Diffusion coefficients of linear trimer particles

We study the diffusive behavior of linear trimer particles via numerical calculations. First, we utilize hydrodynamic bead-shell calculations to compute the microscopic diffusion coefficients for different particle aspect ratios. These values are then used to obtain continuous empirical formulas for said coefficients. As an application example for the empirical formulas, we perform Brownian dynamics simulations of monolayers consisting of a linear trimer surrounded by colloidal spheres. Here, we obtain empirical formulas for the corresponding long-time diffusion coefficients of the trimer. By comparing our data for the microscopic and long-time diffusion coefficients with known results for spherocylinders, we find that the diffusive behavior of both particle geometries is approximately identical. Based on this observation, we introduce simplified equations for the microscopic diffusion coefficients that can be used for arbitrary short rods that are spheres at the minimum aspect ratios. The calculated equations for the diffusion coefficients can be applied to various further numerical and experimental studies utilizing linear trimer particles


I. INTRODUCTION
In systems of colloidal spheres, we are accustomed to quantifying their trembling motion with two simple constants-the well-established translational and rotational diffusion coefficients. However, the general framework to model colloidal diffusion of more complicated particle shapes sharply increases in complexity: 1 To address the shape-dependent diffusive movements of arbitrarily shaped colloids, we usually need a 6 × 6 dimensional diffusion tensor (which contains information on all translational and rotational degrees of freedom, as well as the translation-rotation coupling). 2,3 The knowledge of this quantity is crucial to modeling and understanding the motion of complex particle geometries, for instance, in external fields [4][5][6] or with an effective self-propulsion. [7][8][9] Hence, finding the diffusion tensors of different particle geometries is an essential task of soft matter science up until now. 10 In the case of highly symmetrical colloids such as orthotropic particles (i.e., particles with three pairwise orthogonal symmetry planes 7 ), the diffusion tensor can be simplified to a small set of microscopic diffusion coefficients. 1,11,12 Examples of this are the already addressed translational and rotational diffusion constants of a sphere. To describe the Brownian motion of uniaxial orthotropic particles (i.e., elongated orthotropic particles with axis symmetry), we usually need three distinct microscopic diffusion coefficients. 13 In particular, for cylinder-shaped particles, there are many wellestablished results and approximations for these quantities, such as the theory of Broersma [14][15][16] or the equations of Tirado and García de la Torre. 11,12,17 For prolate spheroids, the relations of Perrin 18,19 can be applied, and, for dumbbell and short spherocylinders, we recently computed empirical formulas in a previous study. 20 To our knowledge, a set of equations for the microscopic diffusion coefficients of linear trimers (which are rod-like particles consisting of three overlapping spherical segments) are still missing.
However, such equations would be highly beneficial for studies based on computer simulations: Apart from "smooth" models such as the Kihara approach, [21][22][23] rod-shaped particles are often implemented via a linear string of rigidly connected (and sometimes overlapping) spherical segments in many-particle simulations. This method (which is often called the "Shish-Kebab model" 24,25 ) was, for instance, utilized to study the cluster formation 26 and the diffusivity 27 of elongated self-propelled particles, anisotropic thermophoresis, 24,25 and the phase behavior of rods with attractive tips. 28 To model simple elongated particles via the Shish-Kebab method in Brownian dynamics (BD) simulations (which is a simulation technique where the microscopic diffusion coefficients are a crucial part of the integrators 29 ), it is often viable to choose an arbitrary set of equations for the diffusion coefficients if the exact diffusive properties do not matter. However, the knowledge of the correct diffusive behavior can improve the transition to real-world applications.
In addition to this generic use of connected spheres to implement elongated particles in simulations, one sometimes aims to study the phase behavior, 30 self-assembly, 31 or the dynamics 20 of particles that possess a geometric shape that resembles overlapping spherical segments. Examples of such studies are BD simulations of dumbbell-shaped particles 32,33 or sedimentation experiments based on "colloidal molecules." 34 Here, the microscopic diffusion coefficients should be exactly matched to the analyzed particle shape to obtain the best results.
