We present a theory for pitch, a matrix property that is linked to the coupling of rotational and translational motion of rigid bodies at low Reynolds numbers. The pitch matrix is a geometric property of objects in contact with a surrounding fluid, and it can be decomposed into three *principal axes of pitch* and their associated *moments of pitch*. The moments of pitch predict the translational motion in a direction parallel to each pitch axis when the object is rotated around that axis and can be used to explain translational drift, particularly for rotating helices. We also provide a symmetrized boundary element model for blocks of the resistance tensor, allowing calculation of the pitch matrix for arbitrary rigid bodies. We analyze a range of chiral objects, including chiral molecules and helices. Chiral objects with a *C*_{n} symmetry axis with *n* > 2 show additional symmetries in their pitch matrices. We also show that some achiral objects have non-vanishing pitch matrices, and we use this result to explain recent observations of achiral microswimmers. We also discuss the small but non-zero pitch of Lord Kelvin’s isotropic helicoid.

## I. INTRODUCTION

Screws are simple machines that are universally used to join parts together and to provide secure enclosures for containers. We often draw a distinction between mechanical screws, which move through solid media, and screws that move through a fluid medium, such as self-propelled swimmers. In all cases, the function and efficiency of the screw are associated with the translation-rotation coupling in the medium, which converts rotation around an axis to linear motion along that axis. The translation-rotation coupling is quantified by the screw’s pitch, which is the distance the screw translates upon completing one revolution. This is an intuitive approach for screws that involve contact between two solid surfaces.^{1} For example, a nut advancing through consecutive threads of a screw travels exactly the distance between threads in a single 2*π*-radian revolution.

For swimmers and other hydrodynamic screws, where physical threads are not explicit, pitch may be an empirically measured quantity that is challenging to obtain.^{2} When the screw operates in a fluid, like a propeller operating in seawater, the translation-rotation coupling decreases because of slip along the screw’s surface.^{3} Schamel *et al.*^{4} and Patil *et al.*^{5} observed this fact when studying helical systems moving in liquid media: the translational motion of the helices in a 2*π*-revolution was less than the standard pitch definition for a helix, which is the distance between consecutive helical turns.

Translational motion can be generated either by rotating the screw itself or by rotating the medium around the screw. Howard *et al.* first observed the translational motion of macroscopic chiral objects induced by the vorticity of a fluid.^{6} In their experiment, they suspended dextro-tartaric acid crystals on one side of a drum that was filled with Isopar H (isoparaffinic hydrocarbons). Then, they rotated the drum and observed the migration of the crystals due to the vorticity of the fluid. Although tartaric acid crystals do not look like traditional screws, left- and right-handed crystals exhibit opposite signs of the translational-rotational portion of their resistance tensors, which govern frictional forces and torques experienced by a body in a fluid. This behavior can be used to separate bodies of opposing handedness since they move in opposite directions in response to rotations. In an earlier paper, we explored the link between parts of the resistance tensor and the translational-rotational coupling, and derived a geometric quantity we called “scalar pitch.”^{7} The scalar pitch is a rotational invariant for rigid bodies, describing how the object will translate in response to rotations.

In this paper, we extend the concept of pitch that was developed in Ref. 7 into a matrix form, and we investigate the physical properties of the characteristic eigenvalues and eigenvectors of this pitch matrix. This results in a method in which chiral (and achiral) objects can be assigned *principal axes of pitch* and three associated *moments of pitch* for motion around those axes. In previous work, the method used for computing resistance tensors was aimed at studying pitch in molecules, so spherical beads representing the atoms were the primary hydrodynamic elements.^{7} In this paper, we develop a symmetrized boundary element method using triangular surface patches to evaluate resistance tensors for arbitrary shapes.

## II. FORMALISM

### A. Resistance and mobility tensors

Consider an arbitrarily shaped rigid body moving in a fluid at a low Reynolds number. This rigid body will feel a force and torque in response to its velocity and angular velocity in the fluid. For example, a propeller placed in a flowing fluid experiences a torque, while a screw rotating through a quiescent medium experiences a linear force.

*O*, is moving with the body. From Brenner’s fundamental work on hydrodynamics,

^{8,9}the relationship among net force $f$, torque $\tau $, velocity $v$, and angular velocity $\omega $ at

*O*is

*O*indicates the quantities that depend on the location of the reference point

*O*.

^{10}

^{,}

**also comprises four blocks that are analogous to the blocks of the hydrodynamic resistance tensor Ξ defined in Eq. (1). Multiplying the mobility tensor by**

*μ**k*

_{B}

*T*yields the diffusion tensor, which is a generalization

^{9,10}of Einstein’s relation,

^{11}connecting the resistance and diffusion tensors.

### B. The pitch matrix

*ϕ*around its long axis (

*z*), it moves linearly along the same rotation axis,

*P*of the screw in terms of a full 2

*π*rotation in

*ϕ*.

*z*),

**) and the resultant motion may also be a linear velocity vector (**

*ω***v**). In this case, the relationship between

**and**

*ω***v**is mediated by a 3 × 3 pitch matrix,

**f**= 0 in Eq. (1).

^{12,13}We can then equate the drag force from translational motion to the rotational contribution of the force on the object

*O*.

