J. Phys. Chem. B 2009, 113, 10117–10120
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Self-Assembly of Rod-Shaped Particles in Diblock-Copolymer Templates Qi-yun Tang and Yu-qiang Ma* National Laboratory of Solid State Microstructures and Department of Physics, Nanjing UniVersity, Nanjing 210093, China ReceiVed: February 9, 2009; ReVised Manuscript ReceiVed: May 1, 2009
We investigate the self-assembly of hard rod-shaped particles with an affinity for A block in diblock AB copolymer templates. The results are consistent with a series of recent experimental findings. Furthermore, we construct the phase diagrams of the mixture with different aspect ratios of particles by changing the particle concentration φp and the volume fraction f of the A block. The variation of the aspect ratio in different particle concentrations will significantly influence the effective volume of the A component and consequent phase behavior of nanoparticle/copolymer hybrid systems, which may account for the emergence of some unexpected phenomena of morphological transitions. The present study provides insightful guidance to control the nanometer-length-scale structures of shaped particles for potential applications as functional devices. Block copolymers could act as powerful templates to direct the self-assembly of inorganic particles into nanometer-lengthscale structures.1 Recently, significant progress has been achieved experimentally2-5 and theoretically6 on the control of spherical nanoparticle location in diblock copolymer templates. The spherical particles were selectively confined in one block domain and assembled into quantum dots,2,4 nanowires,3,5 and nanoring structures4 by varying the relative composition of copolymers. In previous theory, by combining the self-consistent field theory of copolymers and density functional theory of spherical particles, Thompson et al.6 obtained the delicate center distribution of spherical particles in copolymer domains. However, when the shape of particles changes from isotropic sphere into anisotropic rod, which will significantly affect their electronic and optical properties,7,8 the density functional expression becomes extremely complex9 but the detailed information (e.g., orientation) of particles10,11 could still not be included. Particularly, density functional approximation is inaccurate for describing densely packed nonspherical particles concentrated within one copolymer domain, where the excluded volume effect becomes important. So far, the phase behavior of rod particles in copolymer templates and the influence of the aspect ratio (R) on the precise control of particles into ordered nanostructures have not been systematically explored. In this paper, we study a self-assembly of rod-shaped particles with an affinity for the A block in AB copolymer templates with an extension of the hybrid-particle field (HPF) method.12 The work is the first attempt to incorporate translational and rotational motions of nonspherical particles and account well for their excluded volume effects in copolymers. By systematically varying the particle concentration φp and the block-A composition f, we observe a rich variety of confined nanostructures of particles in the copolymers at different aspect ratios and make direct comparisons with recent experiments with the consideration of square particles (equivalent to the sphere particles) and rod particles in a unified framework. The phase diagram determines the parameter regions under which different structures would occur. This provides useful guidance as to how to explore uncharted territory experimentally. The results may * Author to whom correspondence should be addressed. E-mail:
[email protected].
Figure 1. Illustration of rod-shaped particle (a) and its boundary (b) with interfacial width w. The center of mass and direction for the jth rod is rj and θj, and its length and width are represented by llength and lwidth, respectively.
