Orientation-Dependent London–van der Waals Interaction Energy

Article pubs.acs.org/Langmuir

Orientation-Dependent London−van der Waals Interaction Energy between Macroscopic Bodies Hideatsu Maeda*,† and Yoshiko Maeda‡ †

National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi, Tsukuba, Ibaraki, 305-8566 Japan The University of Tsukuba, 1-1-1 Tennodai, Ibaraki, 305-8574 Japan

‡

ABSTRACT: The aim of this work is to derive formulas for numerical calculations of the orientation-dependent London− van der Waals (vdW) interaction energy (VA) between two rectangular bodies with arbitrary dimensions, arranged at arbitrary relative angles (θ) and separations in twisted and coplanar rotational modes. The formulation is made using a simple volume-element-integration method in the framework of the microscopic approach, in which VA is the sum of the local vdW energy (Vp) between body 1 and each thin plate constituting body 2. Examples of the calculation results are the following: (1) The θ values that give maximal and minimal values of VA depend on their shapes and relative positions. (2) As the bodies come close to each other, the variations of VA with θ and thus vdW dispersion torques generated are drastically intensified. (3) Upon increasing the length of crossing rods in twisted configurations, the VA values become constant beyond a critical length (depending on θ and separation), where the length effect on VA disappears. (4) The distribution curves of Vp show that the region in body 2 which interacts effectively with body 1 (i.e., the effective interaction region) is more sharply localized in the vicinity of the surface (closest to body 1) as the separation is decreased.

1. INTRODUCTION The London−van der Waals (vdW) dispersion force1 acting between nonpolar atoms is the weakest interatomic force and is nondirectional. It involves frictional and adhesive phenomena and contributes to the stability of molecular crystals, or the tertiary and quaternary structures of proteins. The theory and experiments on the vdW interactions are described in detail in various books (e.g., refs 2−4.). In recent years, the vdW forces have been utilized to create new functional materials and devices in nanotechnology.5,6 The vdW dispersion energy between macroscopic bodies (e.g., colloids) is intensified by the accumulation of interatomic dispersion energy, and the interaction range is on the order of the dimensions of the colloids. Geckos7,8 and insects9 can move freely on ceilings and vertical walls without dropping; this can be considered to be an adhesion effect due to vdW forces acting between macroscopic bodies. Geckos can generate enough vdW force to support their weight by placing their spatula-like pads (∼200 nm wide) formed at the terminal of the millions of foot hairs (“setae” of ∼2 μm in diameter) in contact with wall surfaces, and they can easily detach their feet by changing the angle between the seta shaft and the wall surface. The vdW force can aggregate colloidal particles, but the electrical double layer formed around the particle surface acts as a repulsion to stabilize their suspensions (DLVO theory10,11). Anisotropic colloids in stable suspensions can form various types of liquid crystals (LC).12,13 The LC structures can be © XXXX American Chemical Society

generated purely by hard core repulsions between rods in their high-density systems.14 Generally, for highly charged colloids (which have a high barrier for avoiding aggregation in the DLVO potentials), the relative influence of the vdW attractions between them can be enhanced by increasing the salt concentration in their suspensions, considerably deepening the secondary minimum in the potentials. Thus, even in lowdensity systems, highly charged rods can form long- or wideranged stable, ordered structures by side-by-side clustering of the rods due to enhanced vdW forces, such as strings and smectic sheets (multiply aligned strings) in narrow gaps,15 and 2D arrays and 3D smectics (multiply layered 2D arrays) in bulk suspensions.16,17 These are examples of one-, two-, and threedimensional architectures with colloidal building blocks in the submicrometer to nanometer range. There are two methods for calculating the vdW energy between macroscopic bodies. One is the microscopic theory developed by Hamaker,18 and the other is the macroscopic theory developed by Lifshitz.19 The London dispersion forces are attractive, acting between two electric dipoles induced by asymmetric fluctuations of orbiting electrons of atoms, and are proportional to r−7, where r is the separation between the atoms. Casimir et al.20 predicted Received: April 27, 2015 Revised: June 2, 2015

A

DOI: 10.1021/acs.langmuir.5b01459 Langmuir XXXX, XXX, XXX−XXX

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Langmuir that when r becomes so large that the propagation time of the induced dipole field of atom 1 to and from atom 2 is not negligible compared to the fluctuating period of the dipole, the r dependence of the vdW energy changes asymptotically from r−7 (nonretardation) to r−8 (retardation). This was confirmed experimentally.21,22 Hamaker approximated the vdW energy between two macroscopic spheres by summing all of the London dispersion energy between a pair of atoms which belong to different spheres (the assumption of pairwise additivity). However, this assumption is insufficient for condensed bodies in which the atoms are closely packed because the polarization of atom 2 (induced by atom 1) is influenced by its nearby atoms (the screening effect). Thus, in the microscopic approach which adopts the two-body approximation, the Hamaker constant should be treated as an undetermined coefficient to be determined using other theories taking account of the manybody effect or experiments as the condition of constraint. On the other hand, the additivity assumption is avoided in the macroscopic approach of Lifshitz, where the atomic structures of macroscopic bodies are neglected (the continuum approximation). The spontaneous fluctuations of the electron density of the atoms and their polarizability which appear in the microscopic approach are replaced with fluctuations of the electromagnetic field and the permittivity of the bodies in the macroscopic approach. Lifshitz first formulated the vdW energy between continuous bodies using Maxwell’s stress tensor, and since then, various macroscopic approaches have been proposed.2−4 In applying the macroscopic approaches to continuous bodies with various geometries, it has been found that the r dependences of the vdW energy are identical to ones calculated with the microscopic approach in both retardation and nonretardation regimes. This indicates that if the original Hamaker constant is replaced with the Lifshitz vdW constant then the microscopic approach can be a good approximate method for calculating the geometrical component of the vdW energy between macroscopic bodies. However, to determine the Lifshitz vdW constants, the complex permittivity and refractive index of macroscopic bodies over a wide frequency range are required, but they are not always available. In recent years, the vdW forces have been measured directly using SFA,21,22 AFM,23−25 AFS,26 and TIRM.27 In the microscopic approach, the vdW energy between macroscopic bodies is expressed by a simple multiple integral, and the body shapes and dimensions can easily be taken into the integral (although analytical solutions are not always derived). In fact, the separation dependence of the vdW energy for various geometrically simple bodies has been calculated and tabulated.2−4 Moreover, analytical and numerical calculations of the vdW energy between bodies of various complex geometries have been reported, where the bodies are approximated as a collection of small spheres,28 cubes of different sizes,29 and finite cylinders30 and are represented by NURBS surfaces.31 In the macroscopic approach, Langbein formulated the vdW energy between spherical bodies, but it is computationally intensive and not practical to use.32,33 Several approximations to this Langbein formula have been proposed (cf., ref 4). The vdW energy has been calculated mostly for semi-infinite bodies (infinitely long cylinders, half spaces, layered semi-infinite slabs, etc.).4 However, most recently, the vdW energies between infinitely thin ribbons with finite dimensions34 and between irregular geometries and surfaces at all separations35 have been calculated.

