Reconstruction of three-dimensional anisotropic structure from small

PHYSICAL REVIEW E 96, 022612 (2017)

Reconstruction of three-dimensional anisotropic structure from small-angle scattering experiments Guan-Rong Huang,1,2,3 Yangyang Wang,4,* Bin Wu,3 Zhe Wang,3 Changwoo Do,3 Gregory S. Smith,3 Wim Bras,3 Lionel Porcar,5 Péter Falus,5 and Wei-Ren Chen2,3,† 2

1 Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Shull Wollan Center, The University of Tennessee and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3 Biology and Soft Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 4 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 5 Institut Laue-Langevin, Boîte Postale 156, F-38042 Grenoble Cedex 9, France (Received 14 June 2017; published 28 August 2017)

When subjected to flow, the structures of many soft-matter systems become anisotropic due to the symmetry breaking of the spatial arrangements of constituent particles at the microscopic level. At present, it is common practice to use various small-angle scattering techniques to explore flow-induced microstructural distortion. However, there has not been a thorough discussion in the literature on how a three-dimensional anisotropic structure can be faithfully reconstructed from two-dimensional small-angle scattering spectra. In this work, we address this issue rigorously from a mathematical perspective by using real spherical harmonic expansion analysis. We first show that, except for cases in which mechanical perturbation is sufficiently small, the existing small-angle scattering techniques generally do not provide complete information on structural distortion. This limitation is caused by the linear dependence of certain real spherical harmonic basis vectors on the flow-vorticity and flow-velocity gradient planes in the Couette shear cell. To circumvent the constraint imposed by this geometry, an alternative approach is proposed in which a parallel sliding plate shear cell is used with a central rotary axis along the flow direction. From the calculation of rotation of the reference frame, we demonstrate the feasibility of this experimental implementation for a fully resolved three-dimensional anisotropic structure via a case study of sheared polymers. DOI: 10.1103/PhysRevE.96.022612 I. INTRODUCTION

The spatial arrangement of the constituent particles of a soft-matter system is driven out of the equilibrium configurations under flow and deformation. There has been much scientific as well as technological interest in understanding how structural distortion at the particle level is connected to macroscopic rheological properties [1,2]. Two-point correlation functions, such as the pair distribution function g(r) in real space or the interparticle structure factor S( Q) in reciprocal Q space, are of particular interest for rheology of complex fluids because these quantities can be probed experimentally. They have been used extensively in theoretical and computational investigations in the past as a vehicle for exploring the microscopic origin of viscoelastic behavior in nonequilibrium liquids. Recent studies of experiments and theory on colloidal dispersions were also able to investigate three-dimensional (3D) structural deformation using confocal microscopy experiments, Brownian and molecular-dynamic simulations, and mode-coupling theory [3–7]. To quantify the structural distortion under different flow conditions, the anisotropic pair distribution function g(r) has been expressed in terms of real spherical harmonics (RSH) in a series of computational studies in the past [1,8–12]. The coefficients associated with different basis vectors, which are related to a number of flow properties such as shear stress and normal stress difference brought about by interparticle interaction, can be straightforwardly computed due to the orthogonality of RSH functions in 3D space. * †

[email protected] [email protected]

2470-0045/2017/96(2)/022612(10)

In conjunction with these computational studies, there has been a considerable amount of experimental effort, mainly via small-angle scattering (SAS) measurements, to investigate the structural evolution under flow and deformation in the nonlinear regime [13–20]. Unlike computer simulations, 3D structural information has to be reconstructed from twodimensional (2D) spectra in small-angle scattering experiments. In the case of uniaxial stretching, we have recently demonstrated that reconstruction can be accomplished by SAS measurement on a single plane parallel to the stretching direction, due to the cylindrical symmetry of the deformation [21]. However, the reconstruction problem for shear geometry has not been completely solved. Because of the low symmetry of shear deformation, it is still unclear whether SAS experiments on conventional flow-vorticity (xz) and flow-velocity gradient (xy) planes can always provide the full information about microstructural changes. In this work, we address the aforementioned challenge from a mathematical perspective. We demonstrate that SAS experiments on the xz and xy planes of a Couette cell in general only yield partial information on the 3D anisotropic structure. The linear dependence among certain RSH basis vectors on these two planes inherently prohibits full reconstruction when the structural perturbation is large. To bypass this intrinsic constraint of Couette geometry, we propose an alternative parallel-plate shear cell with a central rotary axis along the flow direction. We show that additional information among the coefficients of the relevant RSH functions can be obtained by tilting the parallel plates, which makes up the underdetermined linear equations originating from the linear dependence of the basis vectors and thus permits full restoration of the 3D distorted structure.

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©2017 American Physical Society

GUAN-RONG HUANG et al.

PHYSICAL REVIEW E 96, 022612 (2017)

The paper is organized as follows: In Sec. II, we present a detailed analysis of the anisotropic structure obtained from scattering measurements with the Couette cell, and we demonstrate the validity and limitation of this geometry for structural reconstruction via the RSH analysis. We then proceed to introduce the RSH analysis on the nonequilibrium structure factor obtained from scattering measurements with parallel plates with a rotary axis, and we demonstrate its ability to extract the complete information of structural distortion for sheared polymers in Sec. III. Discussions and conclusions are presented in Sec. IV. II. REAL SPHERICAL HARMONICS EXPANSION OF STRUCTURE FACTOR

First, let us briefly review some of the basics of the real spherical harmonic expansion technique (RSHE). To investigate the anisotropic properties, in Ref. [11] the structure factor of the sample in a rheo-SAS experiment is expanded as a linear combination of RSH: S( Q) =

∞  l 

Slm (Q)Ylm (θ,φ),

satisfy the identities Plm [cos (π − θ )] = (−1)l+m Plm (cos θ ), sin [m(φ + π )] = (−1)m sin(mφ), cos [m(φ + π )] = (−1)m cos(mφ). Therefore, the structure factor is simplified as  S( Q) = Slm (Q)Ylm (θ,φ).

