Polarizability and alignment of dielectric nanoparticles in an external electric field: Bowls, dumbbells, and cuboids

Transcription

1 Polarizability and alignment o dielectric nanoparticles in an external electric ield: Bowls, dumbbells, and cuboids Bas W. Kwaadgras, Maarten Verdult, Marjolein Dijkstra, and René van Roij Citation: J. Chem. Phys. 135, (211); doi: 1.163/ View online: View Table o Contents: Published by the American Institute o Physics. Additional inormation on J. Chem. Phys. Journal Homepage: Journal Inormation: Top downloads: Inormation or Authors:

2 THE JOURNAL OF CHEMICAL PHYSICS 135, (211) Polarizability and alignment o dielectric nanoparticles in an external electric ield: Bowls, dumbbells, and cuboids Bas W. Kwaadgras, 1,a) Maarten Verdult, 2 Marjolein Dijkstra, 1 and René van Roij 2 1 Sot Condensed Matter, Debye Institute or Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, the Netherlands 2 Institute or Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, the Netherlands (Received 18 May 211; accepted 19 August 211; published online 3 October 211) We employ the coupled dipole method to calculate the polarizability tensor o various anisotropic dielectric clusters o polarizable atoms, such as cuboid-, bowl-, and dumbbell-shaped nanoparticles. Starting rom a Hamiltonian o a many-atom system, we investigate how this tensor depends on the size and shape o the cluster. We use the polarizability tensor to calculate the energy dierence associated with turning a nanocluster rom its least to its most avorable orientation in a homogeneous static electric ield, and we determine the cluster dimension or which this energy dierence exceeds the thermal energy such that particle alignment by the ield is possible. Finally, we study in detail the (local) polarizability o a cubic-shaped cluster and present results indicating that, when retardation is ignored, a bulk polarizability cannot be reached by scaling up the system. 211 American Institute o Physics. [doi:1.163/ ] I. INTRODUCTION Monodisperse (colloidal) particles with a wide variety o shapes can nowadays be synthesized in the nanometer to micrometer size regime. 1, 2 These particles can serve as building blocks or new materials and devices with great technological potential. Sel-assembly o the particles is an important process by which large-scale nano-structures can be ormed. This sel-assembly process can be spontaneous in the case o avorable thermodynamic conditions and suitable eective particle-particle interactions, 3 1 but can also be steered and urther manipulated by external ields. Rodlike particles in a liquid dispersion, or instance, can spontaneously align at suiciently high concentrations solely due to their excluded volume interactions, 11 but their sel-organisation has also been driven by external magnetic or electric ields, by substrates that preerentially orient the rods, 17, 18 or by luid low. 19 Also, more complicated shapes have been synthesized and studied, or instance, dumbbells, 2, 2 22 cubes, 23 21, 24, 25 and bowls. In this article we study the electric-ield assisted alignment o anisotropic nanoparticles by calculating their polarizability tensor starting rom a microscopic picture o polarizable units that we call united atoms or just briely atoms, even though also larger units could have been taken. 26, 27 We only consider dielectric particles, and not metallic particles From the polarizability tensor o a nonspherical cluster o atoms, the orientation-dependent electrostatic energy ollows, and hence the cluster s ability to align in an external electric ield. We ocus here on cuboid-shaped (rods and platelets), bowl-shaped, and dumbbell-shaped particles, or which we consider various shape and size parameters. The polarizability o similar shapes have in recent years a) Author to whom correspondence should be addressed. Electronic mail: b.w.kwaadgras@uu.nl. been under study using continuum theory and on several occasions we briely compare those results to ours. Our calculations are based on the coupled dipole method (CDM), a ormalism introduced by Renne and Nijboer in the 196s. 36, 37 In the CDM, each atom in a cluster is treated as a Lorentz atom (or Drude oscillator), in which the electron is bound to the nucleus by a harmonic orce. The atoms have no permanent electric dipole moment, but their dipole moments are induced by the local electric ield. This model has been shown to yield the van der Waals interaction between two atoms. 36, 38, 39 In the CDM, many such dipolar atoms interact and the potential energy, including all many-body eects, can be calculated rom the eigenrequencies o a set o coupled harmonic oscillators, yielding orces equivalent to Lischitz s quantum-electrodynamic luctuation theory. 4 Due to its ull many-body nature the CDM is more accurate than continuum theories based on pairwise summations o atomic van der Waals interactions, as employed in, e.g., the Hamakerde Boer approach 41, 42 that underlies the treatment o dispersion orces in DLVO-theory. 43, 44 In this article we do not consider solvent eects explicitly. II. FORMALISM OF THE CDM: POTENTIAL ENERGY, POLARIZABILITY, AND ORIENTATIONAL ENERGY A. Introduction In the Lorentz model or atoms in an external electric ield, an atom is modeled as a dipole consisting o a nucleus and an electron bound to it by a harmonic orce. Thus, i the electron is at a nonzero distance rom the nucleus, the atom is eectively an induced electric dipole. In this simple model, the displacement u o the electron with respect to the nucleus, and thus the atom s polarization p = eu, is linearly dependent /211/135(13)/13415/15/$3. 135, American Institute o Physics

