Letter pubs.acs.org/macroletters
Uniaxial Extension of Surfactant Micelles: Counterion Mediated Chain Stiffening and a Mechanism of Rupture by Flow-Induced Energy Redistribution Subas Dhakal† and Radhakrishna Sureshkumar*,†,‡ †
Department of Biomedical and Chemical Engineering and ‡Department of Physics, Syracuse University, Syracuse, New York 13244, United States S Supporting Information *
ABSTRACT: We study the configurational dynamics in uniaxial elongational flow of rodlike and U-shaped cationic surfactant micelles of cetyltrimethylammonium chloride (CTAC) in the presence of sodium salicylate (NaSal) counterions in water using molecular dynamics simulations. Above the critical strain rate, approximately equal to the inverse of the micelle relaxation time, hydrodynamic forces overcome the conformational entropy of the micelle and a configurational transition from a folded to a stretched state occurs. As the accumulated strain exceeds a critical value of O(100), the micelle ruptures through a midplane thinning mechanism facilitated by the advection of the counterions toward the micelle end-caps. The change in the total pair-potential energy as a function of micelle elongation is well described by a Hookean spring model that allowed to estimate the stretching modulus of the micelle. Micelle stiffness depends greatly on the degree of screening of electrostatic repulsion among the CTA+ head groups by the Sal− counterions condensed on the surface. A moderate increase in the counterion concentration makes the molecular assembly tighter and more immune to deformation by hydrodynamic stresses, resulting in an order magnitude enhancement in the stretching modulus.
C
where τ is the longest relaxation time of the polymer chain, 17−26 or equivalently, the Weissenberg number 1 Wi ≡ ϵ̇τ > 2 . However, to date, the extensional flow dynamics of wormlike micelles are gleaned indirectly from macroscopic rheological experiments such as filament stretching rheometry.27−30 Microstructure deformation under such conditions could cause chain rupture27−34 and alter viscoelastic properties of the fluid. Understanding extensional flow behavior of micellar fluids is of practical interest in many applications. Examples include porous medium flows such as the one encountered in enhanced oil recovery in which micellar fluids are pumped through subterranean rock formations to improve oil extraction. Based on the above-mentioned motivations, we study micelle deformation in uniaxial elongational flow using MD simulations with explicit counterions and solvent, as illustrated in Figure 1a. To understand the influence of the equilibrium micelle shape on the deformation behavior, we analyzed the configurational evolution of a U-shaped (Figure 1b) and a rodlike micelle (Figure 1c) at different values of the accumulated strain ϵ ≡ ϵ̇t for a salt to surfactant concentration ratio R = 0.7. We then 1 , 2τ
ationic surfactants can self-assemble into a variety of shapes referred to as micelles.1−11 Addition of small amounts of salicylate or benzoate counterions, which have high binding affinity toward the surfactant head groups, screen both the inter- and intramicelle electrostatic repulsion and has a pronounced effect on the micelle shape and therefore on the solution rheology.5−11 Specifically, structures such as spheres, rods, wormlike chains, and those with loops and branches have been observed.6−8 Such morphological diversity renders surfactant micelles valuable in numerous applications such as turbulent friction drag reduction in pipelines for fluid transportation,12 enhanced oil recovery, hydraulic fracturing,12 and manufacture of personal care products.13 Though much has been understood about the self-assembly, dynamics, and rheology of micellar fluids, mechanical properties of an individual surfactant micelle has neither been measured experimentally nor calculated by theoretical or computational means at a molecular level. This Letter reports, for the first time, the results of a molecular dynamics (MD) study on the deformation of rodlike and U-shaped micelles under elongational strain. Configurational dynamics of macromolecules in equilibrium and under uniaxial elongational flow have been extensively studied.14−25 A remarkable result of these investigations is the discovery of a coil to stretch transition in the polymer configuration when the strain rate ϵ̇ exceeds a critical value of © XXXX American Chemical Society
Received: October 26, 2015 Accepted: December 22, 2015
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DOI: 10.1021/acsmacrolett.5b00761 ACS Macro Lett. 2016, 5, 108−111
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ACS Macro Letters
sufficiently large value of ϵ. However, as seen from Figure 2, thinning of an initially folded micelle is delayed as compared to
Figure 2. Dt of a U-shaped (circles) and a rodlike (squares) micelle as a function of ϵ. Solid lines represent exponential fits. L (solid line) and Ree (dashed line) evolution of the initially U-shaped chain. Contour length of the rodlike micelle (square). Wi = 50.