On the other hand, the calculations of the orientationdependent vdW (OD-vdW) energy for dielectric bodies are few. With the macroscopic approach, formulas for the OD-vdW energy between infinitely long cylinders has been proposed,36,37 by which the body-size effects on the vdW energy cannot be evaluated. The numerical calculations of the vdW energy between nonspherical particles oriented differently with respect to surfaces have recently been reported.35 In the microscopic approach, the vdW energy between circular disks and an infinite cylinder arranged at relative angles of 0 and 90° have been calculated.38Also, the OD-vdW energy between an ellipsoidal particle and a semi-infinite slab has been investigated.39 Formulas for calculating the vdW energies between ellipsoidal particles of arbitrary size at any mutual orientation have been obtained and are valid at large separations.40,41 Also, a rotational torque between convex bodies has been calculated using a generalized Derjaguin approximation.42 Thus, in this work, based on Hamaker’s approach, we have derived formulas for quickly numerically calculating the vdW energy between rectangular bodies of arbitrary dimensions, arranged at arbitrary relative angles and separations in two different rotational modes.

2. VDW ENERGY BETWEEN PARALLEL-ORIENTING RECTANGULAR BODIES Formulas for calculating the vdW energy between parallelorienting rectangular bodies in the range of short43 and all44,29 separations have been derived. In this section, we express the formulas in a generalized full form that is convenient for deriving formulas for numerically calculating the vdW energy between tilted rectangular bodies. Figure 1 shows rectangular bodies 1 and 2 in a configuration oriented parallel to each other (where their faces are parallel to one of the xy, yz, or xz planes in the orthogonal xyz coordinate system in the inset, respectively). This configuration of bodies 1 and 2 is called here the parallel-orienting configuration. The x-, y-, and z-directional face-to-face distances between bodies 1 and

Figure 1. Parallel-orienting rectangular bodies 1 and 2 in an orthogonal xyz coordinate system. The x-, y-, and z-directional faceto-face distances are denoted by dx, dy, and dz (short blue lines). The length, thickness, and width of the bodies are denoted by 2L, 2t, and 2w, respectively. B

DOI: 10.1021/acs.langmuir.5b01459 Langmuir XXXX, XXX, XXX−XXX

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By substituting ai, bj, and ck (i, j, k = 1, 4) in eq 3 into Vxyz in eq 2, the final form of VR is obtained as follows,

2 are denoted by dx, dy, and dz, respectively, which are shown with short blue lines in Figure 1. The length, thickness, and width of the bodies are denoted by 2L, 2t, and 2w respectively, and suffix 1 or 2 is the body number. Generally, the London−vdW interaction energy between a pair of macroscopic bodies, VR, in the nonretardation region is given by Hamaker16

4

VR (dx , dy , dz) =

⎞ ⎛ ⎡⎪ ⎧ c (a 2 + b 2)3/2 ⎪ ⎫ ck j ⎟ −1⎜ ⎢⎨ k i ⎬ tan ⎜ ⎪ 2 2 ⎟ ⎢⎣⎪ 24ai 2bj 2 a b + ⎩ ⎭ j ⎠ ⎝ i

V = −(A /π )VR

∬v ,v (1/r 6)dv1 dv2

⎛ bj ⎞ bj ⎞ ⎛ 3 ⎞⎛ c + ⎜ ⎟⎜⎜ k − ⎟⎟tan−1⎜ ⎟ ⎝ 32 ⎠⎝ bj ck ⎠ ⎝ ck ⎠

(1)

1 2

where A is the Hamaker constant, r2 = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2, dv1 = dx1 dy1 dz1, and dv2 = dx2 dy2 dz2, where x1, y1, and z1 are the x, y, and z coordinates of an arbitrary point in body 1 and x2, y2, and z2 are those in body 2. The integrations are simply conducted over the volume of the respective bodies, v1 and v2. Using x = x2 − x1, y = y2 − y1, and z = z2 − z1, the six integrations in eq 1 are carried out in the order (z, z1) → (y, y1) → (x, x1) as follows VR (dx , dy , dz) = 2t1

2w1

2t1+ 2t2 + dy − y1

∫0 ∫2t +dx−y 1

2w1+ 2w2 + dx − x1

∫0 ∫2w +dx−x 1

1

dy1 dy × {Vz(x , y , dz + 2L1 + 2L 2)

1

2w1

2w1+ 2w2 + dx − x1

∫0 ∫2w +dx−x 1

4

dx1 dx

1

2t1

2t1+ 2t2 + dy − y1

∫0 ∫2t +dy−y 1

2w1

2w1+ 2w2 + dx − x1 1

4

4

4

4

j=1 j=1

4

=∑ ∑ ∑ (− 1)i + j + k Vxyz(ck , bj , ai)

(2)

k=1 j=1 j=1

where a1 = dz + 2L1; a 2 = dz + 2(L1 + L 2); a3 = dz + 2L 2 ; a4 = dz ; b1 = dy + 2t1; b2 = dy + 2(t1 + t 2); b3 = dy + 2t 2 ; b4 = dy ; c1 = dx + 2w1; c 2 = dx + 2(w1 + w2); c3 = dx + 2w2 ; c4 = dx

(3)

Parameters ai, bj, and ck (i, j, k = 1, 4) come from the upper and lower limits of six integrals in eq 2. Vz(x, y, z) is the second antiderivative of the integrand, 1/r6, over z. Moreover, Vyz(x, y, z) is the second antiderivative of Vz(x, y, z) over y, and Vxyz(x, y, z) is the second antiderivative of Vyz(x, y, z) over x. Thus, Vxyz(x, y, z) in the last equation of eq 2 is given by eq 4, Vxyz =

⎤ ⎡ x x(z 2 + y 2 )3/2 tan−1⎢ 2 2 1/2 ⎥ ⎣ (z + y ) ⎦ 24z 2y 2

⎛ 3 ⎞⎛ x y⎞ ⎛ y⎞ + ⎜ ⎟⎜ − ⎟tan−1⎜ ⎟ ⎝x⎠ ⎝ 32 ⎠⎝ y x⎠ ⎡ ⎤ ⎛1 ⎛1 ⎞ y 1⎞ ⎥ + ⎜ ⎟y(z 2 + x 2)1/2 ⎜ 2 + 2 ⎟tan−1⎢ 2 ⎝ 24 ⎠ ⎝z x ⎠ ⎣ (z + x 2)1/2 ⎦ ⎛1 ⎞ ⎡ ⎤ ⎛1 z 1⎞ ⎥ + ⎜⎜ ⎟⎟z(y 2 + x 2)1/2 ⎜ 2 + 2 ⎟tan−1⎢ 2 2 1/2 ⎢⎣ (y + x ) ⎥⎦ x ⎠ ⎝y ⎝ 24 ⎠ ⎛1 ⎞ ⎡ ⎤ (y 2 + x 2)3 ⎥ + ⎜⎜ ⎟⎟log⎢ 2 2 2 2 2 32 ⎝ ⎠ ⎣ x (z + y + x ) ⎦

⎞ ⎟ ⎟ ⎠

(5)