In recent years, the large-scale fabrication of linear trimer particles became possible through breakthroughs in colloidal synthesis. [35][36][37][38] With this, they can be used as anisotropic building blocks 35 to study, for example, their self-assembly 36 and their diffusive properties 37 in experiments. The missing relations for the microscopic diffusion coefficients of linear trimers would be an excellent tool to complement such experiments when the knowledge of particle dynamics is fundamental. Furthermore, BD simulations containing the microscopic diffusion coefficients could be used to corroborate the experimental findings.
In this direct follow-up to our previous work, 20 we tackle the issue of the missing equations for the microscopic diffusion coefficients of linear trimers with hydrodynamic bead-shell calculations. 39,40 This means that we model the particles via shells of spherical friction beads for which the corresponding diffusion coefficients can be obtained numerically. Extrapolating to the limit of an infinite number of friction beads with zero diameters then gives excellent approximations for the microscopic diffusion coefficients. 39,40 Finally, we use simple fits through our bead-shell results for different aspect ratios to obtain continuous empirical formulas. This is the same strategy we used previously 20 while calculating the empirical formulas for the diffusion constants of dumbbell-and spherocylinder-shaped particles (which reproduced established theoretical predictions and experimental measurements with great success 20 ).
As an application example for the empirical formulas, we use the obtained diffusion coefficients to perform two-dimensional BD simulations of colloidal monolayers (without hydrodynamic interactions). In detail, we consider simple quasi-two-dimensional systems consisting of a single trimer-shaped particle that can only rotate in the plane of the particle positions surrounded by colloidal spheres. We then compute empirical formulas for the long-time diffusion coefficients of the trimer. This setup is in line with other quasitwo-dimensional systems we studied recently in experiments 30 and simulations 20,30 (i.e., colloids bound to the interface of two fluids, where we assume that these fluids have the same viscosity and all influences of the interface are negligible).
By using the calculated microscopic and long-time diffusion coefficients, we also study if linear trimers are a sufficient approximation for spherocylinders (which are cylinders with two hemispherical caps) regarding their diffusive properties. For the comparison of the microscopic diffusion coefficients, we complement our results for trimers with the empirical equations calculated in Ref. 20. In the corresponding BD simulations for the comparison of the long-time diffusion coefficients, we use soft repulsive spherocylinders (SRS) that are based on a Kihara-like interaction potential. [21][22][23] Here, we find that the translational and rotational diffusive behaviors of both shapes are approximately identical. This suggests that linear trimers can be utilized to approximate short spherocylinder-shaped particles, and vice versa, which is in contrast to previous studies where pairwise interacting dumbbells and SRS showed distinct differences in their rotational long-time diffusion coefficients. 20 Based on the similarity between the microscopic diffusion coefficients of the linear timers and the spherocylinders, we also introduce approximated (and shape-independent) formulas for the microscopic diffusion coefficients of short rods that resemble spheres for small aspect ratios. It shows that these formulas are (at small aspect ratios) also viable for various particle geometries such as trimers, dumbbells, prolate spheroids, or spherocylinders.
This work is structured as follows: First, we discuss the methods such as bead-shell calculations and the BD simulations in Sec. II. Afterward, we present our results in Sec. III. Here, we discuss the computed empirical formulas, compare them to other established results, analyze the long-time diffusion coefficients obtained via the BD simulations, and compare the diffusive behavior of trimers and spherocylinders. We then propose approximated equations for the microscopic diffusion coefficients of short particles. We conclude our study with a summary and a discussion in Sec. IV. Note that we now drop the word "linear" from "linear trimer" for convenience. Throughout this work, we will also call the SRS "Kihara spherocylinders" to emphasize the spherocylindric shape of the particles.

II. MODEL AND METHODS
We study the diffusive behavior of trimer colloids. In three dimensions, this specific particle shape consists of three spherical segments with diameter σ that are arranged as a linear string. The middle segment is perfectly centered between the outer two segments that have a center-to-center distance of l. The distance between the center of the middle segment and the center of one of the outer ones is λ = l/2. In addition to its total length L, a trimer can also be characterized by its aspect ratio q = L/σ. The aspect ratio q of the trimer and the segment distance λ are related by q = 1 + 2λ/σ because L = 2λ + σ. See Fig. 1 for the cross-section of a trimer.