3 Kwaadgras et al. J. Chem. Phys. 135, (211) on the local electric ield E that the atom is subjected to, eu = p = α E, where e is the elementary charge. Here, α is the polarizability o the atom, given by α = m e ω 2, (1) where m e denotes the electron mass and ω is the angular requency associated with the harmonic orce that binds the electron to the nucleus. 45 Since an electric dipole will produce a nonzero electric ield in its surroundings, two o these Lorentz atoms will interact, mediated by the electric ield. It has been shown that this dipole-dipole interaction yields the unretarded Van der Waals interaction ( r 6 ) between the pair o atoms (at large separation r). 36 I N o these atoms are brought together and allowed to interact, the dipole-dipole interactions will inluence the electric properties o the cluster as a whole. In general, the total polarizability o the cluster cannot be expected to equal Nα, but will instead be modiied because the atoms are subject to each other s induced electric ield. B. Static polarizability and orientational energy o dipole clusters in an external electric ield The Hamiltonian o a set o N Lorentz atoms at ixed positions r i (i = 1,...,N) has been given and studied in Res. 27, 36, 38, In the present work, we use the same Hamiltonian, but add an extra term to allow or an externally exerted, spatially homogeneous, electric ield E, such that the complete expression or the Hamiltonian is H = N i=1 k 2 i 2m e + N i,j=1 N i=1 e2 m e ω 2 u2 i 2 e 2 u i T ij u j 2 N eu i E, (2) where we denote the momentum o the electron o atom i by k i.the3 3matrixT ij is the dipolar tensor, given in terms o the separation vector r ij = r i r j o atoms i and j, by i=1 (3r ij r ij / r ij 2 I), i i j T ij = r ij 3,, i i = j where I denotes the 3 3 identity matrix and denotes the 3 3 null matrix. Note that the dipolar tensor is not only symmetric in its indices, T ij = T ji, but also symmetric in its elements, T ij = T T ij. As is clear rom the Hamiltonian equation (2) and the orm o T ij, this model describes interatomic interactions in an instantaneous, nonretarded way. Thereore, the validity o the theory is limited to model systems where the relevant length scales are small enough or the speed o light to be essentially ininite. We now introduce 3N-dimensional vectors K, U, and E, which are built up rom the k i, u i, and N copies o E,respectively. We also introduce a 3N 3N-dimensional matrix T, built up rom the T ij. In terms o these objects, the Hamiltonian equation (2) is given by H = K m e 2 m eω 2 U (I α T ) U eu E, where α is given in Eq. (1), and I denotes the 3N 3N-dimensional identity matrix. Next, we introduce a 3N-dimensional vector U that satisies m e ω 2 (I α T ) U = ee, (3) and use it to complete the square in the Hamiltonian, obtaining with H = H + V E (4) H = K2 2m e m eω 2 (U U ) (I α T ) (U U ), (5) V E = 1 2 α E (I α T ) 1 E. (6) Note that V E is constant with respect to the generalized momenta and coordinates K and U U, respectively. The 3N oscillatory modes associated with the (harmonic) Hamiltonian H are given by (U (k) U ) (t) = (U (k) U ) () exp ( iω k t), (k = 1,...,3N), (7) where the amplitude vectors (U (k) U )() and the requencies ω k are given by an eigenvalue equation or the matrix (I α T ): ω 2 k ω 2 (U (k) U ) () = (I α T ) (U (k) U ) (). (8) I we denote the eigenvalues o the matrix T by λ k,itisseen that the eigenvalues o (I α T ) are (1 α λ k ), and thus that the allowed requencies are ω k = ω 1 α λ k. Assuming the system to be in the electronic ground state, we arrive at the total potential energy V = V + V E, (9) where V is the ground state energy o H, given by the sum o mode requencies, V = 1 2 3N k=1 ω k, (1) where is the reduced Planck constant. We note that V depends solely on the matrix (I α T ) and, thus, only on the relative coordinates r ij o the atoms with respect to each other. It ollows that this term is completely independent o the orientation o the cluster with respect to the electric ield. In the absence o other clusters, V can thereore be interpreted as the sel-energy o the cluster;

4 Polarizability and alignment o nanoparticles J. Chem. Phys. 135, (211) in the presence o other clusters, the term also contains the interaction energy between the clusters. 46 For the analysis o the response o the clusters to an external electric ield, however, we turn to the second term V E o Eq. (9): this term, given in Eq. (6), contains all the orientational potential energy o the cluster in the external electric ield. Clearly, the Hamiltonian o Eq. (4) describes a set o harmonic oscillators with equilibrium positions given by U and a shited ground state energy V. Using this interpretation o U, we show, in the ollowing, that V E is the energy o the total time-averaged dipole moment o the cluster in the external electric ield. We irst rewrite Eq. (3) in terms o the mean polarization vector P eu, (I α T ) P = α E, (11) and use this equation, in combination with the expression or V E given in Eq. (6), to derive where V E = 1 2 E P = 1 2 E p c, (12) p c N p i (13) denotes the total polarization o the cluster and p i eu,i denotes the mean (time-averaged) polarization o atom i, as given by the elements o P. From the orm o Eq. (12), it is clear that V E is the energy o an induced dipole p c in an electric ield E. 45 I we divide the matrix (I α T ) 1 into 3 3 sub-blocks D ij, it can easily be seen rom Eq. (11) that p i = α j D ij E, and thus that p c = α N i,j=1 i=1 D ij E α c E, (14) where we deine the 3 3 cluster polarizability matrix by α c α N i,j=1 D ij. (15) An alternative derivation o Eqs. (11) (14) is given in Re. 26. Note that α c depends solely on the spatial conigurational properties o the cluster, not on the external electric ield. Moreover, one can prove mathematically that α c is a symmetric matrix, as long as each atom has an equal polarizability α. 49 This symmetry o α c implies that its eigenvectors are orthogonal, which in turn implies that it is always possible to transorm the system to an orthogonal basis, spanned by these eigenvectors, in which α c is diagonal. Computationally, Eq. (15) is not a practical way o determining α c, since it involves the very expensive operation o explicitly calculating the inverse o a large matrix. Numerically, the most avorable approach is to use Eq. (14): ater choosing a suitable coordinate system, we apply an electric ield in the x direction and calculate the cluster polarization by solving Eq. (11). Eicient numerical algorithms or solving a set o linear equations are readily available, or example, in the LAPACK package. 