Formation of lamellar structures from spherical particles - University of ...

THE JOURNAL OF CHEMICAL PHYSICS 130, 165102 共2009兲

Formation of lamellar structures from spherical particles Zhidong Lia兲 and Jianzhong Wub兲 Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521-0444, USA

共Received 28 August 2008; accepted 22 March 2009; published online 23 April 2009兲 We report disorder to lamellar transition in a system of spherically symmetric particles where the interparticle potential consists of a short-ranged attraction and a longer-ranged repulsion. The system provides a simplified model for aqueous dispersions of colloidal particles and globular proteins that may exhibit stable/metastable clusters or microscopic phases. By using a non-mean-field density functional theory, we predict that under appropriate conditions, a lamellar phase with alternating condensed and dilute layers of particles is thermodynamically more stable than a uniform disordered phase at the same temperature and molecular number density. Formation of the lamellar structure may prohibit the macroscopic fluid-fluid phase transition. At a given condition, the width of the condensed lamellar layers increases with the overall particle density but the trend is opposite for the dilute lamellar layers. A minimal lamellar periodicity is obtained when the condensed and dilute layers have approximately the same thickness. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3118681兴 I. INTRODUCTION

Formation of microstructures such as micelles, vesicles, and lamellar membranes is a signature of amphiphilic molecular systems including surfactants and block copolymers. These well-defined structures arise from the amphiphilic nature of the constituent molecules, viz. an intermolecular attraction compels microscopic aggregation but the aggregation size and structure are constrained by the molecular topology and by the repulsive component of the intermolecular forces.1 Recent experiments and molecular simulations suggest that similar microscopic structures may exist in colloidal dispersions and in aqueous solutions of globular proteins or spherical nanoparticles where the pair interaction potential consists of a short-ranged attraction and a longerranged repulsion.2–9 These microscopic phases can be utilized for practical applications such as materials synthesis and their formation has strong implication in understanding the kinetics of phase transition in colloidal dispersions including protein crystallization.10 This work provides a theoretical analysis of the disorderlamellar phase transition in systems of spherical particles to mimic proteins or nanoparticles in an aqueous environment. Despite its simplicity, the model system exhibits interesting microphases owing to a competition of the short-ranged attraction and longer-ranged repulsion.11–15 The interparticle potential is represented by a double-Yukawa 共DY兲 function. It has been previously demonstrated that the properties of the Yukawa system can be accurately represented by a density functional theory 共DFT兲 that incorporates the fundamental measure theory 共FMT兲 for the excluded volume effects and a兲

Present address: Reservoir Engineering Research Institute, 385 Sherman Avenue, Suite 5, Palo Alto, CA 94306. Electronic mail: [email protected]. b兲 Electronic mail: [email protected]. 0021-9606/2009/130共16兲/165102/6/$25.00

the direct correlation functions for both attractive and repulsive components of the DY interparticle potential.16,17 In this work, we predict that formation of the lamellar structure may inhibit an otherwise stable fluid-fluid transition. The theoretical predictions are supported by experiments for twodimensional systems and by recent simulation results.18,19 We expect that similar phenomena can be observed in bulk colloids or protein solutions. II. MOLECULAR MODEL AND THEORY

The model system constitutes spherical particles of identical size where the pair interaction is described by a DY potential ␸共r兲 =

冦

r⬍␴

⬁ − ␧A

exp关− ZA共r/␴ − 1兲兴 r/␴

+ ␧R

exp关− ZR共r/␴ − 1兲兴 r/␴

rⱖ␴

冧

.

共1兲

In Eq. 共1兲, r designates the center-to-center distance between two spherical particles, ␴ stands for the particle diameter; ␧A and ␧R are energy parameters that characterize the strengths of short-ranged attraction and longer-ranged repulsion, respectively, and ZA and ZR are dimensionless parameters dictating the ranges of attraction and repulsion. In an aqueous dispersion of nanoparticles or globular proteins, the range of attraction between particles corresponds to that of the van der Waals interactions. The longerranged repulsion is primarily due to the electrostatic interaction related to the particle net charge or due to the steric effects if the particles are grafted with a layer of hydrophilic polymers.5 The range of repulsion may span several times the particle diameter. In general, the strength of repulsion at contact 共r = ␴兲 is much smaller than that of the attraction.