Figure 1. (a) Schematic of the elongational flow simulation. The velocity field is shown by arrows. Color scheme: red (Sal−), yellow (hydrophilic part of the surfactant), cyan (hydrocarbon tail), green (Cl−), pink (Na+). Water beads are shown as purple dots. Simulations of a U-shaped (b) and a rodlike (c) micelle. The micelle in (b) has a fold with an approximate equilibrium contour length L0 ≈ 24 nm and a persistence length lp ≈ 15 nm, while for that in (c) L0 ≈ 13 nm. R = 0.7 is kept constant in the two cases. Configuration evolutions of a Ushaped (left column) and a rodlike (right column) micelle at different values of ϵ̇t. Wi = 50.
that of a rodlike micelle, that is, in the former case, Dt decreases appreciably only after the micelle unfolds and assumes a nearly linear shape (Ree → L). In this exponential thinning regime, micelle diameter decreases as Dt ∼ D0 exp(−t/α) with the thinning rate α = 4.70 ns and is approximately equal to 3τ, as it is the case for elastocapillary thinning of a viscoelastic fluid filament under uniaxial elongational deformation. On the other hand, in the case of the rodlike micelle, Dt decreases exponentially as soon as the strain rate is imposed. It is intriguing that the exponential thinning of the micelle diameter, followed by sudden rupture predicted by molecular simulations parallels the macroscopic behavior of viscoelastic wormlike micellar fluids observed in filament stretching experiments.27−29 As shown below, the conformational entropy and, consequently, the elasticity of a micelle can be represented by a harmonic spring, as it is the case with the Oldroyd-B constitutive model, which predicts an elastocapillary decay rate α of 3τ. Rothstein and co-workers27,28 have hypothesized that such macroscopic rupture is due to the en masse failure of the individual micelles within the fluid. The simulations reported here is the first molecular-level evidence for midplane thinning and rupture of a micelle under uniaxial elongational flow. Another insight gained from Figure 1 is that the critical strain required to break the micelle depends greatly on the micelle contour length and initial configuration. The accumulated strains at the onset of scission for the rodlike and Ushaped micelle are 275 and 475, respectively. This helps rationalize the experimentally observed diversity in the elongational flow behavior of micellar fluids, for example, see Figure 3 in ref 29. For instance, a micellar fluid consisting of a network of rigid linear micelles might exhibit the classical exponential elastocapillary thinning and rupture akin to the molecular picture suggested by Figure 1c. On the other hand, a network of longer and floppier micelles might undergo an initial “lag phase” during which the flow field has to overcome the conformational entropy before entering into the exponential thinning regime. This trend of U-stretch transition, exponential midplane thinning, and micelle rupture at a Wi-independent critical strain was observed in all simulations conducted for 20 ≤ Wi ≤ 50: see SI for details. Midplane scission has been also observed in the case of flexible polymer chains in strong flows.37 When a polymer chain is fully extended by uniaxial elongational flow, the tension along the chain contour is maximum at its center and zero at
considered rodlike micelles of approximately equal length at various R = 0.35, 0.55, 0.7, and 1.0. In the MD simulations, we used coarse-grained (CG) descriptions of cetyltrimethylammonium chloride (CTAC), strongly binding sodium salicylate (NaSal) salt, and water, as described in the MARTINI force field.35 Such CGMD simulations have been shown to accurately predict a variety of equilibrium and dynamical properties of CTAC/NaSal micellar solutions such as (i) salt-induced sphere to rod transition,5 (ii) binary interactions and coalescence of spherical micelles,11 (iii) length distribution and end-cap energy of