate of energy dissipation, i.e., increase the viscosity of the suspension. This model has been confirmed though a numerous experimental studies based on rheo-optical measurements [6–9], neutron scattering [10–13], and stress cessation techniques[14], in addition to the computational studies using Stokesian Dynamics [15–19], Dissipative Particle Dynamics [20,21], and Smooth Particle Hydrodynamics [22]. However, computational studies based on this model can only predict a mild increase in the viscosity, or continuous shear-thickening (CST), compare to experimental studies and not the sometimes observed discontinuous shear-thickening (DST), since it does not consider the confining stress associated with particle stiffness, boundary rigidity, or surface tension in suspensions with free surface [23,24].
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Recently, an alternate theory based on frictional contact interactions in granular materials has been tested in describing the rheological properties of noncolloidal and colloidal suspensions. Early computational and theoretical studies based on this model showed an excellent performance in reproducing the experimental flow curve of suspensions in the DST regime. In this model, there exists a critical stress scale, rc / Frep a2 set by interparticle repulsive forces, Frep . If the stress in the system exceeds rc (independent of the volume fraction), lubricated contacts between colloidal particles in close proximity turn into surface-surface frictional contacts [25–27]. Recently, it has been conjectured that the transition from lubricated contacts to surface contacts also depends on the volume fraction as well as the applied stress. At low volume fractions, the contacts stay mainly lubricated regardless of the Pe/stress values while the surface contacts abate and remain transient. However, at higher volume fractions, the majority of the contacts change to frictional form and DST emerges reminiscent of the rigidity percolation in granular materials. Along these lines, recent experimental studies also suggested that manipulating the frictional contacts through shear reversal or orthogonal shear can indeed modify the rheological properties of colloidal and noncolloidal suspensions indicating the importance of frictional contacts to address the complex nature of the suspension rheology, holistically [28,29]. While this theory can provide remarkable agreement with experimental studies on the viscosity of colloidal [30,31] and noncolloidal suspensions [25,26,32–34] by capturing the discontinuous transition in the DST regime, it also predicts a positive first normal-stress difference, N1, due to the breaking of the fore-aft symmetry at high stresses [26]. Computationally, for the first time it was shown there exists a transition from negative (at low stresses) to positive (at high stresses) N1 for noncolloidal particle [26]. It has the indication that at high stresses anisotropic force chains form similar to the well-known behavior of dilatant materials such as frictional granular systems under shear. Experimentally, it has been a controversial topic due to the difficulties involved in measuring reliable stress signals in the dense suspensions and both negative and positive values for the N1 has been reported [31,35–37]. Another issue with incorporating friction-based models to colloidal suspensions is the lack of understanding of frictional forces at nanoscopic scales and the way one can measure these quantities [38–40]. In addition, how the frictional network forms and their statistics remain poorly understood and the mechanism by which the network of particles can be stabilized at the elevated shear stresses remains unexplored. Therefore, when taken as a whole, the existing experimental and computational data strongly suggest that some combination of the long-standing view of hydrocluster formation and of the frictional model may be able to explain the full dynamics, i.e., shear and normal stresses, in colloidal suspensions. In this paper, first we investigate computationally using Core-Modified Dissipative Particle Dynamics, CM-DPD, the rheological properties of dense and semidense colloidal suspensions with two level of roughness, i.e., smooth (l ¼ 0.01) and rough (l ¼ 1.0). We identify the