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Without the longer-ranged repulsion, the system would exhibit a fluid-fluid coexistence that is stable relative to gelation and fluid-solid transition.20 In this work, we will examine in detail the influences of the range of attraction 共ZA兲, the range of repulsion 共ZR兲, and the relative strength of the interparticle repulsion and attraction 共␭ = ␧R / ␧A兲 on the disorder-lamellar phase transition. Thermodynamic properties of the model system can be accurately described by a first-order mean-spherical approximation 共FMSA兲 for bulk fluids and by a DFT for inhomogeneous systems.16,21 Our theoretical calculations are selfconsistent for the transition from the bulk to microscopic phases because for a uniform system, the DFT gives the Helmholtz energy identical to that from the FMSA. Different from a typical mean-field method, the DFT takes into consideration the interparticle correlations explicitly.16 Except near the critical point of the fluid-fluid equilibrium, it predicts structure and thermodynamic properties in excellent agreement with simulation results for the DY system in the bulk and at inhomogeneous conditions.22 Briefly, the Helmholtz energy functional of the Yukawa system is given by

␤F =

冕

dr␳共r兲关ln ␳共r兲⌳3 − 1兴 + ␤

+

1 4

+ +

冕冕 冕 再

dr␳共r兲f DY关n3共r兲兴

drdr⬘⌬CDY共兩r − r⬘兩,¯␳兲关⌬␳共r,r⬘兲兴2

dr − n0 ln共1 − n3兲 +

冋

冕

n1n2 − nV1nV2 1 − n3

册

冎

共n32 − 3n2nV2nV2兲 1 n3 ln共1 − n3兲 + , 2 36␲ 共1 − n3兲 n23 共2兲

where ␤ = 共kBT兲−1, kB is the Boltzmann constant, and T stands for the absolute temperature; ⌳ represents the thermal wavelength; ␳共r兲 stands for the local density distribution of particles; f DY关n3共r兲兴 and ⌬CDY共r , ¯␳兲 represent, respectively, the excess Helmholtz energy per particle and the direct correlation function due to the DY potential; ¯␳ = 关␳共r兲 + ␳共r⬘兲兴 / 2; ⌬␳共r , r⬘兲 = ␳共r兲 − ␳共r⬘兲; and n␣共r兲, ␣ = 0 , 1 , 2 , 3 , V1 , V2 are scalar and vector weighted densities from the FMT.23 The first term on the right side of Eq. 共2兲 is the reduced Helmholtz energy for an inhomogeneous ideal-gas system and the remaining terms define the excess Helmholtz energy ␤Fex, which arises from intermolecular interactions. The second and third terms in Eq. 共2兲 account for contributions due to the DY potential; they are derived from a quadratic expansion of the Helmholtz energy functional with respect to the local density variance ⌬␳共r兲. For a uniform fluid, the third term disappears and Eq. 共2兲 reduces to the first-order meanspherical equation of state 共FMSA兲.21,22 The last term in Eq. 共2兲 gives the excess Helmholtz energy due to the hard-core repulsion; this term is from a modified fundamental measure theory.24,25 The detailed expressions for f DY, ⌬CDY共r , ¯␳兲, and n␣共r兲 are given in our previous publications.16,22,26 While the Helmholtz energy given by Eq. 共2兲 is applicable to microscopic phases of arbitrary geometry, in this

work we restrict our interest to formation of lamellar structures. For a uniform system 关⌬␳共r , r⬘兲 = 0, ␳共r兲 = ␳av兴, Eq. 共2兲 reduces to

␤F/V = ␳av共ln ␳av⌳3 − 1兲 + ␤␳av f DY − n0 ln共1 − n3兲 +

冋

册

n32 n3 n 1n 2 + , 2 ln共1 − n3兲 + 1 − n3 36␲n3 共1 − n3兲2

共3兲

where ␳av stands for the average density, f DY is the Helmholtz energy density for a uniform system calculated from the FMSA,21,22 and the n␣-dependent terms are the same as those from the Boublik–Mansoori–Carnahan–Starling– Leland equation of state.27,28 In our theoretical calculations, the thermal wavelength is treated as unity 共⌳ = 1兲; it makes no contribution to the density distribution of particles in the lamellar structures or in comparison to the Helmholtz energy per particle between the lamellar phase and the uniform phase. Typically the DFT calculation is based on the grand canonical ensemble where the system under investigation is assumed to coexist with a bulk phase of specific chemical potential and the equilibrium density profile is obtained by minimization of the grand potential. For identification of the lamellar phase at a given average particle density, however, the canonical ensemble is more convenient. In this case, the Helmholtz energy functional is minimized with respect to a one-dimensional density profile within the constraint of fixed average density and the chemical potential appears only as a Lagrangian multiplier affiliated with the constraint. The procedure is similar to calculation of the lamellar structure in a block copolymer system.29,30 Given a lamellar thickness, the equilibrium density profile is obtained from the Euler– Lagrange equation