**I**is the 3 × 3 identity matrix and

**0**is the 3 × 3 null matrix. In terms of the mobility tensor blocks, the pitch matrix is

**= 0) and $\omega =Lv$, where $L$ is a 3 × 3 matrix that mediates the generation of angular velocity from linear velocity. Using the resistance or mobility tensors,**

*τ*^{14}studied this process using a three-sphere (trumbbell) model settling under gravity in a viscous fluid and concluded that the trumbbell rotates as it settles. In Sec. III C, we consider a similar example using an isotropic helicoid falling through a fluid, where its linear velocity induces a small angular velocity.

### C. Center of pitch

The translational (tt) and rotational (rr) blocks of the resistance [Eq. (1)] and mobility [Eq. (2)] tensors are symmetric matrices for any point *O*. However, the blocks that couple translation and rotation are only symmetric at the center of resistance (CR) for the resistance tensor^{7–9} and at the center of diffusion (CD) for the mobility tensor.^{7,9} Therefore, the pitch matrix is not generally a symmetric matrix. However, at one special point, which we call the *center of pitch* (**p**), the pitch matrix does become symmetric.

*M*(separate from the origin) will include a portion of the translational block along the line connecting the origin

*O*to

*M*, while the translation-rotation couplings for the mobility tensor will include a portion of the rotational block.

^{7–9}We can express the new couplings

*O*to point

*M*,

**p**where the pitch matrix is symmetric, we set the right side of Eq. (16) equal to its transpose, and we find the coordinates of the vector

**r**

_{p}connecting the center

*O*to

**p**,

### D. Pitch axes, moments of pitch, and the pitch coefficient

^{15}the symmetric pitch matrix and write

*λ*

_{i}) are

*moments of pitch*—each associated with a

*pitch axis*,

*α*

_{i}—which is one of the column vectors making up $A=\alpha 1,\alpha 2,\alpha 3$. This decomposition into principal axes and moments of pitch is a direct analogy to the decomposition of a moment of inertia tensor into principal axes and moments of inertia. For the pitch matrix, however, moments of pitch may be negative if rotating the body counterclockwise around axis

*α*

_{i}results in translation along the negative

*α*

_{i}direction.

*scalar pitch coefficient*, which is the simplest rotational invariant of the pitch matrix, providing equal contributions from rotation around all three axes of pitch,

*scalar pitch coefficient*is available in Sec. II of the supplementary material. Note that this is functionally equivalent to a pitch coefficient that was demonstrated in our previous paper,

^{7}where the eigenvalues of the (tt) and (tr) blocks were used separately to compute |

*P*|/2

*π*.

Figure 1 shows the principal axes of pitch and their associated moments for the Λ and Δ enantiomers of the $[Ru(bpy)3]2+$ ion. If there are degeneracies in the moments of pitch, a linear combination of the corresponding pitch axes will also form a basis for understanding the translation-rotation coupling of the object.

### E. Pitch properties of enantiomers and achiral objects

^{7}Using this mirror image property, we can deduce the following property of the pitch matrix for the two enantiomers:

At the center of pitch, Eq. (20) implies that the moments of pitch (eigenvalues) of the pitch matrix for the left- and right-handed objects have the same magnitude but flip signs. The pitch axes (eigenvectors), however, are identical for both objects.

For an achiral object, which is identical with its own mirror image, the characteristic eigenvalues of the left and right pitch matrices must be the same, and there are two ways for this to happen:

*λ*_{1}=*λ*_{2}=*λ*_{3}= 0In this case, the achiral object has a pitch matrix, which is the null matrix, and, thus, does not exhibit displacement due to rotation. This situation occurs in objects with a high degree of internal symmetry, e.g., spheres and ellipsoids.

*λ*_{1}= 0 and*λ*_{2}= −*λ*_{3}≠ 0In this case, the achiral object has a pitch matrix that is non-zero. These objects can exhibit displacement due to rotation, and this property helps explain the recent observation of achiral microswimmers, which can be propelled through a fluid via rotation

^{17}(see Fig. 2).For chiral objects, there are two additional cases to consider:

*λ*_{1}=*λ*_{2}≠*λ*_{3}(two degenerate eigenvalues)This class of objects is a chiral body in the

*C*_{n}or*D*_{n}point groups with*n*≥ 3.*λ*_{1}≠*λ*_{2}≠*λ*_{3}This is the general case for most chiral objects without higher symmetry axes.

To explore the pitch matrix properties of chiral objects, we can consider the chiral point groups *C*_{n} and *D*_{n}.^{18}^{,} Table I shows the moments of pitch (the eigenvalues of the pitch matrix) for a set of representative molecules that belong to the *C*_{n} and *D*_{n} point groups.^{19–21} To compute these moments of pitch, we first constructed the molecular resistance tensors, representing atoms with spheres with appropriate van der Waals radii. The molecular resistance tensors were computed using the methods described in Ref. 7. The pitch matrix for each molecule was then constructed and diagonalized to obtain the molecular pitch axes (eigenvectors) and the associated moments of pitch (eigenvalues).