5 Having solved Eq. (11) or P, we calculate the sum in Eq. (13), then divide the resulting vector by the electric ield strength; the result is the irst column o α c. To gain the remaining two columns, this procedure is then repeated in the other two Cartesian directions. I we were to neglect the dipolar interactions within the cluster, its polarization would be simply Nα E, i.e., the cluster polarizability would simply be a scalar Nα. The ratio o the actual polarizability α c and this naive guess or the polarizability, = α c, (16) Nα is a measure o how much the polarizability is enhanced due to dipole-dipole interactions and may thereore be called the enhancement actor o the dipole cluster. 26 In terms o α c, we can rewrite Eq. (12) compactly as V E = 1 2 E α c E. (17) This expression can then be written in terms o the eigenvalues α n and the corresponding normalized eigenvectors v n (n = 1, 2, 3) o α c : V E = 1 2 = n,m=1 (E v n )(v n α c v m )(v m E ) 3 (E v n ) 2 α n. (18) n=1 In the irst line, we inserted twice a complete orthonormal set o eigenvectors, while, in the second line, we made use o the act that α c v m = α m v m and v n v m = δ nm. The inner products obey the rule n (E v n ) 2 = E 2, rom which it ollows that 3 n=1 (E v n ) 2 α n α max E 2, where α max = max ({α n }) and the equality is achieved i and only i E v max, where v max is the eigenvector corresponding to α max. It ollows that V E is minimized by an electric ield in the direction o v max. A similar reasoning leads to the observation that V E is maximized by an electric ield in the direction o the eigenvector with the smallest eigenvalue, α min. The dierence between maximum and minimum orientational energy V E is thus given by = 1 2 (α max α min ) E 2. Here, or later purposes, we intentionally kept the reedom o choosing the sign o. C. Rotationally symmetric clusters The bowl- and dumbbell-shaped nanoparticles considered in this article are clusters with an axis o rotational symmetry. The rotational invariance implies that the polarization that would be induced by an electric ield in the direction o the symmetry axis must lie along this symmetry axis, and thereore that this axis is an eigenvector o the cluster s polarizability matrix α c. Since it is known that α c must be a symmetric matrix, we know that its eigenvectors must be perpendicular to each other. This leads to the conclusion that the preerred direction o any rotationally symmetric cluster must either lie along the rotational symmetry axis or perpendicular

5 Kwaadgras et al. J. Chem. Phys. 135, (211) to it. In this article, we always choose our coordinate system such that this rotational symmetry axis lies along the z axis, and choose the x and y directions such that α c is diagonal (i.e., the Cartesian axes are the eigenvectors o α c ). Moreover, or rotationally symmetric clusters, α xx = α yy, and hence we are let with only two independent entries on the diagonal o α c, one o which will be α max and the other will be α min.for the remainder o this article, we now deine the orientational energy dierence as (α zz α xx ) E 2, (19) where we choose the sign o such that is positive when the preerred direction o the cluster is along the rotational symmetry axis (which is equivalent to α zz >α xx ). Using Eq. (16), we can write Eq. (19) as = 1 2 Nα E 2, (2) where = zz xx. As will be shown or all the cluster shapes in this article, the quantity is largely independent o the cluster size, provided the number o atoms is large enough. In this regime, depends only on the shape o the cluster and on the dimensionless interatomic distance a/α 1/3. This assertion does not state anything about the individual values o zz and xx as a unction o cluster size. From the numerical data, it turns out that these quantities can still depend on cluster size, albeit usually only weakly. It is o interest to compare to the thermal energy k B T, since only i k B T, an electric ield can be used to orient the particle. Equating = k B T we derive the required number o atoms N rom Eq. (2), N = 2k BT α E 2. (21) In Sec. IV, we will use this expression to calculate N or bowls and dumbbells and hence estimate the spatial dimensions required or aligning these particles using an electric ield. D. Fourold rotationally symmetric clusters with a nonzero component only in the x direction. Similarly, the polarizations as a result o electric ields in the y and z directions will also point along the y and z axes, respectively. Because these resulting polarizations are proportional to the columns One o the discussed cluster shapes in this article is a cluster with a cubic shape. I we choose the coordinate axes along the ribs o the cube, it can be easily seen rom symmetry considerations that an electric ield applied in the x direction must induce a total cluster polarization p (cube) c o α (cube) c,itollowsthatα (cube) c must be diagonal in this basis. Moreover, because the cube is invariant under 9 rotations, we do not expect the induced polarization o the cube to be dependent on whether the electric ield is applied in the x, y, or z direction and, thereore, the entries on the diagonal o α (cube) c must be equal. Hence, α (cube) c I. (22) Note that, in this case, both the polarizability and the enhancement actor can be described by a scalar: the ormer by the proportionality actor between α (cube) c and I, the latter by this scalar polarizability divided by Nα. Since in this case α xx = α yy = α zz, we ind, rom Eq. (19), that =. This kind o cluster will thereore not have a preerred orientation within an external electric ield. Physically, this is a surprising result since, apriori, one could expect an anisotropic cluster such as a cube to preer to align one o its eatures (such as its ribs, aces, or vertices) along the electric ield. However, simple symmetry arguments negate this expectation. For cuboidshaped rods and platelets, on the other hand, as is the case or bowls and dumbbells, α xx = α yy α zz and hence. In Sec. III, wewilluseeq.