hydrocluster formation is a precursor of any thickening transition both CST and DST through analyzing both frictional and lubricated bonds. In the case of CST, the role of frictional networks remains negligible since all the bonds are transient in nature and cannot be stabilized under the flow. For dense suspensions, the frictional network can in fact percolate and form a space spanning giant cluster. The statistics of the giant cluster can be modeled using a simple random network theory by Erd}os–Renyi. Defining a nondimensional number based on the frictional bond distribution, we have found that indeed formation of giant cluster is concomitant with DST. The physical interpretation of this nondimensional number is shown to be related to the topological characterization of the giant cluster, which can take the form of a soft Gaussian cluster in the case of smooth particles or rigid elongated structures in the case of rough particles. The present work is organized in the following order: first, we introduce the general description of dissipative particle dynamics and the modifications required to adjust the model to study dense frictional suspensions through introducing short range hydrodynamic interactions and frictional forces. In Sec. II, we show the results on rheological and morphological characterization of semidense and dense suspensions and introduce the idea of a random network theory. A. Model description Dissipative Particle Dynamics, DPD, is a particle-based simulation method introduced by Hoogerbrugge and Koelman [41] to study microscopic hydrodynamic description in colloidal dispersions. Similarly to other force-based particle models, DPD solves the equation of motion for each DPD particle. At every time step, the position (ri ), velocity (vi ), and forces (conservative, dissipative, and random) acting on the DPD particles are calculated and updated according to the following equation: Conservative forces:FC 2
mi
d ri ¼ dt2
X
zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ aij xCij ðrij Þeij
Dissipative forces:FD
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ cij xD ij ðrij Þðvij eij Þeij
j2Ni Random forces:FR
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ Hij þ rij xRij ðrij Þ pffiffiffiffiffi eij ; Dt
(1)
where mi is the particle mass, aij is the maximum repulsion coefficient, cij is the coefficient of friction, rij is the strength R of the random force, and xCij ; xD ij ; xij are weight functions. The conservative force is a soft repulsive potential which allows time steps to be considerably higher than those in molecular dynamics. Dissipative force represents the effect of viscosity in the system and depends on both the relative positions ðrij Þ and velocities ðvij Þ of the interacting particles. The stochastic nature of the random force, which represents the effect of thermal fluctuations in the system, is set by Hij , a white noise with Gaussian statistics, i.e., hHij i ¼ 0; hHij ðtÞHkl ðt0 Þi ¼ ðdik djl þ dil djk Þdðt t0 Þ. All forces are truncated, so they reduce to zero at a cutR off distance, rc set by the weight functions xCij ; xD ij ; xij
MODELING OF DENSE COLLOIDAL SUSPENSIONS
xCij ðrij Þ ¼ xRij ðrij Þ ¼
8 < :
rij 1 rc
rij < rc
(2)
rij rc :
0
Espanol and Warren [42] using fluctuation-dissipation theorems showed that the strengths ðcij ; rij Þ and weight functions R ðxD ij ; xij Þ characterizing dissipative and random forces should satisfy the Fokker-Planck equation 2 9 D xij ¼ xRij = : (3) 2c k T ¼ r2 ; ij B
ij
The constrains keep the system Temperature constant and considering constant Number of particles and constant Volume in our simulation it results in the NVT description for the particle ensemble. Out of equilibrium, however, the DPD thermostat may fail in keeping the temperature constant due to the numerical instabilities and alternative thermostats may be required [43,44]. Due to the conservation of both mass and momentum, DPD can provide a suitable framework to study problems involving hydrodynamic interactions. 1. Frictional CM-DPD
a. Core-modified interactions in dissipative particle dynamics. Although DPD in the original formulation was developed and used to study multiphasic systems such as suspension under equilibrium, it has some limitations when the system is forced out of equilibrium. First, the extensive span of length and time scales in colloidal suspensions results in erroneous prediction of the rheological properties if one assumes the same size for solvent particles and colloidal particles [45]. To introduce size difference between colloidal and solvent particles, some studies suggested freezing DPD particles [46] into a cluster of frozen particles as a representation of a colloidal particle. This methodology is not computationally efficient and also violates conservation of momentum due to the freezing procedure. Also, this method induces some numerical artifacts, such as density fluctuations, which require implementation of extra complex boundary conditions [47]. Another way to address this issue is by using particles with different sizes and masses in order to induce size difference in the colloidal suspension systems, a path initially proposed by Whittle and Travis [48]. We have recently showed that this scheme reproduces correctly the rheological properties of colloidal suspensions and gels [20,49]. In this case, Eq. (1) needs to be modified to account for short range hydrodynamic interactions and also core force interactions between colloidal particles to eliminate any unphysical overlap of colloidal particles under equilibrium and out of equilibrium and also account for the frictional forces and their effect on the dynamics of the system