. | . | Moments of pitch $(\xd710\u22124A\u030arad\u22121)$ . | ||
---|---|---|---|---|

Point group . | Molecule . | λ_{1}
. | λ_{2}
. | λ_{3}
. |

C_{1} | 1-bromo-1-chloroethane | −229.32 | 11.07 | 223.36 |

2,3-dihydrofuran | −210.45 | −17.25 | 240.34 | |

Bromochlorofluoromethane | −156.89 | 22.91 | 134.06 | |

D-alanine | −404.01 | 60.99 | 375.02 | |

D-serine | −384.40 | 2.37 | 338.14 | |

SOClBr | −119.43 | −14.30 | 133.58 | |

C_{2} | 1,3-dichloroallene | −477.66 | −25.93 | 583.70 |

2,3-pentadiene | −250.60 | −182.19 | 405.21 | |

cis-$[Co(en)2Cl2]+$ | −306.88 | 91.10 | 145.42 | |

Hydrazine | −324.38 | −16.05 | 256.58 | |

Hydrogen peroxide | −273.39 | −199.04 | 408.42 | |

Mo(acac)_{2}O_{2} | −1323.32 | 4.85 | 1266.70 | |

Titanium dimer | −366.93 | −212.78 | 600.78 | |

C_{3} | Tris-aminomethane | −119.09 | −119.09 | 178.09 |

Triethylamine | −110.35 | 36.66 | 36.66 | |

Triphenylmethane | −1071.89 | −1071.89 | 2056.39 | |

Triphenylphosphine | −857.58 | −857.58 | 1645.00 | |

C_{4} | Tetra-aza copper(II) | −85.70 | −85.70 | 110.43 |

C_{5} | Fe(Me_{5}-Cp)(P_{5}) | −13.70 | 11.47 | 11.47 |

C_{6} | Alpha-cyclodextrin | −9.79 | −9.79 | 34.49 |

D_{2} | biphenyl | −1730.16 | 605.84 | 1144.82 |

trans-$[Co(en)2Cl2]+$ | −153.73 | 79.32 | 90.63 | |

Twistane | −92.72 | 20.74 | 40.50 | |

D_{3} | Guanidinium cation | −39.67 | 81.99 | 81.99 |

Tris(en)cobalt(III) | −176.32 | −176.32 | 328.17 | |

Tris(oxalato)iron(III) | −402.24 | −402.24 | 953.13 | |

D_{4} | Tetrathiacyclododecane | −218.66 | 325.53 | 325.53 |

D_{5} | Twisted ferrocene | −14.26 | 10.73 | 10.73 |

YbI_{2}(THF)_{5} | −180.88 | −180.88 | 356.50 | |

D_{6} | Bis(benzene)chromium | −4.43 | 3.51 | 3.51 |

. | . | Moments of pitch $(\xd710\u22124A\u030arad\u22121)$ . | ||
---|---|---|---|---|

Point group . | Molecule . | λ_{1}
. | λ_{2}
. | λ_{3}
. |

C_{1} | 1-bromo-1-chloroethane | −229.32 | 11.07 | 223.36 |

2,3-dihydrofuran | −210.45 | −17.25 | 240.34 | |

Bromochlorofluoromethane | −156.89 | 22.91 | 134.06 | |

D-alanine | −404.01 | 60.99 | 375.02 | |

D-serine | −384.40 | 2.37 | 338.14 | |

SOClBr | −119.43 | −14.30 | 133.58 | |

C_{2} | 1,3-dichloroallene | −477.66 | −25.93 | 583.70 |

2,3-pentadiene | −250.60 | −182.19 | 405.21 | |

cis-$[Co(en)2Cl2]+$ | −306.88 | 91.10 | 145.42 | |

Hydrazine | −324.38 | −16.05 | 256.58 | |

Hydrogen peroxide | −273.39 | −199.04 | 408.42 | |

Mo(acac)_{2}O_{2} | −1323.32 | 4.85 | 1266.70 | |

Titanium dimer | −366.93 | −212.78 | 600.78 | |

C_{3} | Tris-aminomethane | −119.09 | −119.09 | 178.09 |

Triethylamine | −110.35 | 36.66 | 36.66 | |

Triphenylmethane | −1071.89 | −1071.89 | 2056.39 | |

Triphenylphosphine | −857.58 | −857.58 | 1645.00 | |

C_{4} | Tetra-aza copper(II) | −85.70 | −85.70 | 110.43 |

C_{5} | Fe(Me_{5}-Cp)(P_{5}) | −13.70 | 11.47 | 11.47 |

C_{6} | Alpha-cyclodextrin | −9.79 | −9.79 | 34.49 |

D_{2} | biphenyl | −1730.16 | 605.84 | 1144.82 |

trans-$[Co(en)2Cl2]+$ | −153.73 | 79.32 | 90.63 | |

Twistane | −92.72 | 20.74 | 40.50 | |

D_{3} | Guanidinium cation | −39.67 | 81.99 | 81.99 |

Tris(en)cobalt(III) | −176.32 | −176.32 | 328.17 | |

Tris(oxalato)iron(III) | −402.24 | −402.24 | 953.13 | |

D_{4} | Tetrathiacyclododecane | −218.66 | 325.53 | 325.53 |

D_{5} | Twisted ferrocene | −14.26 | 10.73 | 10.73 |

YbI_{2}(THF)_{5} | −180.88 | −180.88 | 356.50 | |

D_{6} | Bis(benzene)chromium | −4.43 | 3.51 | 3.51 |

When the symmetry axis of the point groups has *n* ≥ 3, we find that two of the moments of pitch are always degenerate (see Table I). When this degeneracy occurs, the *non*-degenerate moment of pitch is associated with a pitch axis that points directly along the *C*_{n} axis of the molecule. This property is expected from the character tables of the *C*_{n} and *D*_{n} point groups,^{22} since one coordinate (e.g., *z*) forms the basis for a *non*-degenerate irreducible representation, and the other two coordinates (e.g., *x* and *y*) span a doubly degenerate representation.