help experimentalists better control these structures for potential uses as functional materials and devices. We consider a system composed of a volume fraction φp of m rod particles dispersed in a volume fraction (1 - φp) of nR diblock AB copolymer chains with the block composition f (volume fraction of the A block). All polymer chains are of the same polymerization index N and incompressible with a segment volume F0-1. The local volume fraction φp(r) of particles with the aspect ratio R ) llength/lwidth (Figure 1) is m
φp(r) )
∑ g(|r - rj|, θj)
(1)
j)1
where13,14
g(|r - rj |, θj) ) 1, if r is in particle j 1 + cos(dwπ/w) , if r is on the boundary of particle j 2 0, if r is not in particle j (2)
{
We take the incompressibility constraint: φA(r) + φB(r) + φp(r) ) 1 to ensure the exclusion of polymer chains from the interior of rod particles. Here, φA(r) and φB(r) are the local volume fractions of the A and B components. We use the self-consistent
10.1021/jp901170x CCC: $40.75 2009 American Chemical Society Published on Web 07/06/2009
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Tang and Ma
field theory (SCFT) to obtain the equilibrium structures of AB copolymers. The free energy of the system is given by15
∫
NF Q 1 dr[AφA + ) -(1 - φp) ln F0kBTV V(1 - φp) V BφB - χABNφAφB + χApNφAφp - χBpNφBφp + ξ(1 - φA - φB - φp)] (3) where kB is the Boltzmann constant, T is the temperature, and V is the volume of the system. χAB, χAp, and χBp are the Flory interaction parameters between the different chemical species, and ξ(r) ensures the incompressibility of the system. Q ) ∫dr q(r,s)q†(r,s) represents the single chain partition function of polymers subject to the field , and the end-segment distribution functions q(r,s) and q†(r,s) represent the probability of finding the sth segment at position r from two distinct ends of chains, which satisfy the modified diffusion equations ∂sq ) ∇2q (r)q and ∂sq† ) -∇2q† + (r)q†. The equations which describe the equilibrium morphologies of copolymers can be solved by a pseudospectral algorithm provided by Tzeremes et al.16 The free energy NF/F0kBTV in eq 3 depends explicitly on the position of rod j, and there is a force b Fj acting on it:
b Fj
))
1 V
∂(NF/F0kBTV) ∂rj
∫ dr(χApNφA(r) - χBpNφB(r) ξ(r))
(4)
∂g(r - rj, θj) ∂rj
The force b Fj produced by the spatial heterogeneity of polymer fields is weak, and the jth rod has a nonvanishing average Fj:17 velocity b υj, which is linear in b
[
]
1 1 f υj ) b ub u + (I - b u jb u j) b Fj ζ| j j ζ⊥
(5)
where b uj is the orientational vector of rod j, and ζ| and ζ⊥ represent the friction coefficients parallel and perpendicular to the rod orientation. Generally, these coefficients are not equal to each other.17 However, because we focus on the prediction of final equilibrium structures, the coefficients ζ| and ζ⊥ can be set equal for simplicity in our calculation. Therefore, relation 5 Fj/ζ. can be simplified into b υj ) b The spatial heterogeneity of polymer fields along the boundary of each rod is of noncentral symmetry around the rod center, and thus, there exists a torque Nj on the rod j:
Nj )
1 V
∫ dr(χApNφA(r) - χBpNφB(r) ξ(r))
∂g(r - rj, θj) d(|r - rj |) (6) ∂rj
where d(|r - rj|) is the projection along b uj between points r and rj (Figure 1). When considering the rotational motion of thin rods, we could neglect the rotation around b uj and assume uj. Because the torque that both ωj and Nj are perpendicular to b is weak, ωj is linear in Nj,17 ωj ) Nj/ζr, where ζr is the rotational bj + Rj and ∆θj ) (∆t/ coefficient. Then we have ∆rj ) (∆t/ζ)F
Figure 2. Equilibrium structures of mixtures. (a) φp ) 0.082, f ) 0.30; R ) 1.00. (b) φp ) 0.082, f ) 0.45; R ) 1.00. (c) φp ) 0.082, f ) 0.675; R ) 1.00. (d) φp ) 0.096, f ) 0.35; R ) 2.67. (e) φp ) 0.096, f ) 0.475; R ) 2.67. (f) φp ) 0.119, f ) 0.525; R ) 3.33. White and dark regions represent the A and B components, respectively.