Fi2fColloid;Solventg
zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ X i2fSolventgj2fSolventg R ¼ FCij þ FD ij þ Fij |fflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} j2Ni
i2fColloidgj2fSolventg
þ
Fel ij
Coreðn;tÞ
þ FH ; ij þ Fij |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} i2fColloidgj2fColloid g
(4)
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where the conservative, dissipative, and random forces are the same as explained in Eq. (1) and either apply between a pair of solvent particles or a solvent and a colloidal particles. The colloid-colloid interactions in our model consist of first, electrostatic repulsive force, ðFel ij Þ which is modeled as an exponentially decaying function with the maximum repulsion strength at the surface-surface distance equal to zero and a characteristic length scale similar to Debye length in real suspensions. Second, hydrodynamic interactions, ðFH ij Þ which account for circumstances where the interparticle distance between two colloidal particles is smaller than the size of a solvent particle. To retrieve the correct short-range hydrodynamic description in the squeezing mode, we included a pair drag term, fijH ¼ 3pg0 R2 =2hij similar to the term suggested by Ball and Melrose [50], where ðg0 Þ is the solvent viscosity and ðRÞ is the size of a colloidal particle. It should be noted that only the squeezing mode is considered due to the soft convergence of the tangential mode, i.e., logarithmic vs inverse mode. Since the lubrication force diverges to infinity at zero surface-to-surface distances ðhij Þ, a cut-off (d/R ¼ 0.00015) is considered to truncate the force and the final form of the hydrodynamic force included in our simulation is
H FH ij ¼ fij ðvij •eij Þeij ;
8 > 3pg0 R2 > > < 2d fijH 3pg 2 > 0R > > : 2hij
hij < d (5) hij d:
Finally, the Core potential ðFCore Þ is a semihard potential ij used to eliminate any nonphysical overlap between colloidal particles and is modified to include the frictional forces through combination of normal (n) and tangential (t) forces. The normal force is proportional to the overlap size, d ¼ jri rj j Ri Rj between two colloidal particles where ri,j are the position of the particles centers and Ri,j are the particles radii. Due to the elastic deformation, a large elastic force emerges to remove the overlap between the particles. This normal force can be calculated as Fnij ¼
"
# kn jdj ðcn ðvi vj Þ nij Þ nij : |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Elastic
(6)
Dissipative
We are using the same force law with kn ¼ 4 104 for particles with surface-surface distance 0 < hij < d. This consideration is based on our previous model that could explain the CST using the elasto-hydrodynamic model [21]. At the point of surface-surface contact, hij ¼ 0, tangential component of the core force interactions follows the Coulomb interaction rule which determines whether a formed contact is in the static or sliding mode. The appearance of the tangential forces requires the relative tangential velocity between the contacting surfaces to be nonzero. One can calculate the relative tangential velocity at contact as vtij ¼ vij ðnij :vij Þnij ;
(7)
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where the relative velocity vij is calculated as vij ¼ ðvi vj Þ þ ðnij xi ÞRi þ ðnij xj ÞRj ;
n0ij ¼ (8)
where v is the particle velocity, nij ¼ ðri rj Þ=jri rj j is the unit vector connecting the particles centers, and x is the angular velocity of the particle. In the static regime, the frictional forces, static friction keeps the tangential component of the velocity vector equal to zero. The Coulomb law dictates that the sliding mode is activated as soon as tangential force is equal to the static friction coefficient multiply by the normal force. After activation of the sliding mode, the friction forces decrease and stay constant. One can summarize the coulomb law as (
fijStatic ls fN fijSliding ¼ ld fN
vtij ¼ 0 vtij 6¼ 0;
ld < ls :
(9)
To account for the discontinuity in the velocity, one can define a tangential spring with a length and stiffness which follow and satisfy the Coulomb law. The former evolves in time based on the tangential velocity as n0ij ðtÞ ¼
ðt t¼ti
vtij ðt0 Þdt0 ¼ nij þ Dtvtij ðt DtÞ;
(10)
where n0ij ðtÞ is the temporal length of the tangential spring which is created at the initial time ti and evolves through time proportional to the instantaneous tangential velocity vtij ðtÞ. At t ¼ ti when the contact is formed the tangential spring length is considered to be zero. For dense suspension or granular materials near jamming transition, two particles in contact can stay in contact for a long time. This can cause a problem when one is using Eq. (10) where the reference frame for the contact can change. To account for this case, one needs to always calculate the component of the spring displacement vector in the instantaneous tangential frame. This can be performed by rotating the reference frame to the instantaneous coordinate using the following equation: nij ðtÞ ¼ n0ij n0 •nij nij ;