### F. Hydrodynamic model: Determining the resistance tensor from triangulated surfaces

*i*(

**v**

_{i}) is related to the unperturbed velocity of the fluid (

**u**) via hydrodynamic interaction tensors $(Bij)$, which connect triangular plate

*i*to the forces experienced by all of the triangular plates comprising the surface of the rigid body,

^{23}

^{24}we introduce a symmetrized version of $B$, which integrates over both triangular patches to obtain coupling between triangular elements,

*S*

_{j}and

*S*

_{i}of triangles

*j*and

*i*, respectively, and

**x**

_{i}is the centroid (or barycenter) of triangle

*i*. The area of triangle

*j*can be similarly expressed as a surface integral,

**a**and

**b**,

^{23,24}

*η*. The ⊗ symbol indicates the outer (tensor) product of two vectors, in this case, $a\u2212b$ with itself.

*w*

_{k}is a weight associated with the quadrature point $ykj$ on triangle

*j*, and

**x**

_{i}is the centroid of triangle

*i*. Using quadrature points and weights, we can, therefore, rewrite Eq. (22) as

^{25}which exactly integrates polynomials of degree 3 and whose points and weights are available in Quadpy.

^{26}Because the centroid is not a point in this quadrature, there is no singularity in the self interaction $(Bii)$.

*N*stands for the total number of triangular plates, and rewrite Eq. (21) as

**V**,

**U**, and

**F**are 3

*N*-dimensional vectors representing the triangles’ velocities, unperturbed fluid velocity, and forces on all of the triangles. The solution of Eq. (27) to find the force requires the inverse, $C=B\u22121$,

^{27–29}

^{7–9}we also have

*x*

_{Oi},

*y*

_{Oi}and

*z*

_{Oi}are the components of the vector between the origin

*O*and the centroid of the triangle

*i*.

Note that in contrast to bead models,^{7,28–30} a boundary element method does not require a volume correction to the rotational block of the resistance tensor since the boundary element method computes interactions using hydrodynamic elements that have no volume.

*O*, it is possible to reconstruct the blocks at another point

*M*. The (tt) block is invariant to choice of origins, the (tr) block follows Eq. (13), and the (rr) block requires coupling to the other blocks of Ξ,

^{7–9}

In Sec. III of the supplementary material, we apply the boundary element method developed here to objects whose blocks of the resistance tensor are known analytically. The boundary element method shows good agreement with the analytical values.

## III. RESULTS

Applying the triangulated surface boundary element method described in Sec. II F, we have computed the principal axes of pitch and moments of pitch for a wide array of objects. These objects include common chiral entities such as helices, achiral swimmers, and one object of historical curiosity: Lord Kelvin’s Isotropic Helicoid. These objects are swimming at a low Reynolds number, and, if not stated otherwise, we shifted the center of pitch to the origin of the coordinate system. Wherever possible, we compare the predictions from the pitch matrix to experimental results for similar objects in similar fluid conditions. For Lord Kelvin’s Isotropic Helicoid, we also investigate two feasible experiments to assess its rotation-translation coupling.

### A. Chiral objects

#### 1. Helices

The hydrodynamic properties of helices have been studied widely because of their importance in the motion of living cells. Chwang and Wu^{31} and Higdon^{32} used a helix connected to a spherical head to model the swimming of micro-organisms and to find optimum design parameters for efficient propulsion under low Reynolds numbers. Purcell^{33} approximated the blocks of the resistance tensor in Eq. (1) as scalars, reducing the 6 × 6 resistance tensor to a 2 × 2 tensor, and explored the relation between these scalar values in the coupling of translational and rotational motions of helical systems. To study the swimming properties of *Escherichia coli* bacteria, Chattopadhyay *et al.*^{34} utilized the same scalar approach as Purcell and estimated the reduced 2 × 2 resistance tensor using optical tweezers to trap a sample of swimming *E. coli*. Recently, Maffeo *et al.*^{35} have looked at using rotating nucleic acid double helices as turbines, using electric fields to drive the motion of these molecules.

The work on helical molecules is at a length scale where the theory of pitch may help guide design parameters for molecular machines. To test these ideas, it is important to determine if the pitch matrix can reproduce previous work on helical systems in general. In this section, we first discuss the pitch matrix properties of a single microhelix using the hydrodynamic model developed in Sec. II F to compute the 6 × 6 resistance tensor. In Sec. III A 2, we apply this work to three primary structures of DNA double helices.

To test the pitch matrix properties for a simple helix, we constructed a 3 *μ*m right-handed helix with an outside diameter of 0.5 *μ*m and a thickness of 0.2 *μ*m using spherical beads. The helix was constructed through a procedure outlined in the supplementary material of Ref. 7, where the centers of consecutive beads are 0.049 *μ*m apart. To triangulate the surface mesh (see Fig. 3) and compute the resistance tensor and pitch matrix for the helix, we used the MSMS algorithm^{36} with a probe radius large enough (0.2 *μ*m) to smooth the helical surface. Figure 3 shows the principal axes of pitch and the associated moments of pitch for motion around these axes. None of the principal axes of pitch lie along the long axis (*z*-axis) of the helix. Therefore, a rotation around the helical *z*-axis will result in translational drift, whose direction is indicated by the vector **v** in Fig. 3(b).

*z*-axis. With an angular velocity

**= (0, 0, 1) rad s**

*ω*^{−1}and using Eq. (5), the resulting translational velocity for this helix is

**v**= (0.009 70, 0.0120, 0.0242)

*μ*m s

^{−1}. The translational motion of the helix will be along the vector

**v**, and the projected distance of travel

*t*is the total time.