ζr)Nj + Θj, where Rj and Θj are the zero-mean Gaussian random variables. To introduce the steric interaction between hard rods, we adopt a judgment similar to the Monte Carlo acceptance rules, i.e., the updated coordinates and orientation of a rod are accepted until no overlapping with other rods occurs. The simulation is performed on a two-dimensional 128 × 128 grid of side length Lx ) Ly ) 13 (in units of the diblock polymer radius of gyration Rg). The repulsive interaction parameters between the A and B components of copolymers and between B block and particles are given χABN ) χBpN ) 20. The particles are assumed to be chemical neutral to the A block (χApN ) 0), which corresponds to the surface modification of particles with the A component.3,5,10 The width of particles is fixed to be 0.61Rg, and the length varies from 0.61Rg and 1.63Rg to 2.03Rg, which corresponds to the aspect ratio R from 1.00 and 2.67 to 3.33. The width of boundary of each rod (w) is fixed to be 0.11Rg.13 We set ∆t/ζ ) 7.0Rg and ∆t/ζr ) 5.0 (in units of radian) to ensure that the translational and rotational step lengths are about 0.02Rg and 0.05, respectively. The Gaussian variables Rj and Θj are generated by the Box-Muller method.18 In the process of predicting the equilibrium structures, our full algorithm will follow the HPF method.12 Figure 2a-c shows the equilibrium structures of square particles (R ) 1.00) in diblock AB copolymer domains. In
Self-Assembly in Diblock-Copolymer Templates Figure 2a, the particles are localized in the block-A pores to form nanodots with an ordered hexagonal structure.2,4 Figure 2b shows that the particles are confined in the block-A lamellae to form nanowires and some nanowires assemble into superlattice structures in larger length scales.3,5 In Figure 2c where the block A becomes a majority component, the particles therefore form nanoring structures which are similar to those reported in a recent experiment.4 Figure 2d and e demonstrate the equilibrium morphologies of rod particles (R ) 2.67) and copolymers. In Figure 2d, rod particles are confined in the block-A nanopore domains with no preferential orientation, in consistent with recent experiment findings where the poly(ethylene oxide)-covered CdSe nanorods with 20 nm in length and 7-8 nm in diameter (R ranges from 2.50 to 2.86) are confined in the cylindrical domains of copolymer templates.10 On the other hand, Deshmukh et al.11 shown that the poly(ethylene glycol)-brushed nanorods with an average aspect ratio of about 2.62 were incorporated in polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) films and the nanorods were selectively localized in the PMMA domains with their long axis parallel to the lamellar regions. This conclusion is consistent with our calculation result illustrated in Figure 2e with R ) 2.67, in which the rod particles are confined in the lamellae of copolymer templates with the orientation parallel to the lamellar domains. We also calculate the equilibrium structures of the mixture with R ) 3.33, as shown in Figure 2f and the rods tend to orient parallel to the domain walls (A/B interfaces). We should point out that the A/B interfaces in Figure 2e and f are not flat. This is because the distribution of rods in the A domain is not always uniform, i.e., in some places rods link together while in other places no rod exists. This locally uneven distribution of particles will swell or shrink the A/B interfaces and thus result in the fluctuation of the interfaces. To reduce this fluctuation, one may increase the interfacial tension of the A/B interface or enhance the uniformity of the rod distribution in the A domain. Within the present framework, we can obtain not only the particles’ position but also their orientational distribution. For example, in Figure 2b, the squares are confined into the A domain with their center localized at the middle of the lamella, and this is similar to previous theoretical result.6,12 While for rod particles shown in Figure 2e and f, we find that the rods localize at the middle of A domain with preferential orientation, i.e., these rods orient parallel to the A/B interfaces with random deviation from their parallel state. Here, the inclusion of the precise orientation information is an important improvement of the previous theories,6,9,12 and it is easily extended to study the self-assembly of other rigid shaped particles19 in the copolymer templates. Furthermore, we construct the phase diagrams of the hybrid system with different aspect ratio R of particles, as shown in Figure 3. For small values of f, the system undergoes a phase transition from a particles confined in pore (PCP) phase to a particles confined in lamellar (PCL) phase with increasing the particle concentration φp, whereas for large values of f, the system transits from PCL phase into particles confined in inverted pore (PCIP) phase. This implies that the particles provide an effective volume of the A component of copolymers. However, by increasing R from 1.00 to 3.33, the phase boundaries become more and more left-tilted, indicating that the effective volume of the A component contributed by particles is distinct for different shaped (e.g., square and rod) particles, even at the same concentration φp. This may lead to some interesting morphological transitions. For example, at low φp
J. Phys. Chem. B, Vol. 113, No. 30, 2009 10119
Figure 3. Phase diagrams of the system for different aspect ratios: (a) R ) 1.00, (b) R ) 2.67, and (c) R ) 3.33. Data points represent the particles confined in the pore (PCP) phase (dark square), lamellar (PCL) phase (red disk), and inverted pore (PCIP) phase (cyan diamond) and their mixed PCP/PCL (green triangle) and PCL/PCIP (blue inverted triangle) phases, respectively.