␳共z兲 =

␳avH exp关− ␦␤Fex/␦␳共z兲兴 , ex 兰H 0 dz exp关− ␦␤F /␦␳共z兲兴

共4兲

where H is the lamellar thickness, i.e., the width of one repeating unit in the lamellae. We calculate the lamellar thickness by minimization of the Helmholtz energy with respect to H. The computational details can be found in our previous publications.29,30 III. RESULTS AND DISCUSSION

Figure 1 shows how the reduced interparticle potential ␸共r兲 / ␧A is influenced by the parameters for the range of attraction 共ZA兲, the range of repulsion 共ZR兲, and relative strength of the repulsion and attraction 共␭兲. As one would expect for a typical colloidal system, the interparticle potential is strongly attractive near contact but becomes weakly repulsive at large separations. Several common features are shared by increasing ZA or ␭ and decreasing ZR, i.e., the repulsive component of the interparticle potential is strengthened and becomes longer ranged as the range of attraction is decreased. Besides, an increase in ␭ reduces the magnitude of attraction in the contact region. We consider first the thermodynamic stability of the model system by comparing the Helmholtz energies of the lamellar and uniform phases at the same average particle density. The thermodynamically more stable phase corre-

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Lamellar colloids

FIG. 1. 共Color online兲 The reduced interparticle potential ␸共r兲 / ␧A for different sets of Yukawa parameters: 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 共solid兲, 共5.0, 1.0, 0.2兲 共dashed兲, 共3.0, 0.8, 0.2兲 共dotted兲, and 共3.0, 1.0, 0.3兲 共dashed dotted兲.

sponds to the one with lower Helmholtz energy. Figure 2 shows the difference in the reduced Helmholtz energy per particle between the lamellar and uniform phases, i.e., ⌬␤F / N as a function of the average dimensionless density ⴱ ␳av = ␳av␴3 for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲. For each density, we also consider the situation that for the uniform phase a two-liquid state may have a lower Helmholtz energy than the single-liquid state, i.e., the macroscopic liquid-liquid phase separation may occur without the lamellar transition. For systems with a relatively weak attraction 共i.e., a small ␧A / kBT兲, the Helmholtz energy of the lamellar phase is higher than that of the uniform phase, indicating that the microphase is thermodynamically unstable in comparison with the disordered uniform phase. As the interparticle attraction increases 共␧A / kBT ⬎ 2.4兲, however, there exists a region where the Helmholtz energy of the lamellar phase is lower than that of the uniform phase. In this case, the differ-

FIG. 2. 共Color online兲 Difference in the reduced Helmholtz energy per particle between the lamellar and uniform phases ⌬␤F / N for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 at ␧A / kBT = 4.0 共solid兲, 3.5 共dashed兲, 3.0 共dotted兲, and 2.5 共dashed dotted兲.

J. Chem. Phys. 130, 165102 共2009兲

FIG. 3. 共Color online兲 Same as in Fig. 2 but for ␧A / kBT = 4.0 and 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 共solid兲, 共5.0, 1.0, 0.2兲 共dashed兲, 共3.0, 0.8, 0.2兲 共dotted兲, and 共3.0, 1.0, 0.3兲 共dashed dotted兲.