A. Diffusion coefficients of thin uniaxial particles
In numerical models or experimental setups where particle rotations around the symmetry axis are negligible, the shapedependent diffusion of uniaxial orthotropic particles (i.e., particles with rotational symmetry regarding a main axis and three pairwise orthogonal mirror planes) can be described by three microscopic diffusion coefficients, 11-13 D ∥ , D , and D r . The translational coefficients D ∥ and D correspond to the diffusion parallel and perpendicular to the symmetry axis. The constant D r describes the rotational trembling motion regarding an axis of rotation through the center of the particle that is perpendicular to the symmetry axis.

FIG. 1.
Cross sections of a linear trimer particle. It can be characterized by the total length L, the diameter σ, and the distance between the outer segments l. The distance between the center of one of the outer segments and the middle segment is λ = l/2. The elongation of the linear trimer can be characterized via the aspect ratio q = L/σ = 1 + 2λ/σ.
For cylinder-shaped particles, it is well-established that these microscopic diffusion coefficients can be written in the form where T is the temperature, kB is the Boltzmann constant, and η is the dispersion agent viscosity. 1,17 For the corresponding qdependent end-effect corrections γ ∥ (q), γ (q), and δ (q), there are common estimations that give excellent results compared to experimental data. 1,13,17 Recently, we confirmed empirically that the same relations (1)-(3) can be utilized to approximate the diffusion coefficients of similar rod-shaped particles such as dumbbell-and spherocylindershaped particles. 20 Analogously to the form of the equations for cylindrical rods of Tirado and García de la Torre, 11,12,17 we found that the end-effect corrections could be modeled by quadratic functions of 1/q [i.e., γ ∥ (q) ≈ A ∥ + B ∥ /q + C ∥ /q 2 , and so forth]. To determine empirical formulas for the microscopic diffusion coefficients of a trimer, we assume that the same holds for linear strings of three (overlapping) spheres. While our results indicate that this assumption is justified, it is not trivial.

B. Bead-shell calculations
To determine the microscopic diffusion coefficients, we use the bead-shell approach, which is described in detail in Refs. 3, 39, and 40. The basic idea is to model the shape of the particle by a shell of spherical friction beads with diameter σ B . These beads interact with each other via hydrodynamic interactions and with the surrounding liquid via Stokes friction. By using hydrodynamic calculations, it is possible to determine the microscopic diffusion coefficients of the resulting shell model. 3 Afterward, this procedure is repeated for shells based on an increasing number of beads with smaller σ B until the limit of an infinite number of beads with zero diameters can be approximated using linear (or if necessary polynomial) fits. 39 The microscopic diffusion coefficients of this limit are said to be the values of the original particle.
The shells of friction elements for the trimers are built in the same manner as for the dumbbells in our previous work: 20 We start by placing the (non-overlapping) friction beads on the curved part of the surface of an eighth of a sphere and then perform mirror operations to obtain a full spherical shell. This is repeated three times to generate all three segments. The spherical shells are shifted in space to form the trimer, where the center of the middle segment is set to coincide with the center of the used Cartesian coordinate system, and the symmetry axis of the original trimer is chosen to be parallel to one of the Cartesian axes. Furthermore, we ensure that the symmetry planes of the bead model that result from the initial mirror operations coincide with the planes of the coordinate system. Then, we remove all friction elements inside the two overlaps of the segments. A gear-inspired formation is left where the segments meet (i.e., on the interface between two segments, we alternately delete friction beets of the particular spherical shells so that they mesh). See Fig. 2 for an image of a trimer bead-shell model.
The hydrodynamic calculations to obtain the microscopic diffusion coefficients of the shells are performed using the wellestablished public domain program HYDRO++. 3,[39][40][41] It uses noslip boundary conditions, and we choose the option of a thirdorder approximation of the hydrodynamic interactions between the beads. 3,39 See Refs. 3 and 39-41 for the mathematical framework of HYDRO++. We use the parameters T = 25 ○ C, η = 0.89 mPas, and σ = 180 nm as input. Furthermore, we vary the bead diameter σ B between 7 and 30 nm.
By dividing relations (1) and (2) by the translational diffusion coefficient of a sphere with the same diameter σ as the segments, the resulting equations only depend on the aspect ratio q. The same holds for the quotient of Eq. (3) and the rotational diffusion coefficient of a sphere with diameter σ. Note that the relation Ds = 3/τD holds, where τD = σ 2 /Ds is the Brownian time. Consequently, we can apply The Journal of Chemical Physics ARTICLE pubs.aip.org/aip/jcp the bead-shell approach to trimers of different aspect ratios and determine the empirical formulas using a simple one-dimensional fit as the quotients only depend on a single parameter q.
Note that HYDRO++ yields a 6 × 6 dimensional diffusion tensor as output, which is necessary to describe the diffusion of more complexly shaped particles with fewer symmetries. Our placement of the shell in the corresponding Cartesian coordinate system enables us to read the microscopic diffusion coefficients off the diagonal of this tensor. See Ref. 20 for more information.