(0.06-0.09) and f (0.375-0.4), or high φp (0.22-0.25) and f (0.475-0.5), when the shape of particles changes from lowanisotropic square to high-anisotropic rod in small length scales, the system may undergo a phase transition from high-anisotropic lamellar (PCL) phase to the low-anisotropic hexagonal pore (PCP) or inverted hexagonal pore phases in large length scales. To examine these phenomena, we assume that the concentration ratio t of the effective A component and B component,
t)
(1 - φp)f + φp (1 - φp)(1 - f)
(7)
stays constant along the phase boundaries, and define the effective concentration of particles,
φ*p )
t0(1 - f*) - f* (1 - f*)(1 + t0)
(8)
where f* takes the values along the phase boundaries and t0 is taken from the value of reference point (f0, φp0) according to eq 7. The reduced effective volume is defined by υ* ) φ*/φ p p, and the dependence of υ* on φp for different R along the boundary between the PCL phase and PCL/PCIP mixphase is shown in Figure 4.20 The result show that υ* of square particles is larger than that of long rod particles in low particle concentrations but becomes smaller in high concentrations. This can be explained as follows: In low particle concentration where no overlapping among particles appears, increasing the aspect ratio R with fixed rod width decreases the number of particles, which reduces the degrees of freedom of particles and thus their
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Tang and Ma rod-rod interactions are included. However, if real interactions are taken into account, the particles would be more likely to aggregate or disperse in the confined domains depending on the attractive or repulsive interactions, which may result in slightly translational shift of the whole phase boundaries. However the present work will be still available. Our work also provides a feasible framework to consider the self-assembly of copolymers and nanoparticles with various specific (e.g., triangle, ellipsoid, and trapezoid) geometries and anisotropies,19 where two element (translational and rotational) motions are essentially included.
Figure 4. Reduced effective volume υ* of particles vs the particle concentration φp with different aspect ratios.
translational entropy. This will lead to the decrease of υ*. In this case, the PCP-PCL and PCL-PCIP transitions require a larger f for long rod particles and the phase boundaries shift to the right by increasing R. In contrast, in high particle concentrations, square particles are confined densely in the A domain and increasing the aspect ratio R (e.g., two or three squares link together to form a rod) will result in the additional overlapping among particles due to the existence of orientational degree of freedom. This overlapping will orient and repulse the rods and swell the effective domain occupied by rod particles. In this situation, υ* provided by long rod particles becomes larger. For this reason, the system could transit more easily from the PCL phase to the PCIP phase for long-rod particles in lower f and the phase boundaries shift to the left with the increase of R. Both effects are combined together to result in the left tilt of boundaries and some unexpected morphological transitions. Therefore, we find that the variation of the aspect ratio will significantly influence the effective volume of the A component and consequent phase behavior of the system. Here, the excluded volume effect plays a crucial role when such nonspherical particles are densely packed into one copolymer domain. The conclusion can be used to experimentally direct the precise control of particles on larger length scales by varying the particle anisotropy in small scales, to obtain the required