rodlike micelles in solution,6 (iv) shear-induced configuration dynamics and rupture of rodlike micelles, 9 and (v) experimentally observed anomalous variation in zero-shear viscosity as a function of added salt.6 The extensional flow simulations presented in this Letter are performed using the LAMMPS36 molecular dynamics package. Simulation details and the animations showing micelle deformation (Movie1 and Movie2) are provided in the Supporting Information (SI). The folded structure shown in Figure 1b under uniaxial extensional flow exhibits conformational transition from a Ushape to a stretched state when Wi exceeds the critical value of 1 : see SI for details. The longest relaxation time τ of the U2 shaped micelle is determined from equilibrium MD simulations as the inverse of the exponential decay rate of the autocorrelation function of its end-to-end vector. This transition is governed by the interplay between the thermal forces which tends to randomize the configuration to maximize the entropy and the hydrodynamic forces which unfold the structure, as it is the case in coil to stretch transition in flexible polymers. To quantify the evolution of micelle configuration, we calculated the midplane diameter Dt, contour length L, and end-to-end distance Ree as a function of ϵ. As shown in Movie1 in the SI, micelle deformation proceeds through midplane thinning process, followed by an eventual rupture at a 109
DOI: 10.1021/acsmacrolett.5b00761 ACS Macro Lett. 2016, 5, 108−111
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ACS Macro Letters the ends. Hence, the chain would likely break at the midpoint resulting in the formation of two equal segments.32,37−39 To further understand the mechanism of micelle scission, we analyze the distribution of the pair potential energy Φ(l) along the contour of the rodlike micelle shown in Figure 1c at different values of strain. In an ideal case, without counterions and at equilibrium, Φ(l) can be expected to be a square wellshaped function with the well depth related to the end-cap energy. However, when the micelle surface is covered with counterions, the energy distribution differs from this ideal behavior as shown by the open circles in Figure 3. Specifically,
Figure 4. (a) Eb and the change (with respect to the initial configuration) in the total pair-potential energy ΔΦ of the U-shaped micelle as a function of ϵ. Inset shows Δ Φ as a function of Δx.
estimated the dissociation energy per surfactant molecule in the aggregate. For the U-shaped micelle, ΔΦ|scission ≈ 1247 kJ mol−1. Hence, the energy required to remove a surfactant from the micellar assembly, which consists of 495 surfactant molecules is approximately 2.5 kJ mol−1. For comparison, this estimate is about 2 orders of magnitude smaller than the C−C covalent bond dissociation energy (≈350 kJ mol−1). As in the case of charged polymers,15,40 stretching energy of micelles is strongly dependent on the effective surface charge, which, in turn, is controlled by the concentration of the condensed counterions. Therefore, we analyzed the deformation of micelles at various R values. For each case, we performed over 50 simulations, each starting from different initial configurations, at Wi = 50. For a rodlike micelle, the stretching energy is approximately equal to the work done by the tensile force as the conformational entropy is negligibly small. This energy penalty as a function of Δx is shown in Figure 5 for different R values. It is evident that the work
Figure 3. Φ along the contour of the rodlike micelle at different values of ϵ. The distance l(s) is measured from the midplane and is normalized with the instantaneous length L(ϵ̇t) of the micelle. Micelle contour was divided into 31 bins in each case. Each point represents the total energy of surfactants in a bin. Inset: Micelle structure at ϵ = 250 (before scission) color-coded to illustrate the variation in Φ(l).
