CHIN.PHYS.LETT.
Vol. 25, No. 12 (2008) 4469
Brownian Dynamics Simulation of Microstructures and Elongational Viscosities of Micellar Surfactant Solution ∗ WEI Jin-Jia(ddd)1∗∗ , KAWAGUCHI Yasuo2 , YU Bo(dd)3 , LI Feng-Chen(ddd)4 1
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049 Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 2788-510, Japan 3 Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum (Beijing), Beijing 102249 4 School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001 2
(Received 17 July 2008) Brownian dynamics simulation is conducted for a dilute surfactant solution under a steady uniaxial elongational flow. A new inter-cluster potential is used for the interaction among surfactant micelles to determine the micellar network structures in the surfactant solution. The micellar network is successfully simulated. It is formed at low elongation rates and destroyed by high elongation rates. The computed elongational viscosities show elongationthinning characteristics. The relationship between the elongational viscosities and the microstructure of the surfactant solution is revealed.
PACS: 82. 70. −y, 61. 20. −p, 61. 30. Cz, 61. 20. +d Surfactant drag reduction is a well-known flow phenomenon in which turbulent friction drag at the same flow rate can be greatly reduced by adding small amounts of surfactant to the carrier fluid.[1,2] Recently, it is widely accepted that surfactants are the most practical drag-reducing additives in district heating and cooling (DHC) systems for reducing pumping power because they are rather stable and show no mechanical degradation compared with polymers.[3,4] The drag-reducing behaviour of surfactants is closely related to their high elongational viscosities. However, to date there has been no good way to measure the elongational viscosities of dilute surfactant solutions quantitatively even by using the best Rheometrics RFX instrument.[4] It is generally believed that there exist some rod-like or thread-like micellar network structures in the surfactant solution which greatly increase the elongational viscosities of the surfactant solution and thus reduce the drag. Recent developments in Cryo-TEM (transmission electron microscopy) have made it possible to obtain direct images of micellar structures without altering the structures during sample preparation.[5] Rod-like or thread-like micellar network structures have been found in various surfactant/salt systems. Figure 1 shows a cryo-TEM image of a micellar network in CTAC/NaSal surfactant solution. The diameter of rod-like micelles is usually about 5 nm. On a length scale greater than the persistence length, a micelle can be thought of as a flexible structure, whereas at lesser lengths, the micelle appears to be linear. Therefore, the micellar network structure can be treated as a connection of rigid rodlike micelles at the ends. Since all these micelles with dimensions of nanometre order are embedded in a solvent, their dynamics are Langevin rather than New-
tonian. A complete time scale separation between solvent and micelle relaxation confirms that the motion of one micelle can be described by Brownian dynamics. A suitable inter-cluster potential should be used to make the rods combine at their ends to form a micellar network structure. We are developing a new model for describing the microstructure of the surfactant solution and for relating microstructure to rheological behaviour and drag reduction effectiveness. We consider the surfactant solution as a system of N rigid rod-like micelles suspended in an incompressible continuum fluid medium of viscosity ηs . The rod-like micelles interact through both inter-rod forces and hydrodynamic forces mediated via the continuum fluid. The micelles also receive fluctuating Brownian forces arising from the apparently random thermal bombardment by surrounding solvent molecules.
50 nm
Fig. 1. Cryo-TEM image of CTAC/NaSal solution.
The rigid osculating multibead rod model is used
∗ Supported by the National Natural Science Foundation of China under Grant Nos 10602043, 50536020 and 50506017, the Programme for New Century Excellent Talents in University under Grant No NCET-07-0235, and the SRF for ROCS, SEM. ∗∗ Email:
[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd
4470
WEI Jin-Jia et al.