nanostructures in the copolymers. In summary, we demonstrate the self-assembly of rod-shaped nanoparticles into nanometer-length-scale structures within copolymer templates. The results can account well for a series of recent experimental findings. The phase diagrams for different aspect ratios of particles are provided to illustrate the phase transitions by varying the particle concentration and copolymer composition. The influence of excluded volume interactions of shaped particles is fully taken into account, and we, for the first time, show the striking difference in the block-A effective volume provided by square- and rod-shaped particles at different concentrations: in low particle concentration, the effective volume provided by square particles is larger than that by rod particles for the reduction of translational entropy, and in high concentration, the situation reverses because of the strong excluded volume effects of long rod particles. In our work, no
Acknowledgment. This work was supported by the National Basic Research Program of China, No. 2007CB925101, and the National Natural Science Foundation of China, Nos. 20674037 and 10629401. References and Notes (1) Balazs, A. C.; Emrick, T.; Russell, T. P. Science 2006, 314, 1107– 1110. (2) Lin, Y.; Boker, A.; He, J. B.; Sill, K.; Xiang, H. Q.; Abetz, C.; Li, X. F.; Wang, J.; Emrick, T.; Long, S.; Wang, Q.; Balazs, A. C.; Russell, T. P. Nature 2005, 434, 55–59. (3) Chiu, J. J.; Kim, B. J.; Kramer, E. J.; Pine, D. J. J. Am. Chem. Soc. 2005, 127, 5036–5037. (4) Park, S.; Wang, J.; Kim, B.; Russell, T. P. Nano Lett. 2008, 8, 1667–1672. (5) Kang, H.; Detcheverry, F. A.; Mangham, A. N.; Stoykovich, M. P.; Daoulas, K.Ch.; Hamers, R. J.; Muller, M.; de Pablo, J. J.; Nealey, P. F. Phys. ReV. Lett. 2008, 100, 148303–1. (6) (a) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Science 2001, 292, 2469–2472. (b) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Macromolecules 2002, 35, 1060–1071. (7) Hu, J.; Li, L.; Yang, W.; Manna, L.; Wang, L.; Alivisatos, A. P. Science 2001, 292, 2060–2063. (8) Liu, J. S.; Tanaka, T.; Sivula, K.; Alivisatos, A. P.; Frechet, J. M. J. J. Am. Chem. Soc. 2004, 126, 6550–6551. (9) Shou, Z.; Buxton, G. A.; Balazs, A. C. Compos. Interfaces 2003, 10, 343–368. (10) Zhang, Q.; Gupta, S.; Emrick, T.; Russell, T. P. J. Am. Chem. Soc. 2006, 128, 3898–3899. (11) Deshmukh, R. D.; Liu, Y.; Composto, R. J. Nano Lett. 2007, 7, 3662–3668. (12) Sides, S. W.; Kim, B. J.; Kramer, E. J.; Fredrickson, G. H. Phys. ReV. Lett. 2006, 96, 250601–1. (13) Chen, K.; Ma, Y. J. Phys. Chem. B 2005, 109, 17617–17622. (14) Matsen, M. W.; Thompson, R. B. Macromolecules 2008, 41, 1853– 1860. (15) Schmid, F. J. Phys: Condens. Matt. 1998, 10, 8105–8138. (16) Tzeremes, G.; Rasmussen, K.Ø.; Lookman, T.; Saxena, A. Phys. ReV. E 2002, 65, 041806-1. (17) Doi, M.; Edwards, S. F. The theory of polymer dynamics; Oxford University Press: New York, 1986. (18) Press, H. W.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran; Cambridge University Press: New York, 1992. (19) Glotzer, S. C.; Solomon, M. J. Nat. Mater. 2007, 6, 557–562. (20) We take the value of t0 from the reference point of f0 ) 0.65 and φp0 ) 0.067 in Figure 3c to obtain the relation between υ* and φp. The reduced volume for squares and low aspect-ratio rods decreases with increasing particle concentration in Figure 4, but this will not always be the case when the values of t0 at other points are selected. However, the relationship between the reduced volumes at the same φp for different aspectratio rods (i.e., in low φp, υ* provided by square particles is larger than that by rod particles, while in high φp the situation reverses) remains unchanged.
JP901170X