3. ORIENTATION-DEPENDENT VDW ENERGY BETWEEN RECTANGULAR BODIES 3-1. Two Different Rotational Modes. In this section, formulas for calculating the orientation-dependent (OD) vdW energy between rectangular bodies in twisted and coplanar rotation modes, denoted by VATR and VACR, respectively, are derived using VR (eq 5) as the base energy function. Figure 2a shows rectangular bodies 1 and 2 in a twisted configuration; they are on different parallel yz planes, respectively, which are separated by an arbitrary distance dx (≠ 0) in the x direction in the orthogonal xyz coordinate system inset. On its own plane, body 2 freely moves while rotating around the axis parallel to the x direction through its center of mass (COM), denoted by a dotted line in Figure 2a, but body 1 is fixed on its own plane. The positive rotational direction is shown by the arced arrow in Figure 2a. (Incidentally, the parallel-orienting configuration taken by bodies 1 and 2 in Figure 1 corresponds to that of a twisted angle θ of 0 or 90°). 3-1-1. Coplanar Rotation. In Figure 2a, if a face of body 1, which is perpendicular to the x axis, and that of body 2 are coplanar, then ck = 0 (k = 1, 4) in eq 3, where the OD-vdW energy diverges (because VR (eq 5) is used). However, the divergence is avoided by rotating the xyz coordinate system in Figure 2a around the z axis so that the x axis is on each common plane if the bodies do not contact each other on the planes. Figure 2b shows a coplanar configuration of bodies 1 and 2, where the bottom faces are on the same plane, corresponding to the case of c3 = 0 for w1 ≠ w2 and the case of c1 = c3 = 0 for w1 = w2 (in the xyz coordinate system of Figure 2a). In addition, there are anticoplanar configurations, where bodies 1 and 2 are on the different sides of the same common plane, corresponding to the two cases: c2 = 0 and c4 = 0. On each common plane, body 1 is fixed, but body 2 can take any position and angle.

dx1 dx ∑ ∑ (− 1)i + j Vxy(x , bj , ai)

1

⎛ ⎞ ⎛ ⎛ ⎞ ai 1 ⎜1 ⎟ 1 ⎜ + ⎜ ⎟ai⎜⎜ 2 + 2 ⎟⎟ bj 2 + ck 2 tan−1⎜ 2 2 ⎜ 24 ⎟ ⎝ bj ck ⎠ ⎝ bj + ck ⎝ ⎠

dy1 dy

1

i=1

∫0 ∫2w +dx−x

⎞ ⎟ ⎟ ⎠

This equation (VR) diverges at ai = 0 or bj = 0; in these cases, VR is partially modified as shown in the Appendix section.

∑ (−1)i Vx(x , y , ai) =

⎛ bj ⎛1 ⎞ ⎛ 1 1 ⎞ + ⎜ ⎟bj⎜ 2 + 2 ⎟ ai 2 + ck 2 tan−1⎜⎜ 2 2 ⎝ 24 ⎠ ⎝ ai ck ⎠ ⎝ ai + ck

⎛ ⎞ ⎪ ⎧ ⎫⎤ (bj 2 + ck 2)3 ⎪ ⎜ 1 ⎟log⎨ ⎬⎥ ⎜ 32 ⎟ ⎪ c 2(a 2 + b 2 + c 2)2 ⎪ ⎭⎥⎦ j k ⎝ ⎠ ⎩ k i

dx1 dx

+ Vz(x , y , dz) − Vz(x , y , dz + 2L1) − Vz(x , y , dz + 2L 2)} =

4

k=1 j=1 i=1

2

VR =

4

∑ ∑ ∑ (−1)i + j + k

(4) C

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Figure 2. (a) Rectangular bodies 1 and 2 in a twisted configuration in an orthogonal xyz coordinate system. They are on different parallel yz planes separated by arbitrary distances, dx (≠ 0), in the x direction. (b) Rectangular bodies 1 and 2 in a coplanar configuration in an orthogonal xyz coordinate system; their bottom faces are on a common xz plane.

Figure 3. (a) Schema for deriving formulas for numerically calculating the orientation-dependent vdW energy between rectangular bodies 1 and 2. The figure presents the x-directional projections of bodies 1 and 2 which are actually separated by dx (≠ 0) in the x direction. Body 2 is divided into three different domains (by yellow-green dotted lines): upper delta, tetragon, and lower delta. Each domain is further divided into many thin plates parallel to body 1. Parameters pl, zl, and ddy are the length, left end position, and thickness of each thin plate. Parameter q is the y-directional distance from the bottom thin plate in each domain. (b) The angle ranges between bodies 1 and 2: (i) θ = θc = tan−1(t2/L2) (the upper) and (ii) θ < θc (the lower). The angle range for a pair of bodies in panel a corresponds to the range of θ > θc. The heights of the upper delta, tetragon, and lower delta domains are denoted by hdu, 2ht, and hdl, respectively.

3-2. Schema for Calculating the OD-vdW Energy. Figure 3a presents a schema for deriving formulas for the ODvdW energy, VATR, between rectangular bodies 1 and 2 in the twisted rotational mode. In the schema, the origin of the xyz coordinate system is set at the COM of body 1, and the x axis is directed perpendicular to this figure face. The schema presents the x-directional projections of body 1 and inclined body 2, separated by dx (≠ 0) in a twisted configuration, viewed from the negative side of the x axis (e.g., corresponding to the xdirectional view of bodies 1 and 2 in Figure 2a). The rotational axis of body 2 is parallel to the x axis through its COM, whose coordinates are denoted by xG, yG, and zG. As shown in Figure 3a, body 2 is divided into three different domains (with yellow-green dotted lines), which are named from the top: upper delta, tetragon, and lower delta. Each domain is further divided into many thin plates parallel to body 1. (A thin plate in each domain is highlighted by the blue color in body 2). Thus, the vdW energy, VATR, between body 1 and inclined body 2 in twisted configurations is obtained by summing all of the local vdW energy between body 1 and each thin plate in three domains (denoted by Vpu, Vpt, and Vpl,

respectively). Responding to the rotation of body 2, all of the thin plates are coupled to slide in the direction parallel to body 1. 3-2-1. Orientation-Dependent Dimensions and Positions of Thin Plates of Tilted Body 2. To calculate the OD-vdW energy VA, the dimensions of the thin plates and the separations between each thin plate and body 1 are needed. Table 1 shows such calculated parameters, which have different forms in the angle ranges of θ > θc = tan−1(t2/L2) (Figure 3a) and θ ≤ θc (the lower in Figure 3b); the parameters for θ ≤ θc are shown on the upper side of each parameter line, and those for θ > θc are shown on the lower side. At θ = θc, the tetragon domain disappears, and the volume of the upper and lower delta D

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Langmuir Table 1. Parameters for Calculating the OD-vdW Energy in the Twisted and Coplanar Rotational Modesa

a (1) TR(CR): the twisted (coplanar) rotation mode. (2) ddy (ddx): the thin plate thickness in the TR (CR) mode. (3) q(s) = (s − 1)ddy or (s − 1)ddx: the y-directional separation between the sth and bottom thin plates in each domain in TR mode or the x-directional one in CR mode. (4) dy = −2w1 (coplanar)/0 (anticoplanar).

domains is maximal. On the other hand, at θ = 0° and π/2, the upper and lower delta domains disappear, while the volume of the tetragon domain is maximal. The dimensions of rectangular bodies (i.e., the thickness, width, and length) are denoted by 2t, 2w, and 2L, with suffixes that indicate the body number, 1 or 2. On the other hand, the thickness, width, and length of the thin plates in body 2 are denoted by ddy, 2w2, and pl(s), respectively, where s denotes the thin-plate number (e.g., s = 1 is the number of the bottom thin plate and s = n or 2n is the number of the topmost thin plate in the upper and lower delta domains or the tetragon domain, respectively). The parameter zl(s) is the z coordinate of the left end of the sth thin plate in each domain in the xyz coordinate system inset in Figure 3a (where zl(s) of a thin plate only in the lower delta domain is shown). The s dependence of the thin plate length pl(s) has a different form in the three domains as shown in Table 1 (which is the reason that body 2 is divided into three domains). The parameter q(s) is the y-directional distance between the sth thin plate and the bottom one in each domain, which is expressed by (s − 1)ddy. The maximal heights of the three domains are denoted by hdu, 2ht, and hdl, respectively (Figure 3b). The x-, y-, and z-directional face-to-face distances between body 1 and the sth thin plate (dx, dy(s), and dz(s)) are expressed using xG, yG, zG, q(s), and θ, respectively. 3-3. OD-vdW Energy in the Twisted Rotation Mode. The OD-vdW energy between rectangular bodies in the twisted rotation mode, VATR, is given by