to model surfactant solutions of rod-like micelles. The micelle is assumed to be made up of n evenly spaced beads of diameter σ, which are linearly connected as shown in Fig. 2. The micelle can be represented by a cylindrical rod of length nσ, diameter σ and aspect ratio of n. Different from the case of a sphere, the rod is multiscale, so its BD simulation is a challenge. The positions of the beads are given in terms of their distances from the centre of mass along the rod axis. The rods interact via the bead-bead potential interaction. Thus the interaction energy between two rigid linear rods is the sum of pairwise contributions from distinct bead i in rod k and bead j in rod l. Figure 2 shows the inter-rod interaction. As described, the micellar network structure is considered to consist of rod-like micelles connected at the micelle ends. In order to simulate the formation of the network structure, for end-end beads potential between two rods, the Leonard-Jones potential is employed: Uklij (rklij ) = 4E[(σ/rklij )12 − (σ/rklij )6 ],
(1)
where r klij is the vector pointing from the centre of bead i in rod k to bead j in rod l; and E represents the strength of the interaction. 1 σ Rod l 1
i
j n Rod k
n
Fig. 2. Schematic diagram of rod model made up of n beads of diameter σ.
To prevent overlap among micelles, for the interiorinterior beads potential between two rods, a repulsive soft-sphere potential is assumed: Uklij (rklij ) = 4E(σ/rklij )6 .
Vol. 25
motions of a rod are given by n N n X X X d2 r kc p h r = F + F + F ki ki klij , dt2 (3 ) i=1 l=1;l6=k j=1 n 2 X d θk (r ki − r kc ) × F hki + F rki [I]k 2 = dt i=1 N n X X p + F klij , (4)
mk
l=1,l6=k j=1
where mk and [I]k are respectively the mass and momentum of inertia of rod k, and r kc and θk are respectively the position of centre of mass and rotation angle of rod k. F hki is the hydrodynamic force on bead i in rod k and is assumed to be proportional to the relative velocity of the sphere with respect to the macroscopic fluid flow: F hki = −ξ(v ki − [κ]r ki ), (5) where ξ is Stokes’ friction constant, ξ = 3πηs σ, and [κ] is the velocity gradient tensor of the macroscopic fluid flow. F rki is the random Brownian force on bead i in rod k due to the thermal motion of the solvent molecules, and its amplitude is given by the fluctuation-dissipation theorem[6]
r F ki (t)F rki (t0 ) = 2kT ξ[1]δ(t − t0 ), (6) where the brackets denote an average over the probability space on which F rki is defined, [1] is the unit tensor, and δ(t − t0 ) is the delta function. The random Brownian force can be sampled from the random number generated by a computer with a Gaussian distribution of zero mean and variance of 2kT ξ/∆t, where ∆t is the time step. F pklij is the potential force on bead i in rod k arising from bead j in rod l: F pklij =
(7)
where nklij is the unit vector pointing from the centre of bead i in rod k to bead j in rod l:
(2)
We call this kind of combined interrod potential between micelles the Wei-Kawaguchi (WK) potential. This is different from the conventional BD simulation in which only one potential is used for interrod interaction. Here, E is the depth of potential. The selection of E is a problem: if it is too large, the micellar structure will become very rigid but if it is too small, the network structure cannot be formed. These two cases are not consistent with TEM images and rheological measurements. Here we selected E = 10kT for the simulations, where k is the Boltzmann constant. The equations of the translational and rotational
dUklij dUklij = nklij , dr ki drklij
nklij = r klij /rklij .
(8)
The additive elongation stress ∆σαα due to the existence of rod-like micelle is given by ∆σαα
1 = 2V +
X N X n k=1 N X
h Fkiα (rkiα − rkcα )
i=1
p (Fklα rklα )
,
(9)
l=k+1
where F pkl =
n P n P i=1 j=1,
F pklij is the force on the centre of
mass of rod k arising from interactions with rod l, r kl
No. 12
WEI Jin-Jia et al.
is the vector pointing from the centre of mass of rod k to that of rod l, V is the volume of the computational cell, and α is the Cartesian component. For a 3-D steady elongational flow in the xdirection, we have vy = −0.5εr ˙ y,
vx = εr ˙ x,
vz = −0.5εr ˙ z,
(10)
where ε˙ is the elongation rate. In this flow only one normal stress difference can be measured: σxx − σyy = (∆σxx − ∆σyy ) + 3ηs ε˙ = ηE ε. ˙
(11)
In this study, dimensionless quantities are used, and length σ, energy kT and friction factor ξ of beads are set to unity. It follows that time is reduced by ξσ 2 /kT (t∗ = kT t/ξσ 2 ) and elongation rate kT /ξσ 2 . The dimensionless elongation rate is called the bead Peclet number, P e = εξσ ˙ 2 /kT , which is the ratio of the time for a bead to freely diffuse a distance σ to the flow scale 1/ε. ˙ At large P e, the inverse elongation rate is less than ξσ 2 /kT and the structure rearrangement due to the elongation will dominate. Conversely, at small P e, the elongation induced structure will be a perturbation of the Brownian structure.