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l,m:even

In rheo-SAS experiments, one does not have full access to the 3D structure factor, but the cross sections of S( Q) on 2D planes. In other words, one now has to deal with 2D bases that are linearly dependent on each other. Therefore, the information on different planes is generally needed to determine the spherical harmonic expansion coefficients. To further demonstrate this point, in the following sections we will consider the truncation of real spherical harmonic expansion at l = 2 and 4, respectively. For the sake of simplicity, the following shorthand notations will be used in our discussion: Slm ≡ Slm (Q), Ylm ≡ Ylm (θ,φ), ξ ≡ cos θ,

(1)

l,m l,m ≡ Ylm (π/2,φ), Yxz ≡ Ylm (θ,φ = 0,π ), Yxy

l=0 m=−l

where S( Q) is the structure factor of molecules, Q is the momentum transfer between the beam and molecules, l and m are both integers, Q is the magnitude of Q, Slm (Q) are coefficients as a function of Q, θ is the angle with respect to the z axis, φ is the polar angle with respect to the x axis on the xy plane, and Ylm (θ,φ) are RSH defined as ⎧√ m |m| ⎪ ⎨ 2Al Pl (cos θ ) sin(|m|φ), m < 0, m Am Ylm (θ,φ) = l Pl (cos θ ), m = 0, ⎪ ⎩ √ m m 2Al Pl (cos θ ) cos(mφ), m > 0,

(3)

l,m ≡ Ylm (θ,φ = π/2,3π/2). Yyz

Coefficient extraction for real spherical harmonics expansion truncated at l = 2

When the degree of microscopic deformation is small, it suffices to truncate the real spherical harmonic expansion at l = 2: S( Q) = S00 Y00 + S2−2 Y2−2 + S20 Y20 + S22 Y22 .

(2)

√ where Am (2l + 1)(l − |m|)!/(l + |m|)!, and Plm are the l = associated Legendre polynomials. Considering the case of shear flow (Fig. 1), where x, y, and z are the direction of flow velocity, velocity gradient, and vorticity, respectively, the S( Q) has the symmetrical conditions S(Q,θ,φ) = S(Q,π − θ,φ) and S(Q,θ,φ) = S(Q,θ,φ + π ). This puts restrictions on the values of l and m: the l and m must both be even, since the Plm and trigonometric functions

(5)

It is easy to see that the structure factor on the xy plane (θ = π/2) is  l,m Slm Yxy S(Q,θ = π/2,φ) ≡ Sxy (Q,φ) = l,m

√ √ √ 5 0 15 2 15 −2 S2 + S2 cos(2φ) + S sin(2φ), = S00 − 2 2 2 2 (6) where the four unknown coefficients require four linearly independent equations (LIEs) to be solved. These equations can be found by carrying out the weighting integral of l,m Sxy (Q,φ) and Yxy with respect to φ,  2π 1 l,m l,m Sxy ≡ Sxy (Q,φ)Yxy (φ)dφ. (7) 2π 0 As a consequence, √ 0,0 Sxy

FIG. 1. Schematic representation of the Couette cell with the reference frame, where v is velocity, ∇v is velocity gradient, ω is vorticity, and the height L is much larger than the gap r. 022612-2

=

S00

−

5 0 S , 2 2

15 2 S , 8 2 15 −2 S . = 8 2

2,2 Sxy = 2,−2 Sxy

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PHYSICAL REVIEW E 96, 022612 (2017)

0,0 2,0 Since Yxy and Yxy bases are coupled together on the xy plane, 2,0 carrying out Sxy will yield no more information. To get more information on the structure factor, we should seek the 2D signal on other planes. On the xz plane, the structure factor truncated at l = 2 is  l,m S(Q,θ,φ = 0,π ) ≡ Sxz (Q,θ ) = Slm Yxz l,m

√

√ 5 0 15 2 2 2 = + S2 (3 cos θ − 1) + S sin θ. (9) 2 2 2 Similarly, we have the following weighting integral(s) of l,m with respect to θ : Sxz (Q,θ ) and Yxz  1 π l,m l,m ≡ Sxz (Q,θ )Yxz sin θ dθ Sxz 2 0  1 1 l,m Sxz (Q,ξ )Yxz dξ. (10) = 2 −1 S00

Therefore, 0,0 = S00 + Sxz



5 2 S . 3 2

with nine undetermined expansion coefficients. As we shall show below, it is not feasible to extract all Slm coefficients on the convectional planes, namely the xy, xz, and yz planes, where only eight LIEs can be established. The structure factor on the xy plane is √

√ 5 0 9 0 15 2 3 5 2 Sxy (Q,φ) = S + S + S − S cos(2φ) 2 2 8 4 2 2 4 4 √

√ 15 −2 3 5 −2 + S − S sin(2φ) 2 2 4 4 √ √ 3 35 4 3 35 −4 S4 sin(4φ) + S4 cos(4φ), (14) + 8 8 √

S00 −

which is a Fourier series with five linearly independent bases: [1, cos(2φ), cos(4φ), sin(2φ), sin(4φ)]. In other words, no matter what weighting function we use, the resulting weighting integrals can yield at most five LIEs, due to the coupling of RSH on the xy plane. Similar to the situation of l = 2, from the weighting integrals, we have

(11) 0,0 Sxy

We emphasize that the left-hand sides of Eqs. (8) and (11) are quantities that could be accessed experimentally. After some straightforward algebraic calculations, we can explicitly express Slm in terms of the weighting integrals: 8 0,0 2,2 − √ Sxy , S00 = Sxz 3 15 2 0,0 16 2,2 2 0,0 − √ Sxy − √ Sxy , S20 = √ Sxz 15 3 5 5 8 2,2 S , S22 = 15 xy 8 2,−2 S . (12) S2−2 = 15 xy Equation (12) is valid for the systems with small structural perturbation. It should be noted that S22 and S2−2 can be measured on the xy plane alone without additional information from the xz plane. Together, the SAS measurements on the xz and xy planes are sufficient for the extraction of all the Slm , provided S( Q) can be truncated at l = 2. Experimentally, however, it is generally difficult to measure the structure on the xy plane, due to technical challenges such as multiple scattering.