ence in Helmholtz energy exhibits a negative valley-shaped profile with respect to the overall particle density. It indicates that the lamellar phase becomes thermodynamically more stable at intermediate packing densities. The range of density that gives a stable lamellar structure expands and the valley moves toward the lower density as the strength of interparticle interaction is further increased. Figure 3 presents the difference in Helmholtz energy per particle between the lamellar and uniform phases for different sets of parameters 共ZA, ZR, and ␭兲 with ␧A / kBT = 4.0. The valley shifts in the direction of higher density when ZA or ␭ increases or ZR decreases. Interestingly, the average density corresponding to the minimum of the valley increases with the decrease in the attraction range. Formation of the lamellar structure in a colloidal dispersion is expected by considering the electrostatic analogy for surfactant assemblies31 and by the ground-state energy calculations.18 Indeed, a twodimensional equivalence of the lamellar structure 共stripes兲 was observed in experimental systems.19,32 The lamellar phases may also be responsible for the unusual structure factor observed in aqueous of solutions of cytochrome C, a globular protein, at moderate concentrations.5 At the intermediate densities, the lamellar phase is thermodynamically more stable than the disordered phase because by separating the particles into a layer-by-layer configuration, the microphase maximizes the short-ranged attraction among the spherical particles while avoids the repulsion significantly at large separations. The lamellar structure is destroyed at low density due to the entropy effect and at high density due to close separation among the particles. The microscopic structure of the lamellar phase depends on the average density and the strength and range of interparticle potential. As shown in Fig. 4, the densities of both condensed layers and dilute layers vary little with changes in the overall density of the system when the reduced energy parameter 共␧A / kBT = 4.0兲 and the interparticle potential 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 are fixed. In this case, an increase in the average density results in changes primarily in the

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J. Chem. Phys. 130, 165102 共2009兲

FIG. 4. 共Color online兲 Density profiles within the lamellar phase for ⴱ = 0.2 共solid兲, 0.4 共dashed兲, 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲, ␧A / kBT = 4.0, and ␳av 0.6 共dotted兲, and 0.8 共dashed dotted兲.

ⴱ FIG. 6. 共Color online兲 Same as in Fig. 4 but for ␳av = 0.5, ␧A / kBT = 4.0, and 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 共solid兲, 共5.0, 1.0, 0.2兲 共dashed兲, 共3.0, 0.8, 0.2兲 共dotted兲, and 共3.0, 1.0, 0.3兲 共dashed dotted兲.

lamellar thickness. The invariance of the densities of both condensed and dilute layers can be understood in terms of the “lever rule” in phase separation, i.e., while the partitioning of the particles in the condensed and dilute layers depends on the average density, the densities of the coexisting lamellar layers are determined only by temperature. ⴱ = 0.5兲 and On the other hand, when the average density 共␳av the range of the interparticle potential 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 are fixed, an increase in the strength of the interparticle potential has little effect on the lamellar thickness but it leads to significant changes in the densities of both condensed and dilute lamellar layers 共Fig. 5兲. Similar to that in bulk fluid-fluid equilibrium, the densities of the condensed and dilute layers approach each other as ␧A / kBT decreases and the density profile at the interface is flattening near the critical region 共␧A / kBT = 2.5兲. Figure 6 shows the structures of lamellar phases at different sets of parameters 共ZA, ZR, and ␭兲 for the interparticle

ⴱ potential. Here the overall particle density is ␳av = 0.5 and the reduced energy parameter is ␧A / kBT = 4.0. While the dilute layer is relatively insensitive to the change in potential parameters 共ZA, ZR, and ␭兲, the interparticle potential has significant effects on the structure of the condensed layers. The density of the condensed layer increases with the relative strength of the interparticle attraction in the contact region. An increase in ZA or ␭ or a decrease in ZR leads to sharper transition from dilute layer to condensed layer. In that case, thinner condensed layers are observed because the repulsion is enhanced and longer ranged but the attraction is shorter ranged. Figure 7 shows the lamellar thickness as a function of the average density when the reduced energy parameter is ␧A / kBT = 4.0 and the potential parameters are 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲. Also shown in this figure are the widths of the condensed and dilute layers in the lamellar phase calcu-

FIG. 5. 共Color online兲 Same as in Fig. 4 but for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲, ⴱ ␳av = 0.5, and ␧A / kBT = 4.0 共solid兲, 3.5 共dashed兲, 3.0 共dotted兲, and 2.5 共dashed dotted兲.

FIG. 7. 共Color online兲 Variation in the lamellar thickness H 共solid兲 and the widths of the condensed HC 共dashed兲 and dilute layers HD 共dotted兲 of the lamellar phase for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 and ␧A / kBT = 4.0.