(21) to calculate N or rods and platelets and estimate the minimum size o the particles to align them in an electric ield. E. Units o distance Throughout the remainder o this paper, we will usually measure lattice spacings in units o α 1/3. The reason is that, throughout the theory, the matrix T is always multiplied by a actor α. Upon applying this multiplication to the submatrices T ij, we get where α T ij = (3s ij s ij / s ij 2 I) s ij 3 (i j), s ij = r ij /α 1/3. Clearly, the relevant parameters are not the r ij themselves, but rather the dimensionless combinations s ij = r ij /α 1/3.Using these dimensionless distances, we thus eliminate the atomic polarizability as an explicit input parameter. At the same time, the dimensionless combinations are O (1) in magnitude (or typical lattices and atomic polarizabilities), which is convenient or computational purposes. We use the numerical data provided in Re. 26 to calculate the dimensionless lattice spacing ã a/α 1/3 or several substances, the result o which is listed in Table I. For clarity, we note here that all other physical quantities remain unscaled in this paper. TABLE I. Lattice spacings a, atomic polarizabilities α, and dimensionless lattice spacings ã a/α 1/3 o some typical substances. (See Re. 26.) Substance a(å) α (Å 3 ) ã Hexane Silica Sapphire

6 Polarizability and alignment o nanoparticles J. Chem. Phys. 135, (211) III. DIELECTRIC RODS AND PLATELETS A simple (but useul) example illustrating the introduced quantities are dielectric rods and platelets. We consider cuboid-shaped clusters, with the atoms on a simple cubic (sc) lattice with a dimensionless lattice constant ã = 2. Let L be the number o atoms along the edge parallel to the axis o 9 rotational symmetry and l be the number o atoms along the other two edges. Then the shape o the l l L cuboid is deined by the ratio l/l. This cluster shape is rod-like or l/l < 1, cubic-shaped or l/l = 1, and platelet-shaped or l/l > 1. The edge o length L o the cuboid is the axis o ourold rotational symmetry, and we choose the z axis along this edge. In this coordinate system, the polarizability matrix is diagonal with only two independent elements, α xx = α yy and α zz. A well-known property o dipoles is that it is energetically avorable or them to lie head-to-toe and unavorable to lie side-by-side. Thereore, or l/l < 1 (rods), i we apply an electric ield in the z direction, thus inducing a polarization o the dipoles along the head-to-toe direction, we expect the dipole-dipole interactions between the atoms to enhance the induced polarization, because there are more dipoles lying head-to-toe than side-by-side. On the other hand, an electric ield in the transverse direction (x-y plane) would induce more atomic polarizations lying side-by-side than head-to-toe and thereore, in this case, we expect the interactions to reduce the induced polarization. Also, we expect this eect to be stronger or smaller l/l ratios, since the smaller this ratio, the more extreme the dierence in the number o head-to-toe and side-by-side interactions will be. We expect the opposite to happen or l/l > 1 (platelets), or similar reasons. In Fig. 1(a), we plot the elements o the enhancement actor xx = α xx /Nα and zz = α zz /Nα, as a unction o l/l, orl = 1. Our heuristic expectation that the enhancement in the z direction is larger than that in the x direction or l/l < 1 is conirmed by this plot. For small l/l, the interatomic interactions reduce the polarization when the electric ield is applied in the x direction, while they enhance it when it is applied in the z direction. However, xx becomes larger than unity or l/l.73, where atomic interactions enhance the polarization in both directions. We note that xx and zz cross over at l/l = 1, which is the special case o a dielectric cube-shaped particle. Here, the enhancements in both directions equal each other, as was predicted in Eq. (22).Thevalue o the enhancements or l/l = 1is xx = zz The L-dependence o the enhancement actor o cubic clusters will be discussed in Sec. V. Forl/L > 1 (platelets), the cuboid polarizability is more enhanced in the x direction than in the z direction, and zz becomes smaller than unity or l/l 1.17, which means that the interatomic interactions start reducing the z-polarizability or suiciently lat platelets. In Fig. 1(b), we plot the orientational energy dierence [as deined in Eq. (19)], or a typical electric ield strength o E = 1 V mm 1, an atomic polarizability α = 5.25 Å 3,at room temperature (T = 293 K), as a unction o the number o cluster atoms, or several values o l/l < 1 (rods). The lattice is simple cubic with lattice spacing a = 2α 1/ Å. Clearly, is linear in the number o particles. We it the (a) k B T (b) (c) l / L.1 l / L.33 l / L N L 11, 14 L 8 L l / L FIG. 1. Properties o an l l L cuboid-shaped cluster o atoms with atomic polarizability α on a simple cubic lattice with spacing ã = 2. (a) The elements xx (red) and zz (blue) o the enhancement actor matrix as a unction o the shape parameter l/l, orl = 1. Note that xx = 1at l/l.73, zz = 1atl/L 1.17, and xx = zz = at l/l = 1. (b) The dierence = zz xx o the enhancement actor elements as a unction o l/l, or several values o L. (c) The energy dierence (in units o k B T ) o turning the cuboidal rod rom its least to its most avorable orientation in an external electric ield, as a unction o the number o atoms N in a rod, with shape parameters l/l =.1, 1/6.17, and 1/3.33. System parameters are given in the text, and the solid lines are linear its to the data. data to the unctional orm o Eq. (2), with being it parameter. For suiciently large N, we can conirm rom Fig. 1(b) that is constant with respect to the particle size, and that hence depends only on the shape parameter l/l (save or the internal parameter ã). As mentioned beore, the individual values or zz and xx are allowed to vary with size, but the numerical data shows that they do so only slightly in the case o rods.