n

VA TR (xG , yG , zG , θ ) =

∑ Vpu TR(xG , yG , zG , θ , s) s=1

n

+∑ Vpl TR (xG , yG , zG , θ , s) + s=1

2n

∑ Vpt TR(xG , yG , zG , θ , s) s=1

(6)

where VpuTR(s), VplTR(s), and VptTR(s) are the local vdW energy values between body 1 and the sth thin plate in the upper delta, lower delta, and tetragon domains, respectively, in body 2 in the twisted rotation mode, and the upper limit of the summations over s is the total number of thin plates in each domain (n or 2n). VpuTR(s), VplTR(s), and VptTR(s) are obtained by introducing the dimensions of the sth thin plate (2w2, ddy, pl(s)), and the face-to-face separations between body 1 and the sth thin plate (dx, dy(s), and dz(s)) in the respective domains into eq 5; to be specific, they are obtained by substituting the expressions in eq 7 in the respective domains into ai, bj, and ck in eq 5 a1 = dz + 2L1; a 2 = dz + 2L1 + pl ; a3 = dz + pl ; a4 = dz , b1 = dy + 2t1; b2 = dy + 2t1 + ddy ; b3 = dy + ddy ; b4 = dy , c1 = dx + 2w1; c 2 = dx + 2(w1 + w2); c3 = dx + 2w2 ; c4 = dx (7)

where dx, dy, dz, ddy, and pl for the respective domains are shown in Table 1. E

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Langmuir 3-4. OD-vdW Energy in the Coplanar Rotation Mode. In Figure 2b, the bottom face of rectangular body 1 and that of body 2 are on a common x−z plane in the orthogonal xyz coordinate system inset. On the plane, body 1 is fixed, while body 2 can take any angles and COM positions under the condition that the bodies never contact each other. To formulate the OD-vdW energy between the rectangular bodies in coplanar configurations, VACR, the schema shown in Figure 3a can be used again if the y axis in the scheme is regarded as the x axis of the coordinate system in Figure 2b. Thus, to calculate VACR, the parameters in the separation/TR section in Table 1 are altered as in the separation/CR section, and the other parameters shown in the upper sections of Table 1 are common to the twisted and coplanar rotational modes. VACR is formulated using the local vdW energy between body 1 and the sth thin plate in the respective domains of body 2 for the coplanar rotational mode, VpuCR(s), VplCR(s), and VptCR(s), as follows

(ii) The interpolation method: The curves of Vpu(s), Vpl(s), and Vpt(s) are smooth functions of s (e.g., as shown in Figures 6, 9b, and 10b). However, if the sth thin plate is in the divergence region, then the local energy value at s deviates from the expected smooth curve, and the deviation becomes increasingly large as the plate approaches the divergence point. The interpolation method which can obtain almost correct energy values (mainly used in this work) is explainable with Figure 4, which shows a VptTR(s) curve between identical rods

n

VA CR (xG , yG , zG , θ , ) =

∑ VpuCR(xG , yG , zG , θ , s) s=1

n

+∑ Vpl CR (xG , yG , zG , θ , s) + s=1

2n

∑ VptCR(xG , yG , zG , θ , s) s=1

(8)

Figure 4. s dependence of VptTR(s) between identical rods (L1 = L2 = 5, w1 = w2 = 1, t1 = t2 = 1); xG = 3.0 (dx =1) and θ = 0.001° in a twisted configuration. Several data points at around s = 100 deviate from the expected curve.

where the upper limit of three summations over s corresponds to the total number of thin plates in the three domains, respectively, and VpuCR(s), VplCR(s), and VptCR(s) are obtained by substituting the expressions in eq 9 in the respective domains into ai, bj, and ck in eqs A3−A7. One or two of the bj’s (j = 1, 4), however, are zero (e.g., for the coplanar configuration shown in Figure 2b, b3 = 0 if w1 ≠ w2 (resulting in eq A4) and b1 = b3 = 0 if w1 = w2 (resulting in eq A6))

with L1 = L2 = 5 and w1 = w2 = t1 = t2 = 1 in a twisted configuration, where the COM of rod 2 is set at xG = 3.0 (thus dx =1) and yG = zG = 0, and the relative angle is 0.001° (thus, almost parallel). At this angle, the volume of the lower and upper delta domains of rod 2 is negligibly small. In Figure 4, s = 1 and 200 indicate the number of thin plates at both ends of the tetragon domain. The values of VptTR(s) around s = 100 deviate from the expected curve (where some points that are greatly deviated are emphasized with red broken lines), and they can be corrected using the interpolation method with a linear curve. 3-6. Validity and Calculation Accuracies of VR and VA (VATR and VACR). (i) It is confirmed that as θ in VA (eqs 6 and 8) approaches 0 or 90°, the energy values of VA are asymptotically near the values calculated using VR. (ii) It is confirmed that the values of the vdW energy between rectangular bodies in various parallel-orienting configurations calculated by Hallez and using Rocco−Hoover’s formulas (cf., Tables 1 and 2 in ref 29) are in good agreement with those calculated using VR (eqs 5 and A3−A5) for the same body sizes and configurations. (iii) The calculation accuracies for VA depend on the total number of the thin plates, n. With increasing n, the VA(n) values asymptotically approach a constant value denoted by C, and thus their calculation accuracies are defined here as 1 − VA(n)/C. For example, in CR mode the typical accuracies are approximately 4 × 10−5 (n = 500) at 90° and 8 × 10−4 (n = 700−3000, depending on the closest separation) at 45°, whereas in the TR mode they are approximately 3 × 10−6 (n = 50) at 1° and 1 × 10−3 (n = 700) at 45°. At these n values, the computation for the OD-vdW energy VA is quickly done at any angle. In this work, the closest (face-to-face or corner-to-face) separations scaled (divided) by the body width (2w2) are

a1 = dz + 2L1; a 2 = dz + 2L1 + pl ; a3 = dz + pl ; a4 = dz ; b1 = dy + 2w1; b2 = dy + 2(w2 + w1); b3 = dy + 2w2 ; b4 = dy ; c1 = dx + 2t1; c 2 = dx + 2t1 + ddx ; c3 = dx + ddx ; c4 = dx (9)

where ddx is the thickness of the thin plates (Table 1). 3-5. Calculation Methods of the vdW Energy for Divergence Configurations. Assuming ck ≠ 0, VR (eq 5) diverges at ai = 0 or bj = 0; there are various divergence configurations for a pair of rectangular bodies, and the details are shown in the Appendix. Thus, VpuTR(CR)(s), VplTR(CR)(s), and VptTR(CR)(s) diverge at ai or bj = 0 because they are based on eq 5 (hereafter, the double superscript TR(CR) is omitted for simplicity, e.g., VpuTR(CR)(s) → Vpu or VATR(CR) → VA). In addition, VR and these local vdW energy functions have large values at |ai| or |bj| < 10−8−10−7, leading to incorrect energy values. The region immediately next to the divergence point together with the point (i.e., ai ≅ 0 or bj ≅ 0) is called here the divergence region. In the calculations of VA executed in this work, the thin plates scarcely entered the divergence regions, except for θ ≈ 0 or 90°. To calculate Vpu(s), Vp(s), and Vpt(s) in the divergence region with high accuracies, the following methods can be used. (i) The approximate method with modified VR functions: If a thin plate in body 2 enters the divergence region, then its local vdW energy Vp can be approximated using one of the modified VR’s or eqs A3−A7 (Appendix). F