From Chasles’ theorem,^{37} which states that rigid body motion can be decomposed into rotation along an axis and translation parallel to that axis (a screw displacement), the term |**v**|/|** ω**| in Eq. (32) may be interpreted as the pitch projected along the vector

**v**. For the helix in Fig. 3, |

**v**|/|

**| = 0.0287**

*ω**μ*m rad

^{−1}; thus, it will travel a distance d = 180 nm after one complete revolution around the

*z*axis. Because the helix is moving through a fluid, rather than a solid, the distance d is smaller than the designed pitch of the helix, which is 1

*μ*m per turn. (As in physical screws, the designed pitch of a helix will only be equivalent to the travel from one rotation when the helix is advancing through a solid substrate.)

The constructed helix in Fig. 3 is similar to the microhelices propelled with a magnetic field by Patil *et al.*^{5} We note that this group observed the microhelices drifting and estimated an experimental projected pitch of 250 nm. In Sec. IV of the supplementary material, we also provide data on three helices that approximate those in the Patil *et al.* experiments^{5} and find projected pitch values from 138 to 280 nm.

To make a direct comparison to experiments, we can use the scalar pitch coefficient, a rotational invariant defined in Eq. (19), which includes contributions from all three pitch axes. In the helix in Fig. 3, the scalar pitch coefficient is calculated to be 125 nm. For helices with flagellar widths ranging from 0.1 to 0.25 *μ*m, we find scalar pitch coefficients from 95.5 to 194 *μ*m. We also note that the scalar pitch coefficients are all ∼70% of the largest of the three moments of pitch, so we can infer that the helix tends to align to the axis of pitch associated with the largest of the three moments. Drifting was also observed by Ceylan *et al.*^{38} in their experiments with helical microswimmers.

#### 2. Double helices: A-, B-, and Z-DNA

To study a biologically relevant set of helices, we analyzed molecular structures representing the A, B, and Z forms of DNA. The A-DNA sample is a dodecamer with three consecutive CpG steps (PDB code 5MVK),^{39} the B-DNA sample is a Dickerson–Drew dodecamer (PDB code 4C64),^{40} and the Z-DNA sample is also a dodecamer (PDB code 4OCB).^{41} These three DNA structures are all derived from experimental crystal structures.

To triangulate the surface of the DNA samples, we represented the atoms as spheres with appropriate van der Waals radii and used the MSMS algorithm^{36} with a probe radius of 1.41 Å to mimic the surrounding water molecules.^{42} We computed the resistance tensor employing the triangulated surface method described in Sec. II F.

In Fig. 4, the upper panels display the pitch axes along with the associated moments of pitch for the three DNA samples and the lower panels, the resulting translational velocity due to a rotation around the *z*-axis (*ω*_{z} = 1 rad s^{−1}). The three DNA samples manifest translational drift coupled to the rotation around the *z* axis. From Eq. (32), we can compute the pitches projected along the vector **v** displayed in the lower panels, $|v||\omega |\xd72\pi =$ 0.13 nm (A-DNA), 0.23 nm (B-DNA), and 0.14 nm (Z-DNA). These values are smaller than the average structural pitch (per turn) associated with DNA, which is 2.82 nm (A-DNA), 3.38 nm (B-DNA), and 4.50 nm (Z-DNA).^{43} Because the DNA samples are moving in a fluid (and not through a solid substrate), we expect the moments of pitch to be significantly smaller than the structural pitch since the fluid slips along the surface of the DNA molecules.

*D*

_{tt},

^{11}

**v**|/|

**| in Eq. (32) with the scalar pitch coefficient |**

*ω**P*|/2

*π*in Eq. (19). The diffusion coefficient is calculated as $Dtt=13TrDCDtt$,

^{9,29}where the matrix $DCDtt=kBT\mu CDtt$ is defined in Eq. (2) and CD stands for the center of diffusion (see Sec. II C). The term |

**|**

*ω*^{2}

*t*is a threshold value that can aid in the design of propulsion experiments.

Table II shows the diffusion coefficients for the DNA samples and the angular velocity conditions for when the translation-rotation coupling overcomes translational diffusion. The translational diffusion coefficients are computed in dilute water solutions at 298.15 K and *η* = 0.89 mPa s.^{45} From the angular velocity conditions in Table II, a 11.6-day experiment requires angular velocities, |** ω**|, that exceed 1.4 × 10

^{3}rad s

^{−1}(A-DNA), 1.1 × 10

^{3}rad s

^{−1}(B-DNA), and 1.6 × 10

^{3}rad s

^{−1}(Z-DNA) to overcome translational diffusion. In longer experiments, as long as rotations are continuous, translation-rotation coupling can overcome diffusion with smaller angular velocities.