7 Kwaadgras et al. J. Chem. Phys. 135, (211) As an example, or l/l =.1, we ind, or the parameters o Fig. 1(b), that 1.1. By substituting the aorementioned numerical data into Eq. (21), it ollows that N Using the shape parameter l/l =.1, the number o atoms along the long edge o the cuboid is calculated as L = (1N ) 1/ which, using a 3.48 Å, corresponds to a physical length o the long edge o 1.8 μm and corresponding length o the short edges o.18 μm. A simple order-o-magnitude calculation shows that retardation eects, which are not included in our theory, become important when length scales are o the order o 5 nm. The estimate o 1.8 μm or alignment exceeds this length scale and thus, retardation eects are expected to be important. The exact magnitude o the error this produces in our estimate is hard to determine without including retardation eects in the CDM, but the expectation is that the error will be marginal, since our estimated length scale does not dramatically exceed the 5 nm boundary. In Fig. 1(c), weplot as a unction o l/l, or several values o L. Interestingly, these graphs overlap or suiciently large N, again conirming the independence o o L. It appears that or cuboids, a suiciently large N is easily achieved: already or l l 5 cuboids, there is almost perect collapse o the data. A. Dielectric strings A special case o a dielectric rod is a cluster consisting o L Lorentz atoms positioned on a straight line, separated by an interatomic distance a. This shape can be viewed as a L 1 1 cuboid or which l/l becomes arbitrarily small as L increases. We will briely discuss this cluster shape here, because it has been investigated previously. 26 For lattice spacing ã = 2, the enhancement actor matrix elements xx and zz are plotted, as a unction o L in Fig. 2(a). We note that this is not a new result, Kim et al. produced a similar plot in Re. 26.From zz > xx, it is clear that an electric ield applied in the z direction will induce a higher polarization than the one applied in the x direction. The enhancement in the z direction, zz, is greater than unity, meaning that the interactions enhance the induced polarization, as expected. On the other hand, xx is smaller than unity, meaning that the polarizability is reduced by the interactions. The limiting value o zz or L is zz (L ) while, in the transverse direction, xx (L ) In Fig. 2(b), we plot the orientational energy dierence equation (19) or three values o the dimensionless interatomic distance ã, or a typical electric ield strength o E = 1 V mm 1, an atomic polarizability o 5.25 Å 3, and at room temperature (T = 293 K). To the numerical results, linear unctions o the orm = 1 2 α E 2 (L L ) (23) have been itted with and L as it parameters. We note here that Eq. (23) is compatible with Eq. (2) when L L. In this regime, atomic strings will have a negligible end eect. We usually ind that L O (1) so, oten, L L.Asan ii L (a) k B T (b) a Α zz xx a Α a Α L FIG. 2. (a) The diagonal elements xx (red squares) and zz (blue circles) o the enhancement actor matrix o a straight line o L dipoles along the z axis. The spacing between the atoms is ã = 2 (note that or ii, no other parameters are needed to deine this system). (b) The energy dierence associated with turning this string rom its least to its most avorable orientation in an external electric ield E = 1 V mm 1, or three dierent values o the dimensionless interatomic distance, ã = 1.75 (red), ã = 2 (blue), and ã = 3 (yellow). Choosing atomic polarizability α 1/3 = 5.25 Å 3 (silica), these values correspond to spacings o, respectively, a 3.4 Å,3.48 Å, and 5.21 Å. The temperature is T = 293 K (room temperature). example, the it parameters or ã = 2 turn out to be 1.72, L (24) Using Eqs. (23) and (24), we can make a prediction or the length L o an atomic string or which the orientational energy dierence becomes o the order o k B T. Equating = k B T, it is easily seen that L = 2k BT α E 2 + L Since in this case a = 2α 1/3 3.5 Å, we ind a minimum string length o about a meter in order to experience any signiicant eect rom the electric ield, which is clearly unphysical. IV. DIELECTRIC BOWLS AND DUMBBELLS In this section we consider two other shapes o dipole clusters, namely, bowl-shaped and dumbbell-shaped clusters. As mentioned in Sec. I, these shapes have recently been synthesized and show sel-assembly behavior that can be

8 Polarizability and alignment o nanoparticles J. Chem. Phys. 135, (211) FIG. 3. Construction and deinition o parameters or (a) bowl and (b) dumbbell. (a) The bowl diameter σ and the bowl thickness d completely deine its shape, as ollows. I d<σ/2, the shape o the bowl is deined by subtracting one sphere rom another (the blue area). (See Re. 21.) I d>σ/2, we let d σ d and use the same construction, but take the intersection o the spheres instead o the dierence (the orange area). Note that in the latter case, the shape is no longer a bowl in the traditional sense o the word. (b) The shape o the dumbbell is constructed by adding two (overlapping) spheres. The sphere diameter σ and the distance between the sphere centers L completely deine the shape o the dumbbell. Note that i L>σ, the dumbbell, in act, consists o two separate spheres. inluenced by an external electric ield. It is o interest to investigate how the shape and size o such particles inluence their interaction with the electric ield. The shape parameters o the bowl and dumbbell, d/σ (the ratio o the maximum thickness o the bowl and its diameter) and L/σ (the ratio o the center-to-center distance o the composing spheres and their diameter), respectively, as well as their theoretical construction, are given in Fig. 3. For the bowl, d/σ = is the limit o an ininitesimally thin hemispherical shell, d/σ = 1/2 corresponds to a hal sphere, and d/σ = 1 corresponds to a sphere. For the dumbbell, L/σ = reers to a sphere, L/σ = 1 corresponds to two touching spheres, and or L/σ > 1, the spheres are actually separated by a gap. The locations o the atoms in the clusters can be inerred by intersecting the cluster shape with a lattice o our choice. In the present work, we will ocus on a simple cubic lattice. 52 Examples o resulting clusters are depicted in Fig. 4, in which our choice o coordinate system is also deined, such that the cluster polarizabilities are diagonal with α xx = α yy. A. Bowls In Fig. 5(a), weplotα xx and α zz or a bowl-shaped particle, consisting o atoms on a sc lattice with lattice spacing FIG. 4. Examples o the dipole setup or (a) a bowl (shape parameter d/σ =.275) and (b) a dumbbell (shape parameter L/σ =.55), both intersected with a simple cubic lattice. Each sphere corresponds to an inducible dipole (Lorentz atom).