2,2 Sxy = 2,−2 Sxy = 4,4 = Sxy 4,−4 = Sxy

S( Q) = S00 Y00 + S2−2 Y2−2 + S20 Y20 + S22 Y22 + S4−4 Y4−4 + S4−2 Y4−2 + S40 Y40 + S42 Y42 + S44 Y44 ,

(13)

(15)

On the other hand, the structure factor on the xz plane is Sxz (Q,θ ) =

√



5 2 2 S P (ξ ) + 3S40 P40 (ξ ) + + 12 2 2 √ √ 5 2 2 35 4 4 S4 P4 (ξ ) + S P (ξ ). + (16) 10 280 4 4

S00

5S20 P20 (ξ )

0,0 2,0 2,2 4,0 4,2 4,4 On the xz plane, [Yxz ,Yxz ,Yxz ,Yxz ,Yxz ,Yxz ] form a linearly dependent set that can be expressed in terms of three proper bases of them. For example,

√ 15 0,0 1 2,0 Y − √ Yxz , 3 xz 3 1 0,0 2 4,0 2,0 = √ Yxz + Yxz − √ Yxz , 5 5 7 0,0 2 2,0 1 4,0 = √ Yxz − √ Yxz + √ Yxz . 7 35 35

2,2 Yxz =

Coefficient extraction for real spherical harmonics expansion truncated at l = 4

As the microscopic deformation becomes larger, the RSHE truncated at l = 2 can no longer precisely describe the scattering spectrum. For example, the l = 4 term of cubic symmetry can be significant for soft-sphere systems [11]. In such a case, we should consider the RSHE of S( Q) up to l = 4:

=

√ 5 0 9 0 S + S , − 2 2 8 4 √ 15 2 15 3 2 S2 − S , 8 16 4 √ 15 −2 15 3 −2 S − S , 8 2 16 4 315 4 S , 128 4 315 −4 S . 128 4

S00

4,2 Yxz 4,4 Yxz

(17)

Alternatively, this can be verified by computing the rank of the matrix whose elements are the coefficients of ξ power in l,m (ξ ). Since the rank of the matrix is 3, we confirm that Yxz this set contains three linearly independent bases. Therefore, measurements on the xz plane provide the following three

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PHYSICAL REVIEW E 96, 022612 (2017)

equations from the weighting integrals with respect to ξ :    5 2 1 2 7 4 0,0 0 S2 + S4 + S , Sxz = S0 + 3 5 5 4   1 2 4 4 2,0 S + S42 − S , = S20 − Sxz 3 2 7 4 2 1 4,0 = S40 − √ S42 + √ S44 . (18) Sxz 5 35 4,0 However, the equation of Sxz is linearly dependent on the other seven LIEs and should be removed. To get a complete set of nine LIEs, one might want to examine the structure factor on the yz plane to gain the remaining two. In particular, an additional equation is needed to decouple S2−2 and S4−2 . However, this turns out to be impossible because the structure factor on the yz plane (φ = π/2,3π/2) is  √ 0 0 5 2 2 0 Syz (Q,θ ) = S0 + 5S2 P2 (ξ ) − S P (ξ ) + 3S40 P40 (ξ ) 12 2 2 √ √ 5 2 2 35 4 4 S4 P4 (ξ ) + S P (ξ ), − (19) 10 280 4 4 which will produce the LIEs, from the weighting integrals, l,m with the same Slm terms as those of Sxz . Since neither Eq. (16) −2 −2 nor Eq. (19) contains S2 and S4 , these coefficients cannot be determined. In other words, S2−2 and S4−4 are coupled together on the xy plane and disappear on the xz and yz planes. From the yz plane, we obtain an additional LIE for the other coefficients:  1 1 4,0 4,0 Syz (Q,ξ )Yyz dξ Syz = 2 −1

2 1 = S40 + √ S42 + √ S44 . 5 35

(20)

Combining Eqs. (15), (18), and (20), we arrive at the following solutions for Slm : √ √ 7 0,0 4 15 2,2 2 7 5 7 0 S0 = Sxy + S + A1 + A2 − A3 , 9 135 xy 9 18 8 √ √ √ √ 5 0,0 44 3 2,2 5 5 23 0 Sxy + Sxy − A1 + A2 − A3 , S2 = 9 135 9 18 8 √ √ √ √ 15 15 15 0,0 4 2,2 5 3 S22 = − Sxy + Sxy + A1 − A2 + A3 , 9 45 9 18 8 128 4,−4 S S4−4 = , 315 xy √ √ 4 0,0 16 15 2,2 4 2 5 1 S40 = Sxy + Sxy − A1 + A2 + A3 , 9 135 9 9 2 √ √

√ 2 0,0 8 15 2,2 2 5 1 2 S4 = − 5 Sxy + S − A1 + A2 − A3 , 9 135 xy 9 9 4 128 4,4 S , 315 xy √ 16 2,−2 S , 2S2−2 − 3S4−2 = 15 xy

FIG. 2. Schematic representation of the coordinate system setup. (a) Reference frames for plate cell [x  ,y  ,z ] and laboratory [x,y,z]. (b) The rotation of the plate cell about the z axis at an angle γ . (c) The rotation of the plate cell about the x axis at an angle α.

on these three planes, the explicit formulas of Eq. (21) can act as a convenient tool for experimentalists to extract the available information. We see that for RSHE truncated at l = 4, S4−4 and S44 can be determined by SAS measurement on the xy plane alone. All three planes xy, xz, and yz are needed for S00 , S20 , S22 , S40 , and S42 , while S2−2 and S4−2 clearly cannot be separated. We note that in Ref. [19], 12 linear equations are implicitly used to numerically extract all nine coefficients Slm of RSHE truncated at l = 4. However, as we have demonstrated mathematically, this cannot possibly be done without invoking additional assumptions. Equation (21) tells us that S2−2 can be drawn out of the xy plane only when S4−2 is much smaller than S2−2 , which is actually valid for the system with small perturbation. However, once higher-order terms are needed for a full description of the structural changes, e.g., for high shear rates and shear strains, measurements on the conventional planes become inadequate. Moreover, the long path length (> 5 mm) for the xy-plane measurement often leads to multiple scattering, and the cross section of Slm with m < 0 on the yz plane is nearly zero, making it difficult to carry out experiments on these two planes. To overcome these difficulties, we propose using the parallel-plate sliding cell tilted at different angles to obtain information on unconventional planes, as shown in Fig. 2. Consequently, a new data analysis method for this cell geometry must be developed. In the next section, the rotation of the reference frame is introduced to resolve all the Slm coefficients up to l = 4, providing a useful tool for investigating the microscopic mechanism of soft-matter systems under flow and deformation.