the distribution is maximum at the end-caps and is highly nonuniform along the cylindrical part of the micelle. However, as shown in Figure S4, substantial variations exist among the time-averaged Φ for the different configurations for the same value of R. These variations in Φ are due to the nonuniform counterion density on the micelle surface, which depends on how the surfactants and salt molecules coalesce and coarsen to form the micelle. Further, the counterion distribution and hence Φ depend upon the flow field. As shown in Figure S5, the flow field advects the counterions from the midplane toward the end-caps. Resultantly, as the strain is increased, Φ becomes smaller near the micelle ends while it increases at the midplane as shown in Figure 3 and thereby facilitating micelle scission. This analysis shows that midplane scission is energetically favorable. The entropic contribution to the free energy Eb of the chain is given by
l pkBT 2
∫0
L ∂t(̂ s) 2 ds , ∂s
Figure 5. Dependence of ΔΦ on micelle extension for R = 0.35, 0.55, 0.7, and 1.0. Inset shows the stretching modulus as a function of R in the units of kJ mol−1 nm−2.
where kB is Boltzmann’s constant,
required to induce the same amount of relative extension increases rapidly with increasing salt concentration. As shown in the inset of Figure 5, the stretching modulus κA increases by an order of magnitude when R is increased by a factor of ≈3. Stretching behavior of a charged molecular assembly is different from that of uncharged ones since the screening of electrostatic interactions by the condensed counterions may lead to a renormalization of the stretching modulus.40 At low salt concentrations, electrostatic repulsion between the surfactant head groups is considerably large which promotes micelle stretching. On the other hand, at higher salt concentration, micelle surface is abundantly covered by counterions. The resulting reduction in the electrostatic repulsion between the head groups makes the molecular assembly tighter and more cohesive, hence makes it more immune to deformation by hydrodynamic stresses. Consequently, the critical strain ϵc of
T is the temperature, and t ̂ is the unit tangent vector along the contour parametrized by the variable s (see SI for details). The energy cost associated with the flow-induced stretching of the micelle can be estimated as the change in the total pair potential energy ΔΦ of the system. Figure 4 shows the variation of Eb and ΔΦ as a function of ϵ for the U-shaped micelle. It can be seen that Eb decreases exponentially with increasing strain and a crossover between Eb and ΔΦ occurs approximately at 200 kJ mol−1 when the micelle is considerably flow-aligned. We plot ΔΦ versus micelle extension Δx in the inset of Figure 4 for R = 0.7. Interestingly, ΔΦ can be expressed as F0Δx + κA(Δx)2, where F0 is a constant related to the initial length and κA ≈ 11 kJ mol−1 nm−2 is the stretching modulus of the chain. This clearly shows that the micelle behaves as a linear Hookean elastic spring. Using ΔΦ at the point of scission, we 110
DOI: 10.1021/acsmacrolett.5b00761 ACS Macro Lett. 2016, 5, 108−111
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scission increases with increasing R: for R = 0.35, 0.55, 0.7, and 1.0, ϵc is 210, 230, 275, and 390, respectively. In conclusion, we have characterized the dynamical and mechanical properties of rodlike and U-shaped surfactant micelles using MD simulations. We have demonstrated a configuration transition from a folded to a stretched state upon increasing the strain rate. This transition is governed by an intricate interplay between hydrodynamic forces and the conformational entropy of the micelle. Micelle scission occurs through a midplane thinning mechanism facilitated by the advection of counterions toward the micelle end-caps. The molecular picture of micelle deformation that has emerged from this study parallels observations of the thinning and scission of micellar fluid bridges in filament stretching experiments. The change in the total pair-potential energy as a function of micelle elongation can be quantified by a Hookean spring model. The stiffness and the critical strain of scission of a micelle strongly depend upon the micelle surface charge and the density of the condensed counterions. As the concentration of the binding counterions is increased the enhanced screening of the electrostatic repulsion between the cationic head groups of surfactants resists the applied stretching forces. Consequently, the stretching modulus increases by an order magnitude for a moderate increase in salt concentration.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.5b00761. Animation showing temporal evolution of micelle deformation (MPG). Animation showing temporal evolution of shrinking of micelle fragments when the flow field is turned off after scission (MPG). Simulation details (PDF).
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work used the computational resources provided by Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant Number OCI-1053575. The authors acknowledged financial support by National Science Foundation under Grants 1049489 and 1049454.
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REFERENCES
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DOI: 10.1021/acsmacrolett.5b00761 ACS Macro Lett. 2016, 5, 108−111