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Lamellar colloids

FIG. 8. 共Color online兲 Lamellar thickness H for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 and ␧A / kBT = 4.0 共solid兲, 3.5 共dashed兲, and 3.0 共dotted兲.

lated from the interfacial density profiles by using the Gibbs dividing surface. Interestingly, the condensed layers grow monotonically with the particle density but the trend is opposite for the dilute layers. As a result, the overall lamellar thickness exhibits a minimum at an intermediate density where the condensed and dilute layers have an approximately equal width. At a low density, the condensed domains have only a few layers of particles and they are separated by thick dilute layers that minimize the interlayer repulsion. An increase in the particle density leads to additional condensed layers. While the width of the dilute layers falls sharply with the increase in the average density, the condensed layers exhibit little changes in the width. The invariance of the condensed-layer width is affiliated with a competition of the short-ranged attraction with the longer-ranged repulsion, which makes addition of particles to the condensed layers energetically less favorable in comparison to formation of new condensed layers. When the average particle density is further increased, the close separation between neighboring condensed layers enhances the interlayer repulsion, which compels growth of the existing condensed layers. Figure 8 shows the effect of reduced energy parameter ␧A / kBT on the lamellar thickness for 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲. With the decrease in interparticle attraction 共␧A / kBT兲, the curvature narrows and the minimal lamellar thickness moves slightly toward the lower density. Figure 9 presents the lamellar thickness at different sets of parameters 共ZA, ZR, and ␭兲 for the interparticle potential. Here the reduced attraction parameter is ␧A / kBT = 4.0. As expected, the minimal thickness shifts downward when ZA or ␭ increases or ZR decreases. The trend can be explained by thinning condensed layer but relative invariance of the dilute layer. Figure 9 indicates that the depth of minimum is mainly determined by the range of interparticle attraction. IV. CONCLUSIONS

By using a DFT that is accurate for systems with shortranged attraction and longer-ranged repulsion, we predict

FIG. 9. 共Color online兲 Same as in Fig. 8 but for ␧A / kBT = 4.0 and 共ZA , ZR , ␭兲 = 共3.0, 1.0, 0.2兲 共solid兲, 共5.0, 1.0, 0.2兲 共dashed兲, 共3.0, 0.8, 0.2兲 共dotted兲, and 共3.0, 1.0, 0.3兲 共dashed dotted兲.

that lamellar structures with alternating condensed and dilute layers can be thermodynamically more stable than uniform phases even though the pair interparticle potential is spherically symmetric. The structure anisotropy arises from a competition of the short-ranged attraction favoring aggregation and the longer-ranged repulsion that stabilizes the macroscopic separation. The lamellar thickness and microscopic structure are mainly determined by the average particle density and the interparticle energy. At appropriate conditions, the disorder-to-lamellar boundary encompasses the coexistence curve of the bulk fluid-fluid equilibrium, implying that the bulk phase transition can be prohibited by formation of the microscopic phases. While qualitatively our theoretical predictions agree with recent simulation, we are unaware of direct experimental verification of the lamellar phases in bulk colloids or protein solutions. Observation of the lamellar structure may explain the formation of quasiplanar nucleus structures during crystallization of apoferritin, a quasispherical protein that interacts with each other with a shortranged attraction and longer-ranged repulsion in an aqueous environment.2,33 ACKNOWLEDGMENTS

We thank Professor Zhen-Gang Wang and Professor Peter G. Vekilov for insightful discussions. For financial support, the authors are grateful to the U.S. Department of Energy 共Contract No. DE-FG02-06ER46296兲. M. Seul and D. Andelman, Science 267, 476 共1995兲. S. T. Yau and P. G. Vekilov, Nature 共London兲 406, 494 共2000兲. 3 S. T. Yau and P. G. Vekilov, J. Am. Chem. Soc. 123, 1080 共2001兲. 4 A. I. Campbell, V. J. Anderson, J. S. van Duijneveldt, and P. Bartlett, Phys. Rev. Lett. 94, 208301 共2005兲. 5 S. H. Chen, M. Broccio, Y. Liu, E. Fratini, and P. Baglioni, J. Appl. Crystallogr. 40, S321 共2007兲. 6 A. Coniglio, L. de Arcangelis, A. de Candia, E. Del Gado, A. Fierro, and N. Sator, J. Phys.: Condens. Matter 18, S2383 共2006兲. 7 A. Stradner, H. Sedgwick, F. Cardinaux, W. C. K. Poon, S. U. Egelhaaf, and P. Schurtenberger, Nature 共London兲 432, 492 共2004兲. 1 2

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