Structure (PDB code) . | $Dtt108nm2s\u22121$ . | $|P|/2\pi nmrad\u22121$ . | $|\omega |2\xd7t1012rad2s\u22121$ . |
---|---|---|---|

A-DNA (5MVK) | 1.70 | 0.023 | $>1.9$ |

B-DNA (4C64) | 1.65 | 0.028 | $>1.3$ |

Z-DNA (4OCB) | 1.66 | 0.020 | $>2.5$ |

Structure (PDB code) . | $Dtt108nm2s\u22121$ . | $|P|/2\pi nmrad\u22121$ . | $|\omega |2\xd7t1012rad2s\u22121$ . |
---|---|---|---|

A-DNA (5MVK) | 1.70 | 0.023 | $>1.9$ |

B-DNA (4C64) | 1.65 | 0.028 | $>1.3$ |

Z-DNA (4OCB) | 1.66 | 0.020 | $>2.5$ |

In comparison with the DNA samples in Table II, the 3 *μ*m single helix in Sec. III A 1 has *D*_{tt} = 4.39 × 10^{5} nm^{2} s^{−1} when suspended in the same dilute water conditions, with a pitch coefficient of |*P*|/2 *π* = 19.9 nm rad^{−1}. The |** ω**|

^{2}

*t*threshold value points to the minimum frequency of rotation |

*ω*| that the helix must have to overcome translational diffusion. This parameter, |

**|**

*ω*^{2}

*t*> 6.65 × 10

^{3}rad

^{2}s

^{−1}, is also dependent on the time scale (

*t*) for the experiment. In a 100 s experiment, the helix will overcome diffusion when its frequency is held constant at a minimum of 1.30 Hz. For the same helix, in a 10 s experiment, the helix will overcome diffusion when its frequency is held constant at 4.10 Hz. In an experiment with similar helices and solvent conditions, Patil

*et al.*

^{5}applied a rotating magnetic field with frequencies in a range of 5–15 Hz to propel their microscopic helices, which are well above the predicted minimum threshold frequencies to observe propulsion. Patil

*et al.*

^{5}also reported that their helices could overcome diffusion when the rotation frequency was 2 Hz.

### B. Achiral swimmers

In Sec. II E, we showed that achiral objects can be divided into two groups by their moments of pitch. The first group consists of achiral objects for which all moments of pitch are zero, i.e., the pitch matrix is a null matrix. As a result, objects in this group exhibit no translation-rotation coupling, and rotation will not produce displacement. Examples of these achiral non-swimmers are well-known; e.g., spheres, ellipsoids, tetrahedra, and cubes. Interested readers are encouraged to consult Sec. V of the supplementary material for more details.

The second group consists of achiral objects with a special symmetry, where one moment of pitch is zero, while the other two have the same magnitude but opposite signs. For these objects, the pitch matrix is non-zero, and translation-rotation coupling persists. These objects have previously been called *achiral swimmers* because they produce displacements due to rotation. Figure 2 shows an achiral microswimmer: an arrangement of three beads that was experimentally tested by Cheang *et al.*^{17} In their work, Cheang *et al.* reported that a non-vanishing (rt) block of the mobility tensor is required for swimming and, from the symmetry investigations conducted in Ref. 8, concluded that *achiral swimmers* are real.

To triangulate the surface of the three-bead arrangement in Fig. 2, we used the MSMS algorithm^{36} with a probe radius of 10^{−6} *μ*m and a triangulation density of 20 vertices/*μ*m^{2}. In Fig. 2, we also found the principal axes of pitch and their associated moments employing the bead model developed in Ref. 7 in addition to the boundary element approach developed in this work (Sec. II F). As expected, with both methods of finding the resistance tensor, one moment of pitch is zero, and the other two have the same magnitude with a flipped sign.

Utilizing the bead and boundary element models, Table III presents the translational diffusion and the scalar pitch coefficient for the three-bead achiral swimmer along with the angular velocity condition for when the translation-rotation coupling surpasses diffusion. Translational diffusion coefficients were computed at 298.15 K and *η* = 1.0 mPa s, reproducing the experimental conditions of Cheang *et al.* work^{17} for a NaCl solution. In the first second of an experiment, the translation-rotation coupling of the immersed three-bead swimmer will overtake diffusion when the angular velocity, |** ω**|, exceeds 7.6 rad s

^{−1}(bead model) and 8.0 rad s

^{−1}(boundary element model). These are equivalent to rotational frequencies that exceed 1.3 Hz and are comparable to a rotating magnetic field frequency of 1–8 Hz applied by Cheang

*et al.*

^{17}to propel their swimmers. The scalar pitch values in Table III and the moments of pitch in Fig. 2 can be used to generate similar swimming speeds reported in Fig. 3(b) of Ref. 17.

Hydrodynamic model . | $Dtt104nm2s\u22121$ . | $|P|/2\pi nmrad\u22121$ . | $|\omega |2\xd7trad2s\u22121$ . |
---|---|---|---|

Bead (Ref. 7) | 6.07 | 79 | $>58$ |

Boundary element (Sec. II F) | 5.80 | 74 | $>64$ |

Translational drift is expected when the axes of rotation are not the principal axes of pitch. This translational drift can be seen clearly in the supporting videos^{17} displaying the motion of the three-bead swimmers. Hermans *et al.*^{46} also reported translational drift in an experiment with a rotating achiral swimmer. In a Taylor–Couette device, Hermans *et al.* achiral swimmer had one orbital radius when rotating clockwise and another when rotating counterclockwise.

### C. Lord Kelvin’s isotropic helicoid

*isotropic helicoid*is an object for which the blocks of the resistance tensor are isotropic at the center of resistance (CR). That is, the four blocks may be written

^{8}

*ξ*

^{tt},

*ξ*

^{tr}, and

*ξ*

^{rr}are scalars, and

**I**is the 3 × 3 identity matrix. The only difference between these objects and spherically isotropic bodies (i.e., spheres, cubes, and tetrahedra) is that helicoids have non-zero rotation-translation coupling (

*ξ*

^{tr}≠ 0).