(11)
where n0ij is the old spring displacement vector from the last iteration. To calculate the change in the tangential spring, one needs to calculate the test force, ft f t ¼ kt nij ðtÞ ct vtij ðtÞ;
ft ¼ jf t j;
(12)
where kt and ct are the tangential spring stiffness and the tangential dissipation coefficient, respectively. If the modulus of the test force is less than the set value by the Coulomb law, Eq. (9), the static friction is active and the tangential spring length is updated using Eq. (10). However, if the test value is larger than ls fN sliding friction is activated and the tangential force will be equal to f t ¼ ðls fN Þt ct vtij and the length of the tangential spring is fixed to
1 ðl fN Þt; kt s
t¼
f t : ft
(13)
B. Simulation parameters In our model, the electrostatic repulsive force, Fel ij ¼ D eðhij =kÞ eij , is a pairwise interacting force with the strength D ranging between 200 and 5000 kB T=a and the interaction range is set to be, k ¼ 0:05a, where a is the particle radius. The pressure and stress in the system are evaluated using Irving-Kirkwood [51] relation to calculate the transport properties in addition to the complete state of the deviatoric stress tensor. The integration time step is Dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 107 m=kB T rc well below contact time scale to ensure that the dynamics remains independent of the numerical procedure and we ran the simulations over 109 steps. The shear deformation is applied using Lees–Edwards periodic boundary condition [52]. The size ratio of 1.4 between the colloidal particles is introduced to mitigate the unwanted crystallization occurrence in the simulations under flow [26,53,54] Two volume fractions / ¼ 0:48; 0:58 with 1221 and 1477 colloidal particles, respectively, are used with /L =/s ¼ 1:0, where /L ; /s are the volume fractions from large and small particles (a number of large particles used in the simulations are NL ¼ 326; 394). The suspensions are studied over a range of shear rates for the above repulsive strengths and for friction coefficients l ¼ 0:01; 1:0, representing smooth and rough particles, respectively.
II. RESULTS A. Rheological measures for the semidense and dense colloidal suspensions The general flow curve for colloidal suspensions can be broken into three distinct regimes: pseudo-Newtonian at low shear strength, i.e., Pe or stress, shear-thinning at moderate shear stresses, and finally shear-thickening at high Pe [1,2,35]. Figure 1 shows the shear viscosity, g, and first normal-stress difference, N1, as a function of stress for sheared frictional suspensions with volume fractions u ¼ 0.48 (䊊) and u ¼ 0.58 (䉭) (the comparison with experimental systems, as well as frictional Stokesian Dynamics simulations are presented in the Supplementary Material) [55]. We have presented the data for both rough (l¼1.0) and smooth (l¼0.01) particles, where l is the friction coefficient used in the Cundall and Strack friction model [Eq. (9)]. Both suspensions show a short shear-thinning regime at low Pe for the small repulsive interactions, 䉭*¼200½kb T=a [54–59]. Upon increasing the shear stress up to a critical value of 5 kb T=a3 rc ðD ¼ 200½kb T=a ; l ¼ 0:01Þ < 10 kb T=a3 , shear-thickening emerges, independently of volume fraction, as observed experimentally [30]. During the thickening transition, N1 remains negative and large for high stresses for the u ¼ 0.48 suspension, which is the hallmark of hydrocluster formation and dominance of hydrodynamic interactions [2,5,17,18,60,61]. For u ¼ 0.58, N1 follows the same behavior as for u ¼ 0.48 but for stresses higher than the second critical shear stress r0 c > 1000ðkb T=a3 Þ
MODELING OF DENSE COLLOIDAL SUSPENSIONS
FIG. 1. Rheological measures [viscosity (a) and N1 (b)] for semidense, A ¼ 0.48 and dense, A ¼ 0.58 colloidal suspensions as a function of flow stress. The repulsive potential strength is set to D* ¼ 200. The dashed area is between rc1 ¼ 5[kbT/a3] and rc1 ¼ 5[kbT/a3] þ 0.01 * 䉭*, consistent with the scaling proposed in frictional Stokesian Dynamics [54]. The data are measured for semi-dense, / ¼ 0.48 (circles) and dense, / ¼ 0.58 (triangles) suspensions of both smooth (filled symbols, l ¼ 0.01) and rough (open symbols, l ¼ 1.0) particles. The inset in panel (a) shows the individual components of the shear viscosity for rough semidense suspensions at different flow strengths (Pe). Inverse triangle: total viscosity, upward triangle (Hydrodynamic interactions þ Elasticity stresses arising due to the HI), circle (direct frictional contact) and square (Brownian contribution to the stress). The half-filled circles are our previous results based on frictionless DPD model containing only Elastohydrodynamic interactions for particle with rigidity kn ¼ 25 000.