S44 =

√ 128 35 4,4 Sxy , 1575 √ 128 35 4,4 S . For the 11025 xy

III. ROTATION TRANSFORMATION OF REAL SPHERICAL HARMONICS

(21) √ 256 7 4,4 S , 2205 xy

0,0 where A1 = Sxz −

2,0 A2 = Sxz +

4,0 − A3 = Syz

experimental measurements

and

We consider the case in which the incident beam is along the y axis and the parallel-plate cell lies on the xz plane, as shown in Fig. 2(a). The directions of flow velocity, velocity

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PHYSICAL REVIEW E 96, 022612 (2017)

gradient, and vorticity are x, y, and z, respectively. Due to the constraint of the cell and data analysis in rheo-SAS experiments, measurements on the xy and yz planes are intrinsically complicated due to problems such as multiple scattering arising from the long path length. Therefore, it is highly desired to extract structural information from other unconventional planes by rotating the xz plane about the x or z axes by an angle α or γ [Figs. 2(b) and 2(c)]. In experiments, this operation corresponds to the rotation of the shear cell. As we shall see below, this cell rotation approach is not only feasible but also beneficial for rheo-SAS experiments and subsequent data analysis. For the sake of simplicity, we introduce the following vector notations for Slm and Ylm :   Sl=2 = S2−2 ,S20 ,S22 ,   Yl=2 = Y2−2 ,Y20 ,Y22 ,   Sl=4 = S4−4 ,S4−2 ,S40 ,S42 ,S44 ,   (22) Yl=4 = Y4−4 ,Y4−2 ,Y40 ,Y42 ,Y44 . Hence, the structure factor truncated at l = 4 can be rewritten as t t + Sl=4 Yl=4 , (23) S( Q) = S00 Y00 + Sl=2 Yl=2 where t is the symbol of matrix transpose. We note that in this expression, the l  2 coefficients are the anisotropic terms, whereas the isotropic S00 is invariant under rotation transformation. Rotation of the reference frame

First, let us consider the passive rotation in Fig. 2(b) about the z axis by an angle γ , which transforms original coordinates [x,y,z] into new coordinates [x  ,y  ,z ]. The transformation, which does not alter the molecular symmetry of soft-matter systems under steady shear flow, can be expressed by matrix multiplication, ⎡ ⎤ ⎡ ⎤⎡ ⎤ x cos γ sin γ 0 x ⎣y  ⎦ = ⎣− sin γ cos γ 0⎦⎣y ⎦. (24) 0 0 1 z z Correspondingly, the original Ylm in the [x,y,z] frame are transformed into a new set of RSH, Yˆlm , in the [x  ,y  ,z ] frame.  For a given l, Yˆlm and Ylm are related through the following matrix multiplication [22,23]:  t (0,0,γ )(C l )† Y t , Yˆ t = R l (γ )Y t = C l D (25) l

z

l

l

l

 lt (0,0,γ ) is the Wigner D function in the zyz where D convection of Euler angles [24], † is the symbol of conjugate transpose, and ⎡ ⎤ i 0 ··· 0 ··· 0 (−i)2l+1 ⎢0 i · · · 0 · · · (−i)2l−1 0 ⎥ ⎢ ⎥ ⎢ .. .. . . ⎥ .. . .. .. ·· ⎢. . ⎥ . . · . ⎢ ⎥ √ 1 ⎢ l  √ C = 2 ⎢0 0 · · · 2 ··· 0 0 ⎥ ⎥. ⎢. . ⎥ .. .. .. .. ⎢ .. .. ··· ⎥ . . . . ⎢ ⎥ l−1 ⎣0 1 · · · 0 · · · (−1) 0 ⎦ 1 0 ··· 0 ··· 0 (−1)l

Since the z rotation preserves the molecular symmetries [i.e., S(Q,θ,φ) = S(Q,π − θ,φ) and S(Q,θ,φ) = S(Q,θ,φ + π )] in the [x  ,y  ,z ] coordinates, the [l,m] must still be even. The dimensions of Rzl (γ ) can be narrowed down from (2l + 1) × (2l + 1) to (l + 1) × (l + 1) in our calculation. Therefore, the transformation can be written as Yˆ00 = Y00 , Yˆl=2 = Rzl=2 (γ )Yl=2 , Yˆl=4 = Rzl=4 (γ )Yl=4 ,

(27)

where Rzl=2 (γ ) and Rzl=4 (γ ) are 3 × 3 and 5 × 5 matrices, respectively: ⎡

cos 2γ Rzl=2 (γ ) = ⎣ 0 sin 2γ ⎡ cos 4γ ⎢ 0 ⎢ Rzl=4 (γ ) = ⎢ 0 ⎣ 0 sin 4γ

⎤ − sin 2γ 0 ⎦, cos 2γ

0 1 0

0 cos 2γ 0 sin 2γ 0

0 0 1 0 0

⎤ − sin 4γ 0 ⎥ ⎥ 0 ⎥. 0 ⎦ cos 4γ

0 − sin 2γ 0 cos 2γ 0

(28) Next, we consider the rotation in Fig. 2(c) about the x axis by an angle α, which is much more complicated than the z rotation since the molecular symmetries are broken under x rotation. The passive transformation between [x  ,y  ,z ] and [x,y,z] is ⎡ ⎤ ⎡ 1 x ⎣y  ⎦ = ⎣0 0 z

⎤⎡ ⎤ 0 x sin α ⎦⎣y ⎦. cos α z

0 cos α − sin α

(29)

The corresponding x-rotation transformation of RSH is  lt (−π/2,α,π/2)(C l )† , Rxl (α) = C l D

(30)

since a series of zyz rotation, at the angles π/2, α, and −π/2, respectively, yields the rotation about the x axis at an angle α. For the sake of simplicity, cos(nα) and sin(nα) are denoted as cn and sn , respectively, where n = 1,2,3,4. The corresponding x-rotation matrices for l = 2 and 4 are

(26) 022612-5

⎡

0

c1

⎢ ⎢ 0 ⎢ ⎢ l=2  Rx (α) = ⎢ 0 ⎢ ⎢−s1 ⎣

c2

−

√

3 s 2 2

0

− s22  l=4 Rxl=4 (α) = Rx1 0

0

s1

3 s 2 2 3c12 −1 2

0

√

0

√

0 c1

− 23 s12 0  l=4 l=4 , Rx2 Rx3

0

⎤

⎥ s2 ⎥ 2 ⎥ √ 3 2 ⎥, − 2 s1 ⎥ ⎥ 0

c12 +1 2

⎥ ⎦

GUAN-RONG HUANG et al.