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Langmuir Table 2. θ Dependence of Vs and Ls for dx = 0.2 and 1 dx 0.2

Vs Ls

1

Vs Ls

1°

3°

6°

12°

25°

45°

89°

−1478.6 180

−493.0 60

−246.9 25

−124.1 15

−61.0 10

−36.5 5

−25.8 3

−1.21 20

−0.87 10

−16.4 200

approximately 0.04, and the scaled face-to-face separations for stably suspended colloidal particles are larger than 0.2.15 It should be noted that for the case in which the corner of body 2 is in the region near contact (in the scheme of Figure 3a), the convergence of VA(n) becomes very slow. The reason is that the effective interaction region is localized around the corner in this case, where VA becomes very large, and thus the difference between the shape of the corner of the rectangular bodies and our discrete model causes a large difference between their VA values, with numerous plates being required to reduce the difference, resulting in a very high computational cost. However, if the shape of the corner can be approximated by the discrete model, then the present approach is still applicable even in the region near contact.

4. NUMERICAL CALCULATIONS OF THE OD-VDW ENERGY The OD-vdW energy between rectangular bodies 1 and 2 in the twisted and coplanar rotational modes, VATR and VACR, respectively, are functions of θ and (xG, yG, zG) as shown in eqs 6 and 8, in addition to their dimensions, (L1, t1, w1) and (L2, t2, w2). To test these functions obtained, the energy values between identical bodies 1 and 2 of L1 = L2 (variables) and w1 = w2 = t1 = t2 = 1 (fixed) in several configurations are calculated and shown in Figures 5−8 (twisted rotation) and Figures 9 and 10 (coplanar rotation). 4-1. Calculation Results for the OD-vdW Energy in the Twisted Rotation Mode. Figure 5a,b shows the L dependence of VATR between identical rods of w1 = w2 = 1 and t1 = t2 = 1 for L1 = L2 = L = 1−100 at relative angles of 3, 6, and 12° and that for L = 1−50 at 25, 45, and 89°, respectively. The COM of rod 2 is fixed at xG = 2.2 (or dx = 0.2) and yG = zG = 0. The inset shows rods 1 (white) and 2 (gray) arranged at a relative angle in a twisted configuration and separated by dx (in the direction perpendicular to this figure face). The arrows near the rod ends mean that the length of both rods, L, is simultaneously increased from 1 to 100 in Figure 5a and from 1 to 50 in Figure 5b. With increasing L, all the curves of VATR sharply decrease and level off beyond a length Ls (depending on θ), where the size effect to VATR disappears, indicating that the part longer than Ls in rod 2 scarcely contributes to VATR. Table 2 shows the Ls values and the level-off values of VATR (denoted by Vs) at angles of 1−89° for dx = 0.2, together with those at angles of 3, 45, and 89° for dx = 1 (as a reference). With increasing θ, Ls decreases from 180 to 3 while Vs increases from −1479 to −26; these values of Vs are markedly lower, and those of Ls are relatively smaller, being compared to those of Ls and Vs for dx = 1, respectively. Because θ is larger and dx is smaller, the length effect on VATR disappears at smaller Ls. Figure 6 shows the s dependence of VptTR(s) between rod 1 (L1 = 5, and w1 = t1 = 1) and the sth thin plate in the tetragon domain of rod 2 (L2 = 5 and w2 = t2 = 1) oriented at θ = 89° in a twisted configuration; rods 1 and 2 are separated by dx in the

Figure 5. Length dependences of VATR between identical rods of L1 = L2 = L = 1−50 or 100, w1 = w2 = 1, and t1 = t2 = 1 for dx = 0.2 at relative angles of (a) 3, 6, and 12° and (b) 25, 45, and 89°. The inset shows rods 1 (white) and 2 (gray) separated by dx ≠ 0 (perpendicular to this figure face); the arrows near the rod ends express that the length of both rods, L, is simultaneously increased.

x direction (cf. Figure 2a), and the COM of rod 2 is set at yG = zG = 0 and xG = 2.05, 2.2, or 3.0 (i.e., dx = 0.05, 0.2, or 1.0, respectively). At θ = 89°, the volumes of the upper and lower delta domains in rod 2 are negligibly small. The tetragon domain is divided into 200 thin plates (n = 100), which are numbered from the bottom plate (s = 1) to the topmost one (s = 200), and thus the s number of the central thin plate is 100, where zG = 0. In Figure 6, the values of VptTR(s) for xG = 2.2 (dx = 0.2) are magnified to 15 times the original ones, and those for xG = 3.0 (dx =1.0) are magnified to 500 times the original ones. The VptTR(s) curves are minimal at s = 100 (the center) and maximal at s = 1 and 200 (both ends). With decreasing xG (or dx), the valley depth of the curve drastically increases and the valley width approaches the thickness of rod 1 or 2t1 = 2 (corresponding to s = 80 to 120). This indicates that, with G

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a twisted configuration (dx ≠ 0) are shown, and the directions of the positive rotation (θ) and displacement (zG) of rod 2 are indicated by the arced and straight arrows, respectively. For zG = 0, the value of VATR is minimal at θ = π/2 and maximal at θ = 0°. With increasing zG, the angle θm giving the minimal energy value shifts to lower values, and the difference between minimal and maximal energy values becomes smaller; that is, the effect of rotation to VATR is reduced. The vdW dispersion torques (−∂VA/∂θ) change sign at θm. Figure 8a shows the θ dependence of VATR between identical rods 1 and 2 (L1 = L2 = 5) at angles of between 0 and 90°,

Figure 6. s dependence of VptTR(s) between rod 1 (L1 = 5, w1 = 1, t1 = 1) and the sth thin plate in rod 2 oriented at θ = 89°. As dx decreases, the region contributing to VATR in rod 2 converges to the overlapping region indicated by light blue in the inset. (Top, dx =1.0; middle, 0.2; bottom, 0.05).

decreasing dx, the region (in rod 2) which effectively contributes to VATR between rods 1 and 2 (that is, the effective interaction region) converges to the crossing region, highlighted in light blue in the inset (where rods 1 and 2 are separated by dx in the direction perpendicular to this figure face). The relative increase in Ls and the drastic decrease in Vs with decreasing θ, shown in Table 2, can be explained by the increase in the effective interaction region with decreasing θ. With a color map, Hallez showed the local vdW energy distribution between a sphere and a cylindrical pore in a plate of finite thickness.29 Figure 7 shows the θ dependence of VATR between identical rods 1 and 2 (L1 = L2 = 5) at angles of between 0 and 90°. The COM of rod 2 is set at xG = 3 (dx = 1), yG = 5, and zG = 0, 5, or 7.5. In the inset, the x-directional projections of rods 1 and 2 in

Figure 8. (a) θ dependence of VATR between identical rods 1 and 2 (L1 = L2 = 5, w1 = w2 = 1, t1 = t2 = 1) at angles of between 0 and 90°. The COM of rod 2 is set at xG = 3 (dx = 1), yG = 0, and zG = 0, 5, 15/2. (b) θ dependence of VATR between identical rods of three different lengths of L1 = L2 = L = 5, 10, 50 for dx = 0.2.

where the COM of rod 2 is set at xG = 3 (dx =1), yG = 0, and zG = 0, 5, 7.5; the inset shows the x-directional projection of the rods in a twisted configuration for yG = 0 and zG ≠ 0. Because of yG = 0, all the curves of VATR are minimal at θ = 0°. As zG increases, the difference between minimal and