> rc ðD ; lÞ this is reversed and N1 starts to grow, eventually becoming positive and very large, as expected from frictional contacts. It should be noted that this behavior is only observed for high friction coefficients (l > 0.5). This is consistent with the recent measurement of N1 for silica colloidal suspensions [31], as computational studies on noncolloidal suspensions [26]. The figure in the inset of Fig. 1(a) presents the reduced viscosity for the semidense rough particles alongside the different contributions from Brownian, elasto-hydrodynamic interactions (including the results from our previous work considering only the elasto-hydrodynamic interaction) and frictional
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contacts. From the figure, it is clear that HI are dominant and that frictional contacts have a trivial effect at this volume fraction. The dependency of the rheological measures (viscosity and the first normal stress) on the interparticle interaction, D*, is depicted in Figs. 2 and 3 for the semidense and dense suspensions, respectively. First, it can be argued from Fig. 2 that similarly to experimental studies, shear-thinning becomes more apparent for higher interparticle repulsion strengths [59,62], with the critical stress at which the shear-thickening emerges shifting to higher values. This can be explained through the fact that formation of either hydroclusters or frictional clusters becomes less frequent at lower stresses and a higher confining pressure is required to keep the clusters stable at higher repulsion strengths. Second, for the semidense suspensions (smooth and rough particles), N1 remains negative and large regardless of the interparticle interactions, while the onset stress at which N1 becomes negative shift to higher stresses for the same reason explained before. The magnitude of first normal stress, jN1 j, which is inversely proportional to the surface-surface distance of hydro-dynamically interacting colloidal particles, also decreases upon increasing the interparticle interactions. Figure 3 depicts the viscosity for the dense suspension for smooth (a) and rough (b) particles as a function of the measured stress. It should be noted that our simulations are rate-controlled and not stress-controlled and consequently, the stress is a measured quantity. For the u ¼ 0.58 suspension (Fig. 3), the frictional and hydrodynamic bonds both are negligible at low shear stress/Pe; at low shear stresses there are no strong driving forces to bring the particles into close proximity to form lubricated layers and/or frictional bonds. At higher stresses, lubricated layers form where large hydrodynamic stresses will initiate the shear-thickening transition. And viscosity starts to increase as g / rb with b 0:45 [Fig. 3(a)] for smooth and b ! 1:0ðb 0:8Þ [Fig. 4(b)] for the rough particles which are characteristics of continuous and DST transitions, respectively [24]. It should be noted that during breakage of the lubrication layers and formation of (transient) contact networks, a shearthickened state emerges where N1 3:0, it can be speculated (for the moment) that a percolating network can form and the rheology becomes dominated by the frictional response, with N1>0 and b ! 1:0ðb 0:8Þ, indicative of DST emergence (Figs. 3 and 4). In fact, formation of particles with coordination number hZpp i ¼ h2Nl =Ni 6:0 is the prerequisite for the formation of rigid structures in 3D for frictionless spheres; the presence of the frictional forces will further stabilize the structure even at the lower coordination numbers around 4.0 [63]. As mentioned before, since the direct contact can be formed upon overcoming the stress scale set by interparticle interactions, it is not surprising to note that frictional bonds (Nl) for both smooth and rough particles show similar trends when presented vs the stress in the system. Nl increases in both systems until it reaches the second stress scale, after which it reaches a plateau (Fig. 5), i.e., he thickened state. In this state, the number of frictional bonds differs between the smooth and the rough systems. One reason for this can be attributed to the percentage of the marginally stable frictional bonds that form and break continuously under flow at elevated stresses. For mechanically stable (MS) frictional packing of spherical particles, it has been shown previously that population of marginally stable bonds decrease in systems with higher friction coefficients. The average number of bonds formed under the same stress values is