9 Kwaadgras et al. J. Chem. Phys. 135, (211) 4 Α xx sc Α zz sc 1.2 Α ii Α (a) d Σ ii (b) xx sc zz sc d Σ k B T (c) d Σ d Σ d Σ N (d) a 1.5a 13a,15.5a,18a d Σ FIG. 5. Quantities associated with a bowl-shaped cluster o atoms on a simple cubic (sc) lattice with dimensionless lattice constant ã = 2, shape parameter d/σ, and diameter σ [illustrated in Fig. 3(a)]. For σ/a = 2, panel (a) shows the elements α xx (red) and α zz (blue) o the polarizability tensor, and (b) the elements xx (red) and zz (blue) o the enhancement actor tensor, both as a unction o d/σ. The light red and blue crosses in panel (b) indicate the enhancement actor elements o a hemisphere as calculated using continuum theory in Re. 33. Panel (c) shows the energy dierence o turning a bowl rom its least to its most avorable orientation in an external electric ield E = 1 V mm 1, as a unction o the number o atoms in the bowl, or d/σ =.25,.4, and.75. The atomic polarizability is 5.25 Å 3 (yielding lattice constant a 3.48 Å) and the temperature is T = 293 K. Panel (d) shows the dierence = zz xx o the enhancement actor elements in the z and x directions, as a unction o d/σ,orσ/a = 7.5, 1.5, 13, 15.5, and 18, showing a strong dependence on the shape parameter d/σ and a weak dependence on the size parameter σ/a. ã = 2 and ixed bowl diameter σ/a = 2, as a unction o the shape parameter d/σ, deined in Fig. 3. As expected, both α xx and α zz rise as d/σ increases rom d/σ = (a hemispherical shell) to d/σ = 1 (a sphere), because the number o atoms increases. Clearly, however, α xx >α zz or all d/σ except d/σ = 1, meaning that the bowl is more polarizable in the x direction and, hence, rom Eq. (12), has a lower orientational energy when the ield is along the x direction. Plotting the diagonal elements o the enhancement actor matrix in Fig. 5(b), we note that xx increases upon decreasing d, but that zz reaches a minimum at around d/σ.5, where the interactions diminishing eect on the polarizability in the z direction is largest. In the same igure, we indicate the results or a hemisphere (corresponding to d/σ =.5), as presented in Re. 33. Considering that the theoretical approach presented in that work is completely dierent rom ours, the agreement is excellent. The orientational energy dierence or bowls composed o atoms on a sc lattice is plotted in Fig. 5(c) as a unction o the number o atoms N or several values or d/σ, or an electric ield strength E = 1 V mm 1,an atomic polarizability α = 5.25 Å 3, and at room temperature (T = 293 K). To the numerical results, linear unctions o orm (2) have been itted to determine, which we now ind to be negative. This implies that the axis o rotational symmetry is, or bowl-shaped particles, the least avorable direction or the external ield. The numerical data show that the individual values o xx and zz vary (slightly) with cluster size but that the dierence is essentially constant. The latter is illustrated in Fig. 5(d) whereweplot as a unction o the shape parameter d/σ, or several dierent bowl diameters σ. These graphs clearly overlap or higher values o σ or d/σ, which means that or those values, is indeed independent o the particle size parameter σ. As an example, or d/σ =.25, we have as it parameter.442, which we can use to predict the size o a nanocluster or which the orientational energy dierence = k B T ; using Eq. (21) with E = 1 V mm 1 and α = 5.25 Å 3 yields N Extrapolating rom the phenomenological dependence o N on σ (N σ 3 ), we expect to reach this number o particles or σ /a (as a comparison, the presently shown data goes up to σ/a = 37). Since a = 2α 1/3 3.5 Å in this case, we can estimate the diameter σ 1. μm beyond which the electric ield becomes capable o aligning the bowls.

10 Polarizability and alignment o nanoparticles J. Chem. Phys. 135, (211) B. Dumbbells In Fig. 6(a), weplotα xx and α zz or a dumbbell particle with ixed sphere diameter σ/a = 2, consisting o atoms on a sc lattice with spacing ã = 2, as a unction o the shape parameter L/σ, deined in Fig. 3. With increasing L/σ (and thus increasing L, since σ is ixed), the number o atoms increases, resulting in a rising trend o α ii or L/σ < 1. For L/σ > 1 (two separate spheres), an increase in L no longer increases the number o particles and instead only increases the distance between particles in both spheres, reducing their interactions. The result is that α zz decreases, while α xx, which beneits rom less interactions, keeps increasing. We note here that α zz already starts decreasing beore L/σ = 1, which can be explained by the act that close to L/σ = 1, the number o particles does not increase enough with increasing L to make up or the larger distance between particles in dierent spheres. We plot the enhancement actor elements xx and zz in Fig. 6(b). We notice here again that zz already stalls at L/σ.7 and decreases beore L/σ reaches unity, while xx displays the opposite behavior. Also note that, or L >σ, as the distance between the separate spheres increases, the enhancement actors decay to that o a single sphere. This was to be expected as the enhancement actor o two spheres at ininite separation is equal to that o a single sphere. Again, we indicate in the graph results that are calculated in Re. 34 using continuum theory or two touching spheres (L/σ = 1) and note the excellent agreement with out work. The energy dierence between the most and least avorable orientations is given by Eq. (19) and is plotted in Fig. 6(c) as a unction o N or three dierent values o L/σ,or E = 1 V mm 1, α = 5.25 Å 3, ã = 2, and room temperature T = 293 K. Using Eq. (2), the it parameter or dumbbells with L/σ =.85 turns out to be.323, such that the number o particles or which the energy dierence becomes comparable to the thermal energy is N This corresponds to σ /a , whereas the presently shown data goes up till σ/a = 2.5. Inserting a 3.5Å,we obtain σ.6 μm as an estimate or the typical dumbbell diameter or which the orientational energy becomes important. In Fig. 6(d), weplot as a unction o the dumbbell shape parameter L/σ, or several size parameters σ. Clearly, since the graphs overlap, also or dumbbells, is largely independent o the overall size, and depends only on the shape Α ii Α 9 8 (a) 7 6 Α zz sc Α xx sc L Σ ii zz sc xx sc (b) L Σ L Σ a 8a 1a,12a,14a k B T L Σ L Σ (c) N (d) L Σ FIG. 6. Quantities associated with a dumbbell-shaped cluster o atoms on an simple cubic (sc) lattice with dimensionless lattice constant ã = 2, with shape parameter L/σ, and dumbbell sphere diameter σ [illustrated in Fig. 3(b)]. For σ/a = 2, panel (a) shows the elements α xx (red) and α zz (blue) o the polarizability matrix and (b) the elements xx (red) and zz (blue) o the enhancement actor matrix as a unction o L/σ. The light red and blue crosses in panel (b) indicate the enhancement actor elements o two touching spheres as calculated using continuum theory in Re. 34. Panel (c) shows the energy dierence o turning the dumbbell rom its least to its most avorable orientation in an external electric ield E = 1 V mm 1, as a unction o the number o atoms in the dumbbell, or L/σ =.25,.85, and.95. The atomic polarizability is 5.25 Å 3 (yielding lattice constant a 3.48 Å) and the temperature is T = 293 K. Panel (d) shows the dierence = zz xx o the enhancement actor elements in the z- andx directions, as a unction o L/σ,orσ/a = 6, 8, 1, 12, and 14, again showing a strong shape- and a weak size-dependence.