⎡

c1 (c12 +1) 2

⎢ ⎢ 0 ⎢ ⎢ √ ⎢− 7 c s 2 ⎢ 2 1 1 ⎢ ⎢ 0 ⎢ ⎢ l=4 ⎢  Rx1 = ⎢ 0 ⎢ √ ⎢ 14 3 ⎢ 4 s1 ⎢ ⎢ 0 ⎢ ⎢ √ 2 ⎢ ⎣ 4 s3 0 ⎡ 0 ⎢ √7(2c2 −c −1) 2 ⎢ 2 ⎢ 8 ⎢ ⎢ 0 ⎢ ⎢ 28c4 −27c2 +3 ⎢ 1 1 ⎢ 4 √ ⎢ l=4 10s2 (3−7c12 ) Rx2 =⎢ ⎢ 8 ⎢ ⎢ 0 ⎢ ⎢ √ ⎢ 2(2s2 −7s4 ) ⎢ 16 ⎢ ⎢ 0 ⎣ √

⎡

l=4 Rx3

PHYSICAL REVIEW E 96, 022612 (2017)

3c12 +4c14 −3 4

0

√ 7(2c22 −c2 −1) 8 √ 3 70s1 c1 4

0

√ − 14 s1 c13 2

0

√ − 2(14s2 +s4 ) 32

0

√ − 70s13 c1 4

0

√

10s2 (7c12 −3) 8 35c14 −30c12 +3 8 √

0 5s12 (7s12 −6) 4 √

0

7(c14 −1) 4

0

√

⎤

For α = 0 or γ = 0, Yˆl=2 = Yl=2 and Yˆl=4 = Yl=4 . The structure factor is invariant under rotation transformation, which involves the transformation between bases in different coordinates. For convenience, we can introduce the following notations: S = [S00 ,Sl=2 ,Sl=4 ], R = [1,R l=2 ,R l=4 ], and Y = [Y00 ,Yl=2 ,Yl=4 ]. The S( Q) can thus be written as S( Q) = SR Y t . LIEs on the xz plane can be established by taking l,m the weighting integral of S( Q) and Yxz (ξ ) with respect to ξ , 1 l,m l,m Sxz = 12 −1 Sxz (Q,ξ )Yxz (ξ )dξ. (33)

⎥ ⎥ 0 ⎥ ⎥ 2 c1 (7c1 −5) ⎥ ⎥ 2 ⎥ ⎥ 0 ⎥ ⎥ ⎥, 0 ⎥ √ ⎥ 2s1 (7s12 −6) ⎥ ⎥ 4 ⎥ ⎥ 0 ⎥ √ ⎥ 14(s3 −2s1 ) ⎥ ⎦ 4 0 √ ⎤ − 14s 3

Rotation method for l = 2

For RSHE up to l = 2, we have the following LIEs on the xz plane:  5 2 0,0 0 S , Sxz = S0 + 3 2

1

4

⎥ ⎥ ⎥ √ 2 ⎥ 2s1 (6−7s1 ) ⎥ ⎥ 4 ⎥ ⎥ 0 ⎥ ⎥ ⎥, 0 ⎥ ⎥ c1 (7c12 −3) ⎥ ⎥ 4 ⎥ ⎥ 0 ⎥ √ ⎥ 3 7c1 (c12 −1) ⎥ ⎦ 4

35s14 8 √ − 2s3 4

14(2s2 −s4 ) 32

⎢ √14s c3 1 1 ⎢ ⎢ 2 ⎢ ⎢ ⎢√ 0 ⎢ 2(7s −2s ) 4 2 ⎢ ⎢ √ 16 ⎢ 5s 2 (7s 2 −6) 1 1 =⎢ ⎢ 4 ⎢ ⎢ 0 ⎢ ⎢ 7c4 −6c2 +1 ⎢ 1 1 ⎢ 2 ⎢ ⎢ ⎣ √ 0

√ 7c1 (c12 −1) 2

0

0 14s1 (2−3s12 ) 4

0 0

√ 3 7c1 (c12 −1) 4

0 c1 (9c12 −5) 4

0

0

0 0

2,0 = S20 . Sxz

⎤

⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 7(s4 √ −2s2 ) ⎥ ⎥ 14 16 √ 4 ⎥ 35s1 ⎥ ⎥. 8 ⎥ 0 ⎥ ⎥ √ 4 ⎥ 7(c1 −1) ⎥ ⎥ 4 ⎥ 0 ⎥ ⎦ 14s2√ +s4 16 2

(31)

c14 +6c12 +1 8

Coefficient extraction by rotation transformation

Having established the basic language for describing the rotation of the reference frame, let us now turn our attention to a determination of the expansion coefficients. In the laboratory frame of Fig. 2, the beam direction is along the y direction, collecting the structural information of the sample on the xz plane. Considering a parallel sliding plate cell tilted at an angle with respect to the z axis or the x axis, one can rotate [x,y,z] coordinates about the z axis or the x axis into [x  ,y  ,z ] coordinates to place the shear cell right on the x  z plane. In the [x  ,y  ,z ] frame, the structure factor is S( Q) =

S00 Yˆ00

There are different ways to rotate the shear cell in order to obtain the remaining LIEs for solving all the expansion coefficients. Rotating the parallel sliding plate cell about the x axis by π/4 leads to the following structure factor:  π  t t Y = S00 Y00 + Sl=2 Rxl=2 α = S( Q) = S00 Yˆ00 + Sl=2 Yˆl=2 4 l=2  0 √ 2 √ S − 3S2 2,0 2 −2 2,1 0,0 Yxz + S Y = S00 Yxz + 2 4 2 2 xz  2 √ 0 3S2 − 3S2 2,2 Yxz , + (35) 4 and hence the corresponding LIEs   √  1 2,0 π = S20 − 3S22 , Sˆxz 4 2   √ 2,1 π ˆ = 2S2−2 , Sxz (36) 4 l,m l,m where Sˆxz (α) is the weighting integral of S( Q) and Yxz with respect to ξ on the xz plane for rotating the shear cell about the x axis by an angle α. Therefore, the coefficients of RSHE can be extracted as √   15 ˆ 2,0  π  0,0 2,0 2Sxz − Sxz , S00 = Sxz + 9 4 √ 2 ˆ 2,1  π  −2 S , S2 = 2 xz 4 2,0 S20 = Sxz ,    1 2,0 20 π Sxz − 2Sˆxz . (37) S22 = 3 4 On the other hand, rotating the sliding plate cell about the z axis by π/4 leads to the following structure factor:

t t + Sl=2 Yˆl=2 + Sl=4 Yˆl=4

t t = S00 Y00 + Sl=2 R l=2 Yl=2 + Sl=4 R l=4 Yl=4 .