^{8}

In 1871, Sir William Thomson (widely known as Lord Kelvin) proposed one design for an isotropic helicoid using a sphere with 12 projecting vanes arranged in a systematic way.^{47} A generalization of his approach is shown in Fig. 5 and is described below:^{48,49}

Center a sphere at the origin, and locate three circles at the intersections of the sphere with the

*xy*-,*yz*-, and*xz*-planes [Fig. 5(a)].Using the six intersection points of the three circles, place the centers of semi-oblate vanes midway between these intersection points. In the end, there will be four semi-oblate vanes per circle [Fig. 5(b)].

The orientation angles

*θ*of the vanes are related to their positions in the circles and define the handedness of the isotropic helicoid. The isotropic helicoid is right-handed for 0° <*θ*< 90° and left-handed for −90° <*θ*< 0°. This definition comes from the sign of the angular velocity*ω*in Eq. (37) and the direction of rotation of a right-handed screw.

We constructed three left-handed isotropic helicoids (*θ* = −30°, −45°, −60°) and two spherically isotropic bodies (*θ* = 0°, −90°) utilizing the procedure in Ref. 48. Each of these isotropic objects is composed of a sphere with 12 semi-oblate vanes, whose dimensions are shown in Figs. 5(c) and 5(d). Triangulation of these objects was performed in OpenSCAD,^{50} and we analyzed them employing the boundary element method described in Sec. II F with a 120-point quadrature developed by Xiao and Gimbutas.^{51} This quadrature exactly integrates polynomials of degree 25, and its points and weights are available in Quadpy.^{26} Figure 6 shows the pitch axes and the moments of pitch for these isotropic objects.

For all five objects in Fig. 6, both the center of resistance and the center of pitch are located at the origin. For the two spherically isotropic bodies, $\Xi CRrt=\Xi CRtr=0$. From Eq. (8), this implies that the pitch matrix itself is zero and no translation-rotation coupling is possible. The isotropic helicoids (*θ* = −30°, −45°, −60°) have non-zero moments of pitch, but our calculations indicate that these are quite small. From the definition of the pitch matrix, these small moments of pitch are related to small rotation-translation coupling values (*ξ*^{tr}), and we can use these moments of pitch to demonstrate rotation-translation coupling of isotropic helicoids. For example, in an experiment where an isotropic helicoid from Fig. 6 with *θ* = −45° is suspended in a viscous fluid and a rotation frequency of 10 Hz is imposed along one of the pitch axes, the helicoid will move 6.3 cm in a 100 s observation time. On the other hand, the spherically isotropic bodies (*θ* = 0°, −90°) will not move at all since there is no rotation-translation coupling. In the supplementary material (Sec. VI), we compute pitch axes and moments of pitch for a related set of left-handed isotropic helicoids, and we show the pitch coefficients as a function of the vane angle (*θ*).

Table IV presents the scalar values associated with the blocks of the resistance tensor [Eq. (35)] for the isotropic objects in Fig. 6. These scalar values were computed in silicon oil with *η* = 490 mPa s^{52} and employing the triangulation and boundary element methods described above. For *θ* = −30°, −45°, and −60°, the scalar values are related to the isotropic helicoids where *ξ*^{tr} ≠ 0. For *θ* = 0° and −90°, the scalar values are related to the spherically isotropic bodies where *ξ*^{tr} = 0.

. | ξ^{tt}
. | ξ^{tr}
. | ξ^{rr}
. | v . | ω
. |
---|---|---|---|---|---|

Angle . | $kgs\u22121$ . | $kgm/(srad)$ . | $kgm2/(srad2)$ . | (cm s^{−1})
. | (rad s^{−1})
. |

0° | 0.284 | 0 | 3.73 × 10^{−4} | 46.8 | 0 |

−30° | 0.283 | 2.65 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −3.34 × 10^{−3} |

−45° | 0.283 | 2.72 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −3.42 × 10^{−3} |

−60° | 0.283 | 1.89 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −2.38 × 10^{−3} |

−90° | 0.283 | 0 | 3.73 × 10^{−4} | 47.0 | 0 |

. | ξ^{tt}
. | ξ^{tr}
. | ξ^{rr}
. | v . | ω
. |
---|---|---|---|---|---|

Angle . | $kgs\u22121$ . | $kgm/(srad)$ . | $kgm2/(srad2)$ . | (cm s^{−1})
. | (rad s^{−1})
. |

0° | 0.284 | 0 | 3.73 × 10^{−4} | 46.8 | 0 |

−30° | 0.283 | 2.65 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −3.34 × 10^{−3} |

−45° | 0.283 | 2.72 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −3.42 × 10^{−3} |

−60° | 0.283 | 1.89 × 10^{−6} | 3.73 × 10^{−4} | 47.0 | −2.38 × 10^{−3} |

−90° | 0.283 | 0 | 3.73 × 10^{−4} | 47.0 | 0 |

^{8,52}

*m*

_{b}and

*m*

_{f}are the masses of the body and the displaced fluid, respectively,

**g**is the gravitational vector field and

**0**is the null vector. The term

*m*

_{b}

**g**represents the gravitational force and the term −

*m*

_{f}

**g**represents the force due to buoyancy.

For chiral objects, we know from our previous work in Ref. 7 that only *ξ*^{tr} will reverse sign. This implies that v in Eq. (37) is the same for both a helicoid and its mirror image, but the angular velocity *ω* will flip sign for the mirror image (enantiomeric) version of the object.