11 Kwaadgras et al. J. Chem. Phys. 135, (211) V. THE ENHANCEMENT FACTOR OF DIELECTRIC CUBES In the nonretarded limit o present interest, the interaction energy between dipoles separated by a distance r is r 3. In three dimensions, this is a long-range interaction which does not decay suiciently quickly to ignore system size and boundary eects. Thereore, we should not expect to be able to determine bulk quantities rom calculations such as those done in this paper. The enhancement actor dierence (discussed in Secs. III and IV), which is ound to be essentially independent o size, seems to be an interesting exception. In Re. 26, the enhancement actor o one-dimensional lines and two-dimensional squares o atoms was determined and investigated in detail. For these low-dimensional systems the r 3 interaction is short-ranged, such that well-deined bulk behavior is ound in the middle o these clusters, independent o the boundary layers. In this section, we aim to do the same or a cubic-shaped particle with atoms on a cubic lattice. We will conirm that the polarizability does not seem to reach a bulk value or this three-dimensional object. Interestingly, since the ully retarded interaction energy decays asymptotically as r 4, we may thus note that the existence o a well-deined polarizability (and hence permittivity) or real-lie bulk substances can be qualiied as a retardation eect: the act the speed o light is inite results in substances having well-deined bulk permittivities. A. Theoretical predictions In Re. 26, the (scalar) enhancement actor or a cubic L L L cluster o atoms on a cubic lattice is plotted as a unction o the rib length L. The enhancement actor is seen to increase with increasing cube size, seemingly approaching some limiting value greater than unity. In contrast, when it is assumed that all dipoles have the same polarization, it is possible to prove that 26, 45 ( ) = 1. However, by assuming the same polarization or all dipoles, we neglect the eect o the suraces o the cube, which is questionable given the long range o the dipole-dipole potential ( r 3 ). One might argue that an ininite lattice o atoms without any surace is the best model or a bulk substance imaginable, but realistic substances are never ininite and, as will be shown, their suraces do have a signiicant eect on the polarizability even when their proportional number o atoms becomes low. InRe. 26 the enhancement actor is plotted or cube sizes up to This is still ar rom the regime where the surace can be expected to be negligible; the ratio o dipoles at the surace is, or this cube size, still ( )/ In the present work, we will thereore consider larger cubic clusters, o sizes up to For these clusters, the raction o surace dipoles is.5. Another prediction or the limiting value o (L ) can be obtained rom the Clausius-Mosotti relation. For our purposes, a convenient orm o this relation is given by 53 p c = Nα E 1 nα /3, where n is the number density o atoms. For a simple cubic lattice, where the number density equals n = 1/a 3, the enhancement actor is thus given by 1 = I. (25) 1 α /3a3 The Clausius-Mosotti relation is expected to give a better prediction or lower densities. 45 Below, we will compare the prediction o Eq. (25) to numerical results as a unction o the dimensionless lattice constant ã = a/α 1/3. B. Numerical methods Special optimization techniques were used in the case o large dielectric cubes. Because the number o atoms in the cluster increases rapidly with the rib length o the cube, we encounter practical problems such as memory limitations. However, two techniques can be used to reduce this problem, which we will now briely discuss. 1. Exploiting symmetry In this technique, we use the symmetries o the cube to reduce the order o the linear equation to be solved. It is possible to express the polarizations o all the dipoles in terms o only those in one octant o the cube. I we insert these relations into the set o Eqs. (11), we reduce the number o dipoles by a actor o 8. Note that in this way, we also increase the computational cost o calculating a matrix element by (roughly) a actor o 8, but since the cost o solving a set o linear equations scales with the square o the order o the equation set, we gain an overall actor o 8 in computation speed. 2. The Gauss-Seidel method The second technique uses the Gauss-Seidel method 54 or solving a set o linear equations, trading computation speed or less memory use. The Gauss-Seidel method is an iterative method or solving P rom an equation o orm (11).The method starts with a guess (discussed below) or P, which we shall call P (). The next approximation or P, P (1), is calculated using the ollowing ormula: p (k+1) i = 1 z ii e i j>i z ij p (k) j i>j z ij p (k+1) j, (26) where the p (k) i are the elements o P (k),thee i are the elements o E, and the z ij are the elements o the matrix (I α T ). Note that in our case z ii = 1, and that we can write Eq. (26) in terms o more amiliar quantities, 51 p (k+1) i = E j>i Z ij p (k) j i>j Z ij p (k+1) j, where the Z ij are 3 3 blocks in the matrix (I α T ).Note that with this technique, it is not necessary to store a new and an old copy o P, because only elements are needed that have been calculated previously; i.e., it is no problem to simply keep overwriting the elements o P.