Figure 7. θ dependence of between identical rods 1 and 2 (L1 = L2 = 5) at relative angles of between 0 and 90° for dx =1. In the inset, the directions of the rotation (θ) and displacement (zG) of rod 2 are indicated by the arced and straight arrows, respectively. VATR

H

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Langmuir maximal energy values becomes smaller. The effect of rotation to VATR is reduced as the separation between the rods increases. Figure 8b shows the θ dependence of VATR between identical rods of three different lengths of L1 = L2 = 5, 10, 50. The COM of rod 2 is set at xG = 2.2 (dx =0.2) and yG = zG = 0. The inset shows the x-directional projection of rods 1 and 2 in a twisted configuration for yG = zG = 0, where the arced arrow shows the positive rotational direction of rod 2, and the straight arrows around the rod ends indicate that the length of both rods is increased at the same time (L1 = L2 = 5, 10, 50). Because of yG = 0, all of the curves of VATR are minimal at θ = 0°, and the minimal value decreases with increasing L. Comparing the curve of VATR for L = 5 and dx = 0.2 in Figure 8b with the bottom curve in Figure 8a (where L = 5 and dx = 1) suggests that the variations of VATR with θ are drastically intensified with decreasing dx. Also, with increasing θ these three curves of VATR are close enough to one another to be identical at 90°. In other words, as seen from the inset of Figure 6, their effective interaction regions are identical at 90°. In Figure 8a,b, VA curves of crossing rods for yG = 0 in the TR mode are minimal at θ = 0°, and the vdW dispersion torques are negative, indicating that the torques align the rods to be parallel. This agrees with previously reported results.39,42 Figure 8a,b also shows that the torques dramatically increase with decreasing dx and increasing L whereas they become minimal at around 90°. 4-2. Calculation Results for the OD-vdW Energy in the Coplanar Rotation Mode. Figure 9a shows the L dependence of VACR/L (L = 1 to 30) between identical rods in the coplanar configuration for θ = 0°. The COM of rod 2 is set at yG = zG = 0 and xG = 2.1 or 2.5 (i.e., dx = 0.1 or 0.5, respectively). At θ = 0°, the upper and lower delta domains disappear. The inset shows rods 1 and 2 in a coplanar configuration for θ = 0°, where the COM of rod 2 is set at yG = zG = 0. Both rods are shown in white to distinguish them from rods in twisted configurations, and the arrow around the rod ends indicates that the length of both rods is increased simultaneously (from L = 1 to 30). With increasing L, VACR/L steeply decreases and levels off at a length of Lc. The value of Lc increases with dx (e.g., Lc = 3 for dx = 0.1 and Lc = 10 for dx = 0.5). The decrease in VACR/L in the range of L < Lc is due to the end effect of the rods, and the increase in Lc with the separation distance is due to the broadening of the effective interaction region. Figure 9b shows the s dependence of the local vdW energy between rod 1 and the sth thin plate in the tetragon domain of rod 2 in the coplanar rotational mode, VptCR(s). Rods 1 and 2 (L1 = L2 = 10) are aligned almost parallel (θ = 0.01°) in a coplanar configuration, and the COM of rod 2 is set at xG = 2.1, 2.2, or 2.7 (i.e., dx ≈ 0.1, 0.2, or 0.7, respectively, because of θ = 0.01°) and yG = zG = 0. At θ = 0.01°, the volumes of the upper and lower delta domains of rod 2 are negligibly small. The inset shows rod 2 of yG = zG = 0, almost parallel to rod 1. The tetragon domain is divided into 400 thin plates (n = 200), and thus the s numbers of the bottom and top thin plates are 1 and 400, respectively. For dx ≈ 0.7, 0.2, and 0.1, VptCR(s) approaches 0 at around s = 250, 150, and 100, respectively. This indicates that as dx is decreased, VptCR(s) rises more sharply to approach 0 at a smaller s value. For dx ≈ 0.1, the effective interaction region is localized between the thin plates of s = 1 and 100 in rod 2, that is, from the surface to the depth of approximately one quarter the thickness of rod 2 (2t2 = 2). The width and depth of the

Figure 9. (a) L dependence of VACR/L at L = 1 to 30 for parallelaligned identical rods 1 and 2 of yG = zG = 0 as in the inset. (The rods in coplanar configurations are shown hereafter in white.) The values of VACR/L for dx = 0.5 are magnified to 30 times the original ones. (b) s dependence of VptCR(s) between rod 1 and the sth thin plate of rod 2 with dx = 0.1, 0.2, or 0.7. Rods 1 and 2 (L1 = L2 = 10, w1 = w2 = 1, t1 = t2 = 1) are aligned to be almost parallel (θ = 0.01°). The values of VptCR(s) for xG = 2.2 (dx = 0.2) are magnified to 7 times the original ones, and those for xG = 2.7 (dx = 0.7) are magnified to 100 times the original ones. The inset shows almost parallel-aligned rods 1 and 2, and s denotes a thin plate number of between 1 and 2n = 400 in the tetragon domain.

effective interacting region can be estimated by VpTR and VpCR, respectively, as seen in Figures 6 and 9b. Figure 10a shows the θ dependence of VACR between identical cubes 1 and 2 of L1 = L2 = t1 = t2 = w1 = w2 = 1, and the COM of cube 2 is set at xG = 3, yG = 0, and zG = 0, 1.5, or 3. The inset shows cube 2 of xG = 3 and yG = 0 oriented at θ = π/ 4 in a coplanar configuration, where the tetragon domain disappears. For zG = 0, cube 2 is the closest to cube 1 at θ = π/4, where VACR between the cubes is minimal. With increasing zG, the minimum peak shifts to lower angles, and the difference between the maximal and minimal values of VACR decreases. In other words, the effect of rotation to VACR is relatively reduced with increasing separation distance. As two bodies come close each other, the effective interaction region is localized, and the I

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increase in the thin plate volume (pl(s) × ddx × 2w2) with s and of the decrease in VplCR(s) due to the increase in dx(s) with s. For xG = 3.0 and 2.7, the minimum peak of VplCR(s) appears at around s = 150 and 100, respectively, in the lower delta domain, but the right end part of the curves enters the upper delta domain. For xG = 2.5, the minimum appears at around s = 50, and the values of VplCR(s) at s > 600 are negligibly small in the lower delta domain. Thus, with decreasing xG, the peak depth increases drastically, and the peak width narrows sharply. The effective interaction region is localized around the corner of cube 2 closest to cube 1, which extends from the thin plate of s = 1 to that of s = 600, that is, from the corner to a distance (depth) of half the lower delta domain height (hdl in Table1). This indicates that the rest of the region in the lower delta domain and the entire upper delta domain scarcely contribute to VACR. As the cubes come close to each other, the effective interaction region is more sharply localized to the vicinity of the corner of cube 2. In Figure 10a, two cubes 1 and 2 are closest to each other at θ = 45° for zG = 0, where VA is minimal in the CR mode. Also, in Figure 7, two rods are closest at θ = 90° for zG = 0 in TR mode, where VA is minimal. These examples indicate that the vdW forces orientate body 2 to the direction in which body 2 is closest to body 1, according to the fixed COM position of body 2; the end-to-end orientation for plates or laths is another example.40

5. CONCLUSIONS In this work, we have derived formulas for numerically calculating the OD-vdW energy between macroscopic dielectric rectangular bodies in the nonretardation regions. These formulas enable the systematic evaluation of the θ dependence of the vdW energy for anisotropic bodies with finite sizes. The effect of rotation on the vdW energy between the bodies is drastically enhanced by their approach. The OD-vdW energy between rods reveals two different critical lengths above which the effect of the length to the vdW energy disappears. The localization behavior of the effective interaction region can be estimated from the distribution curves of the local vdW energy. Our formulas enable quick numerical calculations of the energy with high accuracies and can be a useful model for evaluating the actions of the OD-vdW forces appearing in various systems of anisotropic dielectric bodies.