(34)

(32) 022612-6

t S( Q) = S00 Yˆ00 + Sl=2 Yˆl=2

 π  t Y = S00 Y00 + Sl=2 Rzl=2 γ = 4 l=2 2,0 2,2 = S00 Y00 + S20 Yxz − S2−2 Yxz ,

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and hence the corresponding LIE  π  5 −2 0,0 0 S˜xz = S0 − S , (39) 4 3 2 l,m l,m where S˜xz (γ ) is the weighting integral of S( Q) and Yxz with respect to ξ on the xz plane for rotating the cell about the z axis by an angle γ . From this analysis, we see that rotation along the z axis by a single angle does not yield sufficient information for extraction of all the l = 2 coefficients. Therefore, we recommend the x-rotation method for l = 2.

S2−2

=

S20 =

S22 =

Rotation method for l = 4

For the sake of simplicity, Rz (γ ) and Rx (α) are denoted as the z rotation at an angle γ and the x rotation at an angle α, respectively. Similarly, take the integrals with respect to ξ ,  1 1 l,m l,m (γ ) = SRz (γ ) dξ, S˜xz Y t Yxz 2 −1  1 1 l,m l,m Sˆxz (α) = SRx (α) dξ. (40) Y t Yxz 2 −1 Here we only consider the rotation method about the x axis. An alternative approach involving rotations about both the x and z axes will be given in the Appendix. By measuring the SAS spectrum on the xz plane and subsequently rotating the cell about the x axis by π/6 and π/4, we find the following nine LIEs through the weighting integrals, after removing the linearly dependent ones:      5 1 7 t 0,0 ,0,0,0, , Sxz = 1,0,0, S, 3 5 5    1 −2 t 2,0 Sxz = 0,0,1, − ,0,0,0,1, √ S , 3 7  −2 1 t 4,0 = 0,0,0,0,0,0,1, √ , √ Sxz S, 5 35 √ √  √  π 1 − −3 1 −19 3 5 7 t 2,0 = 0,0, , ,0,0, , , Sˆxz S, 4 2 2 8 4 56   √  √ 6 5 3 2,1 π ˆ = 0, 2,0,0, , ,0,0,0 St , Sxz 4 2 14 4 √ √   1 −3 17 π 5 35 t 4,0 = 0,0,0,0,0,0, , , Sˆxz S, 4 4 10 140  −3 4,1 π ˆ Sxz = 0,0,0,0, √ ,1,0,0,0 St , 4 7  √ √  π 13 5 21 3 2,1 = 0,1,0,0, , ,0,0,0 St , Sˆxz 6 56 8 √ √   π 9 −3 41 5 35 t 4,0 = 0,0,0,0,0,0,0, , , Sˆxz S . (41) 6 16 8 560  Solving this set of LIEs yields the final solutions to S, √ 5 2,0 65 4,0 √ ˆ 2,0  π  0,0 S + Sxz + 5Sxz − S00 = Sxz 2 xz 8 4     13 4,0 π 4,0 π + 14Sˆxz , + Sˆxz 2 4 6

S4−4 = S4−2 = S40 = S42 = S44 =

√ √ 3 ˆ 4,1  π  4 ˆ 2,1  π  7 2 ˆ 2,1  π  Sxz + S − Sxz , 6 4 4 xz 4 3 6 √ √ 3 2,0 7 5 4,0 ˆ 2,0  π  13 5 ˆ 4,0  π  Sxz − Sxz − Sxz − Sxz 2 2 4 4 4 √ 4,0  π  , + 7 5Sˆxz 6 √ √ 3 2,0 5 15 4,0 √ ˆ 2,0  π  S − Sxz − 3Sxz 2 xz 2 4 √     √ 11 15 ˆ 4,0 π 4,0 π Sxz + 5 15Sˆxz , − 4 4 6 √ √ √ 42 ˆ 2,1  π  7 ˆ 4,1  π  2 21 ˆ 2,1  π  S − S + Sxz , − 9 xz 4 4 xz 4 9 6 √ 2 2,1  π  6 ˆ 2,1  π  1 ˆ 4,1  π  Sxz + Sxz + √ Sˆxz , − 3 4 4 4 6 3   33 4,0 13 ˆ 4,0  π  4,0 π Sxz + Sxz − 7Sˆxz , 8 4 4 6 √ √ 4,0  π  √ 4,0  π  7 5 4,0 Sxz + 2 5Sˆxz − 4 5Sˆxz , 4 4 6 √ √ 3 35 4,0 3 35 ˆ 4,0  π  √ ˆ 4,0  π  Sxz − 35Sxz . (42) Sxz + 8 4 4 6

To summarize, when the real spherical harmonic expansion is truncated at l = 2, we can measure the SAS spectrum from the xz plane, rotate the shear cell about the x axis by π/4, and carry out additional SAS measurements on the tilted sample. Equation (37) is the working equation for extracting all four expansion coefficients from these two planes. In the case of truncation at l = 4, our method is to perform SAS

FIG. 3. The profile of Slm vs Rg,0 Q in the affine deformation model of the polymer predicted by Eq. (44) with g = 0.3. (a) S00 and (b) Slm with l = 0. 022612-7

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S( Q) under shear deformation is 2 [exp (−η) + η − 1], η2   (43) 2 (1 + g 2 )Q2x + Q2y + Q2z + 2gQx Qy , η = Rg,0

S( Q) =

FIG. 4. The 2D SAS spectra of a rotating sample with g = 0.5 on the xz plane of a laboratory frame: (a) Sˆxz ( π4 ) and (b) Sˆxz ( π6 ). Here, the spectra are computed using Eq. (44) with Qy = 0.

measurement on the xz plane and subsequently rotate the cell about the x axis by π/6 and π/4, respectively. Equation (42) can be used to fully resolve the nine coefficients. Tests with the affine deformation model of a polymer