12 Polarizability and alignment o nanoparticles J. Chem. Phys. 135, (211) TABLE II. An overview o techniques used or calculating the polarizability o a cubic cluster o atoms on a simple cubic lattice, and their associated acronyms, as used in the caption o Fig. 7. IntheLAPACK methods, we load the elements o the matrix in memory to the numerical precision speciied in the Precision column and use the routines in the LAPACK package to solve the relevant set o linear equations. The Gauss-Seidel methods involve (re-)calculating the elements o the matrix on the ly and, starting rom an initial guess, using 2 iterations o the Gauss-Seidel method to solve the set o linear equations. The Symmetries column reers to whether or not the symmetries o the dielectric cube were exploited. The L max column lists estimates or largest easible rib lengths that each method can handle, given our available resources (a) 1 3 a Α a Α a Α a Α a Α a Α L Acronym Method Symmetries Precision L max NDL LAPACK No Double 2 SDL LAPACK Yes Double 4 SSL LAPACK Yes Single 48 NGS Gauss-Seidel No Double 6 SGS Gauss-Seidel Yes Double As an initial guess we construct P () as ollows: we sum the Z ij horizontally and then solve the equation N Z ij p () i = E j=1 or each p () i. Using this guess, the enhancement actor could be calculated to a precision o 1 digits within 2 iterations. Since Z ij can be (re-)calculated on the ly as needed and the elements o E can be inerred using only a threedimensional vector, we only need to store 3N numbers, i.e., the elements o P. This eectively eliminates the memory problem. However, as a consequence, the resulting calculation is much slower than the one that uses the LAPACK routine. By combining the two techniques (symmetry exploitation and the Gauss-Seidel method), we were able to calculate the enhancement actor or cubes as large as The data points presented in Subsection V C have been calculated using various methods, corresponding to combinations o applying the two aorementioned techniques. In Table II, we give an overview o these methods, and deine the acronyms that are used in the caption o Fig. 7. The dierent methods have all been tested or consistency and the agreement between them is excellent. Computationally, the most practical techniques were the SDL-method or small rib lengths, because o its speed and simple implementation, and the SGS-method or large rib lengths, because o its negligible memory usage. C. Numerical results In Fig. 7(a), the (scalar) enhancement actor o L L L cubes, as calculated numerically, is plotted as a unction o L, or several (dimensionless) lattice spacings ã. For all o the lattice spacings, the qualitative behavior o the enhancement actor is the same: or low L, it increases rapidly as a unction o L, but starts to level o at L 1, seemingly reaching a limiting value (L) > 1 around L 2. This agrees reasonably well with prediction (25) made using the Clausius-Mossotti relation, and contradicts the pre (b) L FIG. 7. The enhancement actor o a cubic L L L cluster with atoms on a cubic lattice, as a unction o the number o atoms along the rib L, in (a) or the dimensionless lattice spacings ã = 1.75 (blue), ã = 1.8 (red), ã = 2. (yellow), ã = 2.5 (green), ã = 2.64 (light blue), and ã = 3. (purple), and in (b) a zoom-in (along the vertical axis) or ã = 2. with the data points generated by dierent numerical methods (see text and Table II): SGS (dark yellow points and curve), NGS (red points), SSL (green points), SDL (bright yellow points), and NDL (light blue points). Note that SSL slightly underestimates or L 4. diction ( ) = 1 (made by ignoring the edges o the cube). We investigate the behavior with a particular lattice spacing, ã = 2, in ull detail. Figure 7(b) shows a (vertical) zoom-in o Fig. 7(a), including data points as generated by various numerical methods. It can be seen rom Fig. 7(b) that the enhancement actor (or ã = 2) reaches a maximum value o at L = 2, ater which it starts decreasing, where the rate o decrease reaches a maximum at L = 34. This behavior is the same or all other values o ã, albeit with dierent values o. The decrease o observed beyond L = 34 is so slow that it could, conceivably, be caused by systematic rounding errors in the calculation o the elements o the matrix. This is still an open question that may be approached in a number o dierent ways, or example, by increasing the cube size and observing whether the enhancement actor keeps decreasing; another approach might be to increase the precision to which the elements o the matrix are calculated and observe whether this aects the value o the enhancement actor. A relevant observation here is that the single precision method SSL gives slightly lower results (or large L) than the double precision methods. However, the dierence does not appear to be large enough to expect the decline to vanish or asymptotically large precisions. It is clear that we have not been able to determine a limiting value or. The decay observed beyond L = 34 slows down as L increases, but a reliable extrapolation to

13 Kwaadgras et al. J. Chem. Phys. 135, (211) a ~ FIG. 8. The enhancement actor o a cube o atoms, as a unction o the (dimensionless) lattice constant ã = a/α 1/3. In blue, the numerical results are plotted, while in red, the prediction as given by the Clausius-Mosotti relation, Eq. (25), is shown. The dashed green line is Eq. (22)o Re.32, where the enhancement actor is calculated using continuum theory. Note the excellent agreement between the latter and our result, despite the completely dierent approaches. asymptotically large cubes could not be determined. Thereore, bulk behavior was not reached or our cube sizes. We stress again that this conclusion was reached not based on the absolute value o at L = 12, but based on the variation that we observe or large L. In Fig. 8, we plot the enhancement actor o a cube as a unction o its dimensionless lattice constant ã, together with the Clausius-Mosotti prediction o Eq. (25). As expected, the Clausius-Mosotti prediction is better or larger lattice constants (i.e., lower densities). Note that with increasing ã,eq.(25) changes rom an under- to an overestimation o the enhancement actor. In the same igure, we plot the results as presented in Eq. (22) o Re. 32. We note the excellent agreement, despite the act that the two results are calculated in completely dierent ways. 1. Local enhancement actor Apart rom the global enhancement actor discussed until now, it is also o interest to consider its local counterpart in inite clusters. Choosing the z direction to lie along the external electric ield E [see Fig. 9(a)], we deine the local enhancement actor as the z-component o the polarization o the local atom, divided by α E. Note that, using this deinition, the analogy with the global enhancement actor is not complete, because even when the electric ield is applied in the z direction, the polarization o the individual atoms can have nonzero components in the x and y directions. Consequently, i we were to express the local polarization as p = α E, where would be a 3 3 matrix, then would not be diagonal. In Figs. 9(b) and 9(c), we plot the local enhancement actor along two planes cut through the middle o a cube, as illustrated in Fig. 9(a). Fromthe shape o the graph we clearly see that is not a spatial constant and varies most pronouncedly on the aces and in the corners o the cube. In Fig. 9(b) we observe that the sides o the cube normal to the electric ield (z =±6a) experi- FIG. 9. (a) The orientation o the planes, cube, electric ield, and coordinate system with respect to each other: the planes are cut through the middle o the cube, in the x-z plane and the x-y plane, while the electric ield is applied in the z direction. The cube is oriented such that the ribs lie along the Cartesian directions. The (green) line along the x axis denotes the intersection o the two planes and is also represented in panels (b) and (c). (b) and (c) The local enhancement actor (deined in the text) o two sheets o dipoles lying on perpendicular planes, cut through the middle o a cube o dipoles on a cubic lattice, with lattice constant ã = 2. Panel (b) corresponds to the blue and panel (c) corresponds to the red plane as depicted in panel (a). The yellow and green lines appearing in this igure also correspond to the directions along which we plot the local enhancement actor in Fig. 1. ence a clear polarization reduction (i.e., < 1), e.g., reaching a local enhancement actor o in the middle o the ace (at x =, y =, and z =±6a). The interior o the cube turns out to experience a slight enhancement o in the center x = y = z = (not visible rom the graphs). A more dramatic enhancement is experienced by the aces o the cube that lie in-plane with the electric ield (e.g., x = ±6a); or example, at z =, the enhancement on