Figure 10. (a) θ dependence of VACR between identical cubes 1 and 2. The zG values of cube 2 are 0, 1.5, and 3. In the inset, cube 2 is oriented at θ = π/4. The arced arrow indicates the direction of rotation, and the straight arrow indicates the increasing direction in zG of rod 2. (b) s dependence of VplCR(s) between cube 1 and the sth thin plate in the lower delta domain of cube 2 oriented at 44.9°. The xG values of cube 2 are 2.5, 2.7, and 3.0. The values of VplCR(s) for xG = 2.7 and 3.0 are magnified to 10 and 50 times the original ones, respectively. The inset shows cubes 1 and 2 oriented at θ = 45°. The arrow directed downward indicates the decreasing direction of xG.

■

APPENDIX

A1. Divergence Configurations and Base Energy Functions

Assuming here that ck ≠ 0 (k = 1, 4), VR (eq 5) diverges at ai or bj = 0 (i, j = 1, 4). (i) For a1 = 0 (i.e., dz = −2L1 from eq 3), the left-side face of body 2 and that of body 1 are on the same plane (when viewed from the negative direction of the x axis in Figure 1). (ii) For a2 = 0 (thus dz = −2(L1 + L2)), the right face of body 2 and the left face of body 1 are on the same plane. (iii) For a3 = 0 (dz = −2L2), the right face of body 2 and that of body 1 are on the same plane. (iv) For a4 = 0 (dz = 0), the left face of body 2 and the right face of body 1 are on the same plane. The same explanation holds true for bj = 0 (j = 1, 4), if dz, L1, and L2 are replaced with dy, t1, and t2, respectively (compare ai with bj in eq 3), and the “left” and “right” faces are replaced with the “lower” and “upper” faces, respectively. The other conditions causing the divergence of VR between the parallel-orienting rectangular bodies in Figure 1 are shown below (B−D):

vdW energy and its gradient dramatically increase. Thus, the changes in the position and volume of the effective interaction region with θ induce a large change in the vdW energy. This is the reason that the variation of the vdW energy with θ for close bodies is considerably larger than that for remote bodies. Figure 10b shows the s dependence of VplCR(s) between cube 1 (L1 = t1 = w1 = 1) and the sth thin plate in the lower delta domain of cube 2 (L2 = t2 = w2 = 1) oriented at θ = 44.9° (where the tetragon domain mostly disappears). The COM of cube 2 is set at yG = zG = 0 and xG = 2.5, 2.7, or 3.0. The inset shows cubes 1 and 2 of yG = zG = 0 oriented at θ ≈ 45°. The lower delta domain is divided into 800 thin plates (n = 800), where the thin plates are numbered from the bottom plate (s = 1) to the topmost one (s = 800). In Figure 10b, the minimal peak in the three curves is caused by the multiplier effect of the increase in VplCR(s) due to the J

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Moreover, if an and bm = 0 at the same time, then VR (eq 5) is

(A) one-face coplanar: (1) ai = 0, (2) bj = 0; (B) two-faces coplanar: (1) ai = 0 and bj = 0, (2) a1 = 0 and a3 = 0 (L1 = L2), (3) b1 = 0 and b3 = 0 (t1 = t2); (C) three-faces coplanar: (1) a1 = 0, a3 = 0, and bj = 0, (2) b1 = 0, b3 = 0, and ai = 0; (D) four-faces coplanar: (1) a1 = 0, a3 = 0, b1 = 0, and b3 = 0 (L1 = L2 and t1 = t2). Under the above conditions, there are two types of coplanar configurations; one is that two bodies are on the same side of each common plane, corresponding to the case of a1 = 0 or a3 = 0 (b1 = 0 or b3 = 0), and the other is that the respective bodies are on the different sides of each common plane, that is, anticoplanar, corresponding to the case of a2 = 0 or a4 = 0 (b2 = 0 or b4 = 0). For the configurations of ai = 0 or bj = 0, following antiderivative V′xyz or V″xyz is used to calculate VR (eq 5). V ′xyz (x , y , 0) =

summation of Vxyz(ck, bj, ai) over i and j, respectively, in the first term on the right side of eq A5, and the second term with V′zyx(ck, bj, 0) and the third term with V″xyz(ck, 0, ai) are added. 4

VR (dx , dy , dz) =

4

+∑

VR (dx , dy , dz) =

4

∑

( −1)(i + m + k)V ″xyz (ck , 0, ai) (A5)

It should be noted that the term with V′xyz(ck, 0, 0) is absent in eq A5. This is because V′yz(x, 0, 0) = 0. A2. Base Energy Functions for the Coplanar Rotational Mode

The base energy function for the CR mode is VR between parallel-orienting bodies 1 and 2 on the four common planes, corresponding to the case of b1 = 0, b2 = 0, b3 = 0, and b4 = 0 in the xyz coordinate system in Figure 2b. In these cases, the base energy function is given by eq A4; in addition, if w1 = w2 (and b1 = b3 = 0), then VR is modified as in eq A6, where terms j = 1 and 3 are excluded from the summation of Vxyz(ck, bj, ai) over j

4

( −1)(i + j + k)Vxyz(ck , bj , ai)

∑∑ ∑

( −1)(n + j + k)V ′xyz (ck , bj , 0)

k = 1 i = 1(≠ n)

(A2)

4

4

∑

4

+∑

where V′xyz(x, y, 0) is the second antiderivative of Vyz(x, y, z = 0) over x, where Vyz(x, y, z = 0) is the second antiderivative of Vz(x, y, z = 0) over y. Also, V″xyz (x, 0, z) is the second antiderivative of Vyz(x, y = 0, z) over x, where Vyz(x, y = 0, z) is obtained by putting y = 0 into the second antiderivative of Vz(x, y, z) over y. Some examples of the base energy functions are shown in eqs A3−A7. When an = 0, VR (eq 5) is partially modified with V′xyz(x, y, 0), as shown in eq A3, 4

( −1)(i + j + k)

k = 1 j = 1(≠ m)

(A1)

⎛ 1 z2 ⎞ log⎜1 + 2 ⎟ 16 ⎝ x ⎠

4

∑

k = 1 j = 1(≠ m) i = 1(≠ n)

1 ⎛x z ⎞ −1⎛ z ⎞ ⎜ − ⎟tan ⎜ ⎟ 16 ⎝ z x⎠ ⎝x⎠

V ′′xyz (x , 0, z) = −

4

∑ ∑

Vxyz(ck , bj , ai)

⎛ y⎞ y⎞ 1 ⎛x ⎜ − ⎟tan−1⎜ ⎟ 32y ⎝ y x⎠ ⎝x⎠

⎛ y2 ⎞ 1 log⎜1 + 2 ⎟ + 32 x ⎠ ⎝

−

modified so that terms i = n and j = m are excluded from the

and instead the second and third terms with V″xyz(ck, 0, ai) are

k = 1 j = 1 i = 1(≠ n) 4

introduced.