The affine deformation model of a polymer can serve as an ideal tool for testing the validity of our cell rotation approach. When the segments of a Gaussian chain deform affinely on all length scales, the single-chain structure factor (form factor)

where Rg,0 is the radius of gyration in equilibrium, g is the strain since γ has denoted the rotation angle about the z axis, and [Qx ,Qy ,Qz ] = Q[sin θ cos φ, sin θ sin φ, cos θ ]. Since we are dealing with an analytical model, each Slm (Q) term can be directly computed by the integral  2π  π 1 m S( Q)Ylm sin θ dθ dφ. (44) Sl = 4π 0 0 The RSHE spectrum of the affine deformation model predicted by Eq. (44), with g = 0.3 and up to l = 4 order, is plotted in Fig. 3, where the contribution of each Slm term can m be evaluated. Under this condition, the ratio of Sl+2 to Slm is on the order of 0.1, supporting the truncation at l = 4. Next we examine the 2D SAS spectra (Fig. 4) after rotating the sample cell about the x axis by π/4 and π/6, respectively. From the 2D spectra, we can apply Eq. (42) to determine the expansion coefficients Slm . Figure 5 compares these extracted coefficients from l = 4 and 2 RSHE with those obtained directly from the 3D integral [Eq. (44)]. It can be seen that the l  4 terms come into play in the RSHE spectrum at high strains, and RSHE truncated at l = 4 is sufficient to reconstruct the spectrum of S2−2 . IV. CONCLUDING REMARKS AND SUMMARY

In this work, we have restricted our discussions to shear geometry. For uniaxial extension, we have previously shown that the RSHE of the single-chain structure factor of a polymer chain has a much simpler form [21]:  S( Q) = Sl0 (Q)Yl0 (θ ). (45) l:even

FIG. 5. Comparison between the affine deformation model of polymer [Eq. (44)] and the (a) l = 2 and (b) l = 4 RSHE of Eqs. (37) and (42).

Because of the high symmetry of the uniaxial extension, the RSHE coefficient Sl0 of any order l can be obtained in principle from SANS measurement on a single plane parallel to the stretching direction (e.g., the xz plane). In contrast, as we show here, RSHE coefficients are much more difficult to obtain in the case of shear, even for small l’s (2 and 4). Generally speaking, when microscopic deformation is large, reconstruction of the 3D distorted structures from 2D SAS measurement becomes practically impossible for shear geometry. This sharp contrast between shear and extension underscores the fundamental role of the symmetry of flow in the rheo-SANS experiments. On a philosophical level, we generally cannot fully restore the 3D information by examining a finite number of planes. This becomes possible only when there are further restrictions on the molecular symmetry. For uniaxial extension, the cylindrical symmetry of the deformation significantly simplifies the matter, making the determination of expansion coefficients straightforward. For shear, when the perturbation is sufficiently small, we only need to truncate the expansion at l = 2 or 4. Such a truncation

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can be equivalently understood as an additional symmetry requirement. The analysis presented in this work has important implications for rheo-SAS experiments. While shear geometry has been widely used, the pros and cons of different cell designs have not been thoroughly discussed and considered in the context of experimentally accessible information. For example, we have shown that in the case of truncation at l = 4, SAS experiments on the xy, xz, and yz planes yield five, two, and one LIE, respectively. While the relative importance of these planes has been intuitively understood previously, this issue has not been addressed rigorously from a mathematical point of view. For the sample rotation method proposed herein, only certain cells are suitable for this approach. On the other hand, although the current discussion has only touched upon simple shear and uniaxial extension, our real spherical harmonic expansion approach is certainly applicable to other types of deformation, such as planar and biaxial extensions. Without question the consideration of the deformation symmetry is also critical for the proper design of rheo-SAS experiments in these cases. To conclude, we discuss the reconstruction of threedimensional anisotropic structure from small-angle scattering experiments by using real spherical harmonic expansion analysis. Because of the low symmetry of the simple shear geometry, the determination of expansion coefficients requires measurements on different planes. When the structural distortion is large, a reconstruction of the 3D S( Q) becomes impossible in practice. To address this challenge, we propose an approach to rheo-SAS in which a parallel sliding plate shear cell is used with a central rotary axis along the flow direction. By rotating the shear cell, all the real spherical harmonic expansion coefficients can be determined from measurements on two and three special planes, respectively, for l = 2 and 4. For l = 4, our proposal is to perform SAS measurement on the xz plane and subsequently rotate the cell about the x axis by π/6 and π/4, respectively. This design has the potential to bypass the issue of multiple scattering in the traditional Couette flow cell, as it does not require direct measurement on the xy plane. Lastly, we demonstrate this cell rotation method by using the affine deformation model for polymers. Our approach produces excellent results up to (microscopic) shear strain of 1.0 for RSHE truncated at l = 4. We confirm that the higher-order terms cannot be neglected when the mechanical perturbation rises to a certain value. ACKNOWLEDGMENTS

This work was sponsored by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. This research at the Spallation Neutron Source and the Center for Nanophase Materials Sciences of Oak Ridge National Laboratory was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Y.Y.W. gratefully acknowledges the support by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT Battelle, LLC, for the U.S. Department of Energy. G.R.H. acknowledges the support

from the National Center for Theoretical Sciences, Ministry of Science and Technology in Taiwan (Project No. MOST 106-2119-M-007-019), and the Schull Wollan Center during his visit to Oak Ridge National Laboratory. APPENDIX: ROTATION ABOUT THE x AND z AXES FOR l =4

In the main text of this paper, we describe a method for l = 4 involving cell rotation about the x axis by two different angles. Alternatively, the extraction of Slm for l = 4 RSHE can be achieved by rotating a plate cell about both the x and z axes. Similarly, one can find the following nine LIEs:      5 1 7 t 0,0 Sxz = 1,0,0, ,0,0,0, , S, 3 5 5    1 −2 t 2,0 Sxz = 0,0,1, − ,0,0,0,1, √ S , 3 7  −2 1 t 4,0 Sxz = 0,0,0,0,0,0,1, √ , √ S, 5 35 √ √  √  π 1 − −3 1 −19 3 5 7 t 2,0 Sˆxz = 0,0, , ,0,0, , , S, 4 2 2 8 4 56   √  √ π 3 6 5 2,1 Sˆxz = 0, 2,0,0, , ,0,0,0 St , 4 2 14 4 √ √   π 1 5 35 t −3 17 4,0 Sˆxz = 0,0,0,0,0,0, , , S, 4 4 10 140  −3 4,1 π ˆ Sxz = 0,0,0,0, √ ,1,0,0,0 St , 4 7      5 1 7 t 0,0 π ˜ Sxz = 1, − ,0,0,0, − ,0,0, − S, 4 3 5 5   √  √  1 3 3 − 3 1 1 t 2,0 π ˜ Sxz = 0, ,1, − , , ,0, , S . (A1) 6 2 6 7 2 2 7 l,m l,m and Sˆxz correspond to the Here, we emphasize that S˜xz rotation along the z and x axes, respectively. The set of LIEs  are solvable and give the solutions to S, 0,0 2,0 4,0 S00 = −3.732Sxz − 2.955Sxz − 5.152Sxz       2,0 π 2,1 π 4,0 π + 2.304Sˆxz + 2.179Sˆxz − 5.494Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 + 5.643Sˆxz + 4.732S˜xz + 5.702S˜xz , 4 4 6 0,0 2,0 4,0 S2−2 = −1.512Sxz − 2.253Sxz − 1.260Sxz       2,0 π 2,1 π 4,0 π + 0.9717Sˆxz − 1.009Sˆxz − 1.803Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 + 1.513S˜xz + 3.155S˜xz , + 2.813Sˆxz 4 4 6 0,0 2,0 4,0 S20 = 5.291Sxz + 3.553Sxz + 7.018Sxz       2,0 π 2,1 π 4,0 π − 2.576Sˆxz − 2.436Sˆxz + 7.643Sˆxz 4 4 4