significantly higher for dense suspensions and a discontinuity is even observed in the case of very rough particles when the results are presented with respect to flow strength, Pe (Supplementary Material [55]). The critical stress at which the bonds start to be formed is almost independent of volume fraction, which is consistent with a series of experimental and computational studies, rc 10½kB T=R3 : From the rheological measurements, it can be seen that among all the systems studied above, only the dense suspension with very high frictional coefficients results in a discontinuity in the viscosity and positive normal forces. One can argue that formation of an average number of bonds per particle around 3.5 can lead to formation of a percolated network and
FIG. 5. Frictional bonds vs stress for the semi-dense (A) and dense (B) suspensions with repulsive interaction potential 䉭*¼200. The results for each volume fraction represent data for the both smooth (ⵧ) and rough (䉭) particles. The stress at witch stable frictional bonds form is independent of the volume fraction and shift to higher values for stronger repulsive interactions, (see the Supplementary Material for the dependency of the rc on 䉭*). Among all the cases only dense suspensions of rough particles show the prerequisite for formation of stable network with Nl>3.0.
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FIG. 6. Frictional (Nl: square) and lubrication bonds (NL: circle) vs stress for the semi-dense (a) and dense (b) suspensions with repulsive interaction potential 䉭*¼200. The results for semidense suspension (CST) depict different bond types for both smooth (black symbol) and rough (red symbol) particles, while in the dense suspension the bond number is presented for only rough particles (DST). To appreciate the effect of lubrication forces, the same simulations were performed for systems without lubrication interactions (open symbols, see Supplementary Material) it clearly shows that removing short-ranged HI results in the decrease in the frictional bonds network and consequently lower stress values (results are not shown for the stress).
stabilizing the frictional networks by turning a particle to a nonrattler. In fact, at high friction coefficients, the required coordination number for MS systems has been measure to be around z¼ 4.0, less that the isostatic value ziso¼6.0. From contact-counting, our system is close to the MS, which, in this case, means the network of the particles can be considered stable. It should be also noted that since we are performing our measurements under constant finite rate then we would expect our system to be always flowing. The proximity to the MS systems under flow can explain the stress discontinuity; however, we are exploring this point further. } s–Re nyi random network and frictional bonds 2. Erdo in semidense and dense suspensions
To analyze the validity of the assumption regarding the percolating network and the fact that stable frictional contacts can indeed form a percolated network, it is necessary to further analyze the frictional bonds and investigate the bond distribution and distribution of the frictional clusters formed at different flow strengths. To do so, we resort to the concept of a Giant component in the network of nods, i.e., colloidal particles, and links, i.e., frictional bonds. In the network theory, one of the most important quantities describing the robustness of a network is the size of its Giant component, which is the largest set of nodes that are connected to each other. In the rest of this section, we want to define a way to measure and identify the Giant component in the colloidal suspensions and bridging the gap between its formation and the rheological properties of the suspensions. From the network theory, Erd} os and Renyi [64] predicted the properties of a random network G(N,P) with N nodes and probability of formation of bonds equal to P, which are independent of each other. They have shown that for such a network, the probability of finding a node with k degree follows a Poisson distribution: PðkÞ ¼
ð NPÞk eNP : k!
(14)
For the independent bond formation processes, one can estiN mate the number of bonds as P. In our colloidal sus2 pensions network, we have already calculated the number of frictional bonds per particle as Nl . Equating these number results NðN 1Þ NNl P¼ 2 2
[ NP Nl :
(15)
Thus, it is possible to check if the frictional bonds follow the same statistics as the random network. In this case, Eq. (14) becomes PðkÞ ¼
ðNl Þk eNl : k!