4

+ ∑ ∑ ( −1)(n + j + k)V ′xyz (ck , bj , 0)

4

k=1 j=1

VR (dx , dy , dz) =

(A3)

4

4

∑ ∑ ∑ (−1)(i+ j + k)

Vxyz(ck , bj , ai) 4

4

+ ∑ ∑ ( −1)(i + 1 + k)V ″xyz (ck , 0, ai) k=1 i=1 4

4

+ ∑ ∑ ( −1)(i + 3 + k)V ″xyz (ck , 0, ai)

4

k=1 i=1

∑ ∑ ∑ (−1)(i+ j + k)Vxyz(ck , bj , ai) k = 1 j = 1(≠ m) i = 1

4

4

k = 1 j = 1(≠ 1,3) i = 1

where the term of i = n is excluded from the summation of Vxyz(ck, bj, ai) over i in the first term of the right side of eq A3 and instead the second term with V′zyx (ck, bj, 0) is added. Similarly, if bm = 0, VR (eq 5) is modified using V″zyx (ck, 0, ai) as shown in eq 4, where the term j = m is excluded from the summation of Vxyz(ck, bj, ai) over j and instead the second term is added. VR (dx , dy , dz) =

4

(A6)

Similarly, if identical rods 1 and 2 are coplanar and parallel as

4

+ ∑ ∑ ( −1)(i + m + k)V ″xyz (ck , 0, ai)

shown in Figure 9a (i.e., b1 = b3 = 0, and a1 = a3 = 0), then VR is

k=1 i=1

given as follows,

(A4) K

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VR (dx , dy , dz) =

4

∑ ∑

4

∑

(11) Verway, E. J. W.; Overveek, J. Th. Theory of the Stability of Lyotropic Colloids; Elservier: Amsterdam, 1948. (12) Davidson, P.; Gabriel, J. C. P. Mineral liquid crystals. Curr. Opin. Colloid Interface Sci. 2005, 9, 377−383. (13) Lekkerkerker, H. N. W.; Vroege, G. J. Liquid crystal transitions in suspensions of mineral colloids: new life from old roots. Philos. Trans. R. Soc. A 2013, 371, 1471−2962. (14) Frenkel, D.; Lekkerkerker, H. N. W.; Stroobants, A. Thermodynamic stability of a smectic phase in a system of hard rods. Nature 1998, 332, 822−3. (15) Maeda, H.; Maeda, Y. Spontaneous formation of string-like clusters and smectic sheets for colloidal rods confined in thick wedgelike gaps. Langmuir 2013, 29, 10529−10538. (16) Maeda, H.; Maeda, Y. Liquid crystal formation in suspensions of hard rodlike colloidal particles: Direct observation of particle arrangement and self-ordering behavior. Phys. Rev. Lett. 2003, 90, 18303−18306. (17) Maeda, H.; Maeda, Y. Atomic force microscopy studies for investigating the smectic structures of colloidal crystals of β-FeOOH. Langmuir 1996, 12, 1446−1452. (18) Hamaker, H. C. The London-van der Waals attraction between spherical particles. Physica 1937, 4, 1058−72. (19) Lifshitz, E. M. The theory of molecular attractive forces between solids. Soviet Phys. - JETP 1956, 2, 73. (20) Casimir, H. B. G.; Polder, D. The influence of retardation on the London-van der Waals forces. Phys. Rev. 1948, 73, 360−372. (21) Tabor, D.; Winterton, R. H. S. The direct measurement of normal and retarded van der Waals forces. Proc. R. Soc., Sect. A 1969, 312, 435−440. (22) Israelachivili, J. N.; Tabor, D. The measurement of van der Waals dispersion forces in the range 1.5 to 130 nm. Proc. R. Soc., Sect. A 1972, 331, 19−38. (23) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Direct measurement of colloidal forces using an atomic force microscope. Nature 1991, 353, 239−241. (24) Cappela, B.; Dietler, G. Force distance curves by atomic force microscopy. Surf. Sci. Rep. 1999, 34, 1−104. (25) Butt, H. J. Measuring electrostatic, van der Waals, and hydration forces in electrolyte solutions with an atomic force microscope. Biophys. J. 1991, 60, 1438−1444. (26) Leite, F. L.; Mattoso, L. H. C.; Oliveira, O. N., Jr.; Herrmann, P. S. P., Jr. The Atomic Force Spectroscopy as a Tool to Investigate Surface Forces: Basic Principles and Applications. In Modern Research and Educational Topics in Microscopy; Méndez-Vilas, A., Díaz, J., Eds.; Formatex, 2007; Vol. 2, pp 747−757. (27) Prieve, D. C. Measurement of colloidal forces with TIRM. Adv. Colloid Interface Sci. 1999, 82, 93−125. (28) Roth, C. M.; Neal, B. L.; Lenhoff, A. M. Van der Waals interactions involving proteins. Biophys. J. 1996, 70, 977−987. (29) Hallez, Y. Analytical and numerical computations of the van der Waals force in complex geometries: Application to the filtration of colloidal particles. Colloids Surf., A 2012, 414, 466−476. (30) Jaiswal, R. P.; Beaudoin, S. P. Approximate scheme for calculating van der Waals interactions between finite cylindrical volume elements. Langmuir 2012, 28, 8359−8370. (31) Yang, P.; Qian, X. A general, accurate procedure for calculating molecular interaction force. J. Colloid Interface Sci. 2009, 337, 594− 605. (32) Langbein, D. Retarded dispersion energy between macroscopic bodies. Phys. Rev. B 1970, 2, 3371−3383. (33) Thennadil, S. N.; Garcia-Rubio, L. H. Approximation for calculating van der Waals interaction energy between spherical particles: A comparison. J. Colloid Interface Sci. 2001, 243, 136−142. (34) Stedman, T.; Drosdoff, D.; Woods, L. M. Van der Waals interactions between nanostructures: Some analytic results from series expansions. Phys. Rev. A 2014, 89, 012509. (35) Priye, A.; Marlow, W. H. Computation of Lifshitz-van der Waals interaction energies between irregular particles and surfaces at all

( −1)(i + j + k)

k = 1 j = 1(≠ 1,3) i = 1(≠ 1,3)

Vxyz(ck , bj , ai) 4

+∑

4

∑

( −1)(i + 1 + k)V ″xyz (ck , 0, ai)

k = 1 i = 1(≠ 1,3) 4

+∑

4

∑

( −1)(i + 3 + k)V ″xyz (ck , 0, ai)

k = 1 i = 1(≠ 1,3) 4

+∑

4

∑

( −1)(1 + j + k)V ′xyz (ck , bj , 0)

k = 1 j = 1(≠ 1,3) 4

+∑

4

∑

( −1)(3 + j + k)V ′xyz (ck , bj , 0)

k = 1 j = 1(≠ 1,3)

(A7)

Moreover, when coplanar bodies 1 and 2 in the xyz coordinate system in Figure 2b are parallel, if a face, perpendicular to the x axis, of body 1 and that of body 2 are on the same planes, corresponding to the case of c1 = 0, c2 = 0, c3 = 0, and c4 = 0, then VR (eq A4) diverges (even if the bodies are not in contact with each other). This is avoided by rotating the xyz coordinate system in Figure 2b around the y axis so that the x axis is on the same planes, each corresponding to ai = 0 (i = 1, 4) in the xyz coordinate system after rotation. In this case, VR is given by eq A5. Incidentally, eqs 5 and A6 were used to calculate VATR in Figures 4−8 and VACR in Figures 9 and 10, respectively, together with the interpolation method if necessary.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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REFERENCES

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M

DOI: 10.1021/acs.langmuir.5b01459 Langmuir XXXX, XXX, XXX−XXX