ing the simulation period: rx (t) = rx (t = 0) exp(εt), ˙ ry (t) = ry (t = 0) exp(−0.5εt), ˙ rz (t) = rz (t = 0) exp(−0.5εt). ˙
(12)
These equations are obtained by integrating Eq. (10). They ensure that the evolution of cell boundaries is compatible with the particle dynamics and that the system volume remains constant regardless of the motion. A velocity Verlet-like algorithm was used in the simulation which stores positions, velocities and accelerations all at the same time and minimizes the round-off error. Two different micelle volume concentrations, 0.126% and 0.189%, were simulated; these are very dilute surfactant solutions. Figure 3 shows the evolution of micellar structures at P e = 0.01. It can be seen that when steady elongational flow starts, the micellar network structure begins to elongate and orient itself along the axis of elongation. As the network structure orients itself, the rod-like micelles align with the elongation axis. The compressive nature of the flow in the directions transverse to the elongation direction appears to play an important role in the alignment process. This leads to a nonuniform orientation angular distribution of micelle directions, in other words, a preferred alignment angle.
z
(b)
-10 -20
10
20
10 20
0
x
y
(a) 0
10
20
10 20
0
-20 -10
x
20
20 10 0 -10 -20
10
y
0
10
20
20
10
z
(c)
0
-20 -20 -10 -20 -10
x
y0
10
20
20
10
0
-20 -10
x
(c)
0
10
20
10 20
0
20
-20 -10
x
Fig. 3. Snapshots of micellar microstructures in the startup of steady elongational flow with P e = 0.01 at the concentration of 0.126%: (a) t∗ = 0, (b) t∗ = 150, (c) t∗ = 300.
For the steady elongational flow, the Lee-Edwards sliding periodic image boundary conditions are used. The fluid should extend along one dimension (x here) and contracts in the remaining orthogonal directions so as to maintain the constant density. In the most general treatment of this problem, one must follow the time evolution of a fluid element which in simulations is represented by a cubic box containing the micelles. All points on the boundaries of the box change at every time step according to the following equations dur-
10
z
y
0
-10
-20 -10
20 10 0 -10 -20 -20 -10
(b)
z
0
-20 -20 -10 -10
z
y
20 10
(a)
z
20 10 0 -10 -20 -20 -10
4471
0
-10 -20 -20 -10
y
0
10
20
20
10
0
-20 -10
x
Fig. 4. Snapshots of micellar microstructures in the startup steady elongational flow with P et∗ = 1.5 at the concentration of 0.126%: (a) P e = 0.01, (b) P e = 0.1, (c) P e = 1.0.
Figure 4 shows the effects of P e on micellar structures at P et∗ = 1.5, where P et∗ is a dimensionless elongation strain. We can see that when the elongation rate exceeds a critical value, the micellar network structure starts to be destroyed and the rod-like micelles become more and more parallel to the elongation flow direction with increasing elongation rate.
4472
WEI Jin-Jia et al.