(16)
From the average number of frictional bonds calculated before and using Eq. (3), we have calculated the theoretical P(k). Figure 6 depicts this quantity at different Pe regimes for the semidense and dense suspensions, respectively. The actual distributions of the frictional bonds that arise are also presented to confirm the validity of the network statistics. As can be clearly seen from the figure, for both systems with low and high frictional coefficients at two dense and semidense states, the network distribution can be captured using the Poisson distribution. Increasing the friction coefficient in both volume fractions results in broadening the distribution to higher coordinate number, i.e., frictional bond number. Comparing the semidense and dense suspensions at the same level of roughness for the particles shows that the distribution becomes broader, which is a consequence of the higher probability of formation of nodes with higher coordination at the same Pe values by increasing the volume fractions, i.e., decreasing the Euclidian distance between each node. The deviations from the theoretical prediction by the random distribution of the random network at high coordination numbers are probably due to the fact that when the local density, i.e., number of bonded neighbors, increases, the possibility of the formation
MODELING OF DENSE COLLOIDAL SUSPENSIONS
of the new bonds decreases and is not independent anymore. Also, one point which remains for future work would be the probability of bond formation not being uniform in space due to the presence of the compressional and dilatational axis; a more detailed analysis might be required to consider its effect on the final prediction of the random network model, but that is outside the scope of the present work. Following this argument, it is possible to identify the criteria developed for the random network and identify the formation of the Giant component. Molly and Reed [65] identified a critical measure to identify the existence of a Giant component in the network. They have proven for a random network with N nodes and k bonds the P probability of finding a Giant P component is almost 1 if D ¼ k k2 PðkÞ 2 k kPðkÞ > 0. From this criterion and from the bond distribution of the
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frictional bonds we have calculated D for the suspensions with different volume fractions and frictional coefficients. The results are shown in Figs. 7(e) and 7(f). Interestingly, the values calculated for semidense suspensions are all negative regardless of the roughness of the particles. For the highest frictional coefficient D initially becomes negative and large but increasing the stress/Pe results in a reversal and an increase in its values. However, it still remains negative, which is indicative of formation of individual clusters that are not percolated. For the dense suspension, however, the scenario is more complex. Similarly to the semidense suspension, at very small Pe values D ¼ 0, which is due to the fact that there are no frictional bonds formed under the flow and P(k) is 0 for all the k except k ¼ 0 where P(0) ¼ 1. When Pe/stress increases in the systems, D initially grows to be negative and
FIG. 7. Distribution of the frictional contacts measured at different Pe regimes; Pe O(0.1), O(1.0), O(10), and O(100). From the frictional bonds at each flow strength, the Poisson distribution is calculated and represented as the dashed lines (a)–(d). The distribution is measured for the semi-dense suspensions (a) and (c) for the smooth (a) and rough (b) particles. The contact number distribution for the dense suspension of the smooth and rough particles are shown in subfigures (b) and (d), respectively. The Molly and Reed constant is calculated for each system at different flow stresses and are shown in figure (e) for the semidense and (f) for the dense suspensions. The inset in figure (f) is the zoom around the critical stress where MR values show a dip when, at higher stresses, N1 switches sign and becomes positive and large.