For the angular velocity in Eq. (37), Brenner^{8} employed a standard definition of *ω* > 0 for counterclockwise rotations, following the right-handed rule.^{12} This implies *ξ*^{tr} < 0 for right-handed isotropic helicoids since the scalars *ξ*^{tt}, *ξ*^{rr}, and $\xi tt\xi rr\u2212\xi tr2$ are always positive.^{8} We note that this definition is the opposite of the one employed by Gustavsson and Biferale,^{49} and Collins *et al.*^{52}

Table IV also provides the terminal velocities and terminal angular velocities for the bodies in Fig. 6. To compute these values, we used *g* = 9.81 m/s^{2} and the experimental conditions in Ref. 52, i.e., body density *ρ*_{b} = 1.16 g/cm^{3} and fluid density *ρ*_{f} = 0.98 g/cm^{3} (silicon oil). The mass contributions to the gravitational and buoyant forces can be calculated from the volume of the rigid body, *V*_{b} = 75.267 cm^{3}, which is the combined volume of the sphere and the 12 semi-oblate vanes in Fig. 5. The spherically isotropic bodies will not manifest angular velocities because their scalar *ξ*^{tr} = 0. We predict that the left-handed isotropic helicoids will manifest an angular velocity on an order of 10^{−3} rad/s in a clockwise direction.

The scalar friction values, *ξ*^{tt}, *ξ*^{tr}, and *ξ*^{rr} scale as *η L*, *η L*^{2}, and *η L*^{3}, respectively, where *L* is the length of the rigid body. Since the volume *V*_{b} scales with *L*^{3}, we conclude that the terminal velocity v and angular velocity *ω* in Eq. (37) will scale as *L*^{2}/*η* and *L*/*η*, respectively. To compare with Collins *et al.* helicoid,^{52} we can scale the size of our isotropic helicoid (*θ* = −45°) by 0.348 in the same fluid, and obtain a scaled $v\u2032=0.3482v=5.69cms\u22121$, and a scaled *ω*′ = 0.348 *ω* = −1.19 × 10^{−3} rad/s. Collins *et al.*^{52} reported v′ = 4.74 cm s^{−1} and *ω*′ = −0.003 rad/s. Since *ω*′ from Collins *et al.* has 1% uncertainty, our calculation reveals that more sensitive instruments would be required to measure the rotation-translation coupling of the Collins *et al.* helicoid.

## IV. CONCLUSION

We have presented a general theory for the pitch of objects that are interacting with a fluid medium at a low Reynolds number. The pitch matrix, defined in Eq. (5), is diagonalized to yield three pitch axes along with their associated “moments of pitch.” The pitch axes and moments arise out of the *geometry* of the objects’ surfaces, and they have a number of important properties. First, the symmetry of the object defines the number of degenerate and non-zero moments of pitch. Second, chiral objects (molecules, helices) couple rotational and translational motion in the fluid and will move in the opposite direction from their enantiomers (mirror images) under the same rotation. Third, the pitch matrix also provides an explanation for the rotation-translation coupling that allows *achiral* swimmers to migrate when they rotate in a fluid. This theory also helps us to understand the translational drift of rotating helical objects. There are many potential uses of this theory, but the primary interest for chemists is to develop an efficient method for separating enantiomers without the costly synthetic pathways currently in use.

One of our primary observations is that chiral objects with a *C*_{n} axis of symmetry have two degenerate moments of pitch when *n* ≥ 3, and there is no drift for rotations around that axis of symmetry. This appears to be the case for propeller-shaped molecules, and this observation points to a general and efficient design principle.

There are many ways to approximate the hydrodynamic resistance tensor, and we have developed a boundary element method that obeys the symmetry properties of the blocks of the resistance tensor. This was also true of earlier methods that used small beads or atomic spheres to represent the surface of an object or the surface of a molecule, but the method for triangulated surfaces given here is generally applicable to rigid bodies of arbitrary shapes.

Our theory of pitch has been tested against some experiments on microswimmers, and our predictions agree well with experimental observations of translational-rotational coupling. We also show results for an object of historical curiosity, Lord Kelvin’s isotropic helicoid, which can exhibit a small angular velocity as it falls through a fluid. For the collective behavior of a multi-molecule system, consider Ref. 7, where a competition model was developed to study the separation of chiral molecules in solution. Note that even in racemic mixtures, separation can be achieved under sufficiently large solution vorticities.

There are many potential uses for this theory. Projection of molecular dipoles onto the pitch axis with the largest moment of pitch can help design polarized microwave methods that separate enantiomers through molecular rotation in the fluid (instead of rotating the fluid around the enantiomers). In addition, this method also now allows us to identify the geometries of achiral swimmers from the eigenvalue structure of the pitch matrix. We also now have a firmer understanding of the non-axial drift of chiral objects due to projections of angular velocity onto the axes of pitch.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional properties of the pitch matrix and the pitch coefficient, as well as applications of the theory of pitch in isotropic helicoids, achiral swimmers, non-swimmers, and analytically solvable objects. The supplementary material also develops a relationship between pitch and the moment of inertia for a sphere, and this is used to analyze translation-rotation coupling in spheres rotating in non-Newtonian fluids. An accompanying set of text files provides molecular geometries for the enantiomers and DNA structures, as well as triangulated surfaces for the helix, achiral swimmers, and isotropic helicoids. Code that computes the blocks of the resistance tensor and pitch matrices for these objects is also included.

## ACKNOWLEDGMENTS

Support for this project was provided by the National Science Foundation under Grant No. CHE-1954648. Computational time was provided by the Center for Research Computing (CRC) at the University of Notre Dame.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Anderson D. S. Duraes**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). **J. Daniel Gezelter**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

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