Figure 5 shows the calculated dimensionless elongational viscosity (ηE −3ηs )/3(η0n −ηs ) versus dimensionless elongation rate n(n2 − 1)/72P e with surfactant volume fraction αv as a parameter. η0n is the zero shear viscosity in the absence of potential. The molecular weight of micelles is related to the bead number n in a rod-like micelle. Generally, the elongational viscosity increases with increasing bead number or micellar weight. Therefore, by using different values of n, for example, from 5 to 10, we can obtain different elongational viscosity curves. By using the particular abscissa of n(n2 − 1)/72P e, the elongational viscosity curves for different n will collapse to one curve, while the effect of varying n or the molecular weight of micelles can be inferred. In Refs. [6,8], the same abscissa was also used for this purpose. The analytical[8] and numerical results for the case of 0.126% in the absence of potential are also shown. The no-potential model is the same as the potential model except that there is no potential interaction among rod-like micelles. This, of course, allows the micelles to cross over each other. Bird et al.[8] also used this for the theoretical study of Brownian motion of rods and the corresponding rheological characteristics. It can be clearly seen that there exists a transition from a low elongation viscos ity plateau (ηE −3ηs )/3(η0n −ηs ) = 1 at low elonga tion rates to a high one (ηE −3ηs )/3(η0n −ηs ) = 2 at high elongation rates for the case of no potential. For the case of WK potential, (ηE − 3ηs )/3(η0n − ηs ) decreases with elongation rate, levelling out at high elongation rates. The strain-thickening characteristic in the absence of potential is due to the increasing alignment of rods with elongation. The elongation viscosity may be evaluated using the viscous dissipation argument. The rate of viscous dissipation due to the presence of the rods increases as their alignment increases, and takes the maximum value of 0.5N ξ(n−1)2 σ 2 ε˙2 /V when the rods are completely aligned with the flow direction, where V is the volume of the computational box. The strain-thinning behaviour in the presence of potential is related to the increasing destruction of the micellar network structure. The rods in solution interact with each other more during weak elongation than during strong elongation, causing the elongational viscosities to be higher at low elongation rates and to drop as the rods become increasingly aligned in the flow direction at higher elongation rates. This strain-thinning phenomenon has also been found by Cathey and Fuller[9] in their uniaxial and biaxial elongational viscosity measurements of semi-dilute solutions of rigid rod polymers and well explained by Doi and Edwards.[10] At elongation rates of less than 0.1 for the case of 0.126%, (ηE − 3ηs )/3(η0n − ηs ) shows a significant increase in the presence of WK potential interaction between rod-like micelles than those in the absence of potential interaction. We can see that there exists a critical elongation rate of unity above which the elongational viscosity curves become paral-
Vol. 25
lel to the analytical curve for the case of no potential. This indicates that the micellar network structure due to the WK potential is completely destroyed and the contribution of WK potential to the rheology disappears. The difference between the viscosity curve and the analytical curve is the increase of elongational viscosity due to the formation of the micellar network caused by WK potential interaction. The elongational viscosities increase with increasing surfactant concentration. The increase of elongational viscosities by the formation of the micellar network structure in the surfactant solution suppresses vortex stretching, resulting in the reduction of turbulence energy production and friction drag in turbulent flow.
Fig. 5. Elongational viscosities.
In summary, by using a new WK potential in Brownian dynamics simulation, the surfactant micellar network structure is obtained at low elongation rates and is destroyed by elongational flow at high elongation rates, and the alignment of model rod-like micelles increased with increasing elongational strain rate, resulting in elongation-thinning characteristics. The large elongational viscosity due to the formation of the micellar network structure is considered to cause the reduction of turbulent drag.
References [1] Mysels K J 1949 Flow Thickened Fluids U.S. Patent 2492173 [2] White A 1967 Nature 214 585 [3] Gyr A and Bewersdorff H W 1995 Drag Reduction of Turbulent Flows by Additives (Netherlands: Kluwer) [4] Lu B 1997 PhD Dissertation, Ohio State University. [5] Clausen T M, Vinson P K, Minter J R, Davis H T, Talmon Y and Miller W G 1992 J. Phys. Chem. 96 474 [6] Liu T W 1989 J. Chem. Phys. 90 5826 [7] Allen M P and Tildsley D J 1987 Computer Simulation of Liquids (Oxford: Oxford Science) [8] Bird R B, Curtiss C F, Armstrong R C and Hassager O 1987 Dynamics of Polymeric Liquids: Kinetic Theory (New York: Wiley) vol 2 [9] Cathey C A and Fulle G G 1988 J. Non-Newtonian Fluid Mech. 30 303 [10] Doi M and Edwards S F 1986 The Theory of Polymer Dynamics (Oxford: Oxford Science)