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small. These values start to become negative at the critical stress where the shear-thickening emerges. This is an indication of forming separate clusters, i.e., frictional bonds formed inside the hydroclusters. Upon further increase in the flow strengths, the trend reverses and positive values for D are measured, which clearly show that a Giant component emerges and a percolated network is reached. From the rheological measurements, we can see that appearance of the percolated network for the dense system with very rough particles is concomitant with the discontinuity in the viscosity and emergence of positive N1. The results for the topological properties, i.e., radius of gyration, of the Giant cluster formed in suspension with semidense and dense volume fractions for different level of roughness are presented in Fig. 8 (for the calculation method see the Supplementary Material [55]). It should be noted that the radii of gyration are measured for the Giant clusters formed at different strains and different flow rates. It can be seen that the topological properties of these clusters lie between two limits indicated by dashed lines: one indicative of soft/Gaussian clusters, hR2g i / N, and the other of rigid/ elongated clusters, hR2g i / N 2 . Interestingly, it is clear that for all the volume fractions studied here, the distribution of radii of gyration is uniform between these two populations. However, increasing the frictional coefficient results in
skewing the distribution toward that typical of more rigid and elongated particles. This is an indirect confirmation for the theory proposed herein for the random network and the criteria for formation of the percolated/giant cluster. Shifting the population of the particles with the rigid/elongated branch will shift D to be positive and large, while the presence of soft/random clusters results in negative D. From the previous discussion, it was shown and proven that although D>0 is required for the discontinuity in the viscosity, it is not sufficient (comparing dense suspensions of smooth and rough particles shows D>0 for both of them). However, looking at distribution of the radius of gyration it is possible to see that for smooth particles, formation of the soft clusters is as probable as formation of rigid clusters [Fig. 8(b)], while for the very rough particles there are no signs of formation of soft frictional clusters and all the cluster are rigid [Fig. 8(d)]. This in fact can describe the sudden jump in the D value as depicted in Fig. 7(f). To study the temporal properties of the giant cluster, in Fig. 9 we show the relative number of particles that belong to the giant cluster for semidense [Figs. 9(a) and 9(b)] and dense suspensions [Fig. 9(c)] measured at different strains and for different deformation strengths and the results show several interesting traits. First, the size of the giant cluster (Ngiant/Ntotal) fluctuates around a steady-state value that
FIG. 8. Radius of gyration calculated for the Giant component in the semidense (a) and (c) and dense (b) and (d) suspensions for the smooth (a) and (b) and rough (c) and (d) particles. Rg has been calculated for different flow strengths and is represented as the number of particles residing in the clusters. The two dashed lines are the scaling for the for rigid (dash-dotted line) and random, i.e. Gaussian, (solid line) clusters. For the system showing DST (d) all the data are concentrated near the rigid limit, which is concomitant with the percolating network and final DST transition.
MODELING OF DENSE COLLOIDAL SUSPENSIONS
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size (SS-MCS). Finally, it should be noted that at semidense volume fraction MCS fluctuates significantly. For example, for the smooth particles for the highest deformation rates the size fluctuates between 0.01 and 0.1 of the system size, while for the rough particles MSC increases and fluctuates between 0.1 and 0.4 of the system size. For the dense rough systems for the same level of the deformation rate, at the shearthicken state, compare to the semi-dense regime MSC is significantly larger and almost all the particles belong to the giant cluster, i.e., MSC 1, for the strains larger than c > 0.2. Finally, Figs. 10 and 11 show the frictional clusters for systems with different roughness and volume fractions, respectively. For visualization purposes, smaller clusters are represented by the larger particles while the big, i.e., giant clusters are shown with smaller particles. The color code is set to identify clusters with a number of particles less than Nmax¼10 (blue), 50 (green), 100 (yellow), and >100 (red). Consistently with previously discussed points, the frictional bond network for semi-dense suspension shows distinct behaviors at low and high flow strengths for the smooth and rough particles, Fig. 10(a) shows there are no frictional bonds/clusters formed for the smooth particles. At the same Pe, for the rougher particles there are two clusters in the calculation cell. It should be noted that the physical nature of the contact, i.e., slightly rough or very rough, by itself does
FIG. 9. Percentage of the colloidal particles that belong to the giant cluster and its temporal evolution for semidense suspensions (top and middle) and dense suspensions (bottom). The data are presented with respect to the strain for different Pe values.
increases upon increasing the deformation rate (Pe). Second, increasing the friction coefficient can stabilize the frictional bonds at higher deformation rates and in general results in the increase in the steady-state value of the maximum cluster
FIG. 10. Microstructure of frictional clusters for semidense suspensions for smooth (left) and rough (right) particles. The clusters are color coded based on the particle population size detected in each cluster. If less than 10 particles are detected in a single cluster they are shown as blue, if there are more than 100 particles are detected in a cluster they are shown as red. For visual clarity the sizes of particles are shown smaller in the case of large population of particles, i.e., (dred < dyellow < dgreen