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S22

=

0,0 6.545Sxz

π  4

+ π 

PHYSICAL REVIEW E 96, 022612 (2017)

0,0 − 5.291S˜xz

2,0 3.407Sxz

+

π  4

4,0 8.682Sxz

π 

2,0 − 6.375S˜xz

π  6

4,1 + 2.822Sˆxz

,

π 

S42

2,1 4,0 − 3.187Sˆxz − 4.673Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 − 7.806Sˆxz − 6.545S˜xz − 7.887S˜xz , 4 4 6

2,0 + 8.964Sˆxz

0,0 2,0 4,0 S4−4 = 1.155Sxz + 1.721Sxz + 0.9621Sxz π  π    2,0 2,1 4,0 π − 0.2020Sˆxz + 0.7697Sˆxz + 1.377Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 − 2.479Sˆxz − 1.155S˜xz − 2.409S˜xz , 4 4 6 0,0 2,0 4,0 S4−2 = 1.309Sxz + 1.952Sxz + 1.091Sxz π  π    2,0 2,1 4,0 π − 0.229Sˆxz + 0.8727Sˆxz + 1.561Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 − 1.309S˜xz − 2.732S˜xz , − 1.811Sˆxz 4 4 6 0,0 2,0 4,0 S40 = −2.366Sxz − 0.9183Sxz − 2.513Sxz       2,0 π 2,1 π 4,0 π + 1.152Sˆxz + 1.089Sˆxz − 3.865Sˆxz 4 4 4

[1] D. J. Evans, H. J. M. Hanley, and S. Hess, Phys. Today 37(1), 26 (1984). [2] M. Kröger, Phys. Rep. 390, 453 (2004). [3] X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, Science 333, 1276 (2011). [4] J. Zausch and J. Horbach, Europhys. Lett. 88, 60001 (2009). [5] N. Koumakis, M. Laurati, S. U. Egelhaaf, J. F. Brady, and G. Petekidis, Phys. Rev. Lett. 108, 098303 (2012). [6] K. J. Mutch, M. Laurati, C. P. Amann, M. Fuchs, and S. U. Egelhaaf, Eur. Phys. J. Spec. Top. 222, 2803 (2013). [7] N. Koumakis, M. Laurati, A. R. Jacob, K. J. Mutch, A. Abdellali, A. B. Schofield, S. U. Egelhaaf, J. F. Brady, and G. Petekidis, J. Rheol. 60, 603 (2016). [8] D. J. Evans and H. J. M. Hanley, Phys. Rev. A 20, 1648 (1979). [9] H. J. M. Hanley and D. J. Evans, J. Chem. Phys. 76, 3225 (1982). [10] S. Hess and H. J. M. Hanley, Phys. Rev. A 25, 1801 (1982). [11] H. J. M. Hanley, J. C. Rainwater, and S. Hess, Phys. Rev. A 36, 1795 (1987). [12] J. F. Morris and B. Katyal, Phys. Fluids 14, 1920 (2002). [13] N. A. Clark and B. J. Ackerson, Phys. Rev. Lett. 44, 1005 (1980).

=

0,0 −3.023Sxz

π  4

0,0 + 2.366S˜xz

π  4

2,0 + 2.851S˜xz

π  , 6

4,0 − − 4.569Sxz π  π    2,0 2,1 4,0 π + 1.472Sˆxz + 1.711Sˆxz − 4.939Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 + 3.605Sˆxz + 3.023S˜xz + 3.643S˜xz , 4 4 6 2,0 1.173Sxz

0,0 2,0 4,0 S44 = −2.000Sxz − 0.7761Sxz − 3.392Sxz π  π    2,0 2,1 4,0 π + 0.9736Sˆxz + 2.611Sˆxz − 3.267Sˆxz 4 4 4 π  π  π  4,1 0,0 2,0 + 2.385Sˆxz + 2.000S˜xz + 2.409S˜xz . 4 4 6 (A2)

The solutions for Slm are much more complicated than the formulas presented in the main text, since the z rotation contains less information than that of x rotation. Lastly, without tilting many angles about a single axis, it should be feasible to rotate a parallel-plate cell about different axes to gain additional information for extracting high-order terms, e.g., the l = 6 terms.

[14] B. J. Ackerson and N. A. Clark, Physica A 118, 221 (1983). [15] Y. Suzuki, J. Haimovich, and T. Egami, Phys. Rev. B 35, 2162 (1987). [16] N. J. Wagner and W. B. Russel, Phys. Fluids A 2, 491 (1990). [17] N. J. Wagner and B. J. Ackerson, J. Chem. Phys. 97, 1473 (1992). [18] B. J. Maranzano and N. J. Wagner, J. Chem. Phys. 117, 10291 (2002). [19] A. K. Gurnon and N. J. Wagner, J. Fluid. Mech. 769, 242 (2015). [20] Z. Wang, T. Iwashita, L. Porcar, Y. Wang, Y. Liu, L. E. SanchezDiaz, B. Wu, T. Egami, and W.-R. Chen, arXiv:1611.03135. [21] Z. Wang, C. N. Lam, W.-R. Chen, W. Wang, J. Liu, Y. Liu, L. Porcar, C. B. Stanley, Z. Zhao, K. Hong, and Y. Wang, Phys. Rev. X 7, 031003 (2017). [22] J. Ivanic and K. Ruedenberg, J. Phys. Chem. 100, 6342 (1996). [23] M. A. Blanco, M. Flrez, and M. Bermejo, J. Mol. Struct., Theochem. 419, 19 (1997). [24] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, 1st ed. (World Scientific, Singapore, 1988).

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