C. Brownian dynamics simulations
The determined microscopic diffusion coefficients are used in BD simulations of colloidal monolayers to give an application example. In detail, we analyze systems based on a single trimer (or spherocylinder) in a suspension of colloidal spheres.
In these simulations, we base all particle interactions on the purely repulsive Weeks-Chandler-Andersen (WCA) potential. 42 It is given by where ε is the interaction strength. Throughout all BD simulations, we use ε = 1kBT for simplicity. The trimers are implemented analogously to the dumbbell particles of Refs. 20, 30, 32, and 33: The interaction potential between the spheres and the segments is considered pairwise. Hence, for the interaction of the trimer and a sphere, the WCA potential is evaluated three times, where the distances between the sphere center and the segment centers are inserted in Eq. (6). The total interaction force results from the sum of all three interactions. The torque that acts on the trimer due to the interaction with sphere i is Here, ⃗ Fi, j (t) are the interaction forces that act on the two outer segments, and the operator [⋅]z gives the third (i.e., z) component of the inserted vector. Note that the middle segment does not contribute to the torque.
Differently from the trimers, the spherocylinder shape is implemented via the soft repulsive spherocylinders (SRS) model, which utilizes the Kihara approach 21 and was also applied in Refs. 8, 22, 43, and 44. In this model, the smallest distance to the line segment in the middle of the spherocylinder that connects the centers of the hemispherical caps is used for r in the evaluation of the WCA potential between the particles. For the calculation of the corresponding interaction torque, the vector pointing from the spherocylinder center to the point on the line segment that is connected to the sphere center via the shortest distance is used as a lever. See Ref. 23 for a detailed description.
The uniaxial particles and the spheres are overdamped colloids moving in a dispersion agent. Thus, we can use the BD algorithm to model the dynamics. The position ⃗ r(t) and orientation ⃗ e(t) vectors of the uniaxial particle are updated using where ⃗ F(t) and M(t) are the total interaction force and torque acting on the particle, and Δt is the length of the simulation step. The Rj are standard normally distributed random numbers, meaning that To maintain a length of |⃗ ei(t)| = 1, ⃗ e(t) is normalized after every simulation step. More information on the integration of colloidal rods can be found in Refs. 8, 20, 29, 32, and 33.
The position of the ith sphere can be integrated with the following relation: where ⃗ Fi(t) is the interaction force affecting sphere i and ⃗ R consists of standard normally distributed components. This is in line with Refs. 45-47.

III. RESULTS AND DISCUSSION
In the following, we analyze the results of the bead-shell calculations and the resulting empirical formulas before comparing the obtained relations with established results. We also discuss the BD simulations that we use as an application example. Finally, we investigate the differences between trimers and spherocylinders regarding their dynamic properties and introduce approximated equations for short rod-like particles that are spheres for q = 1.00.

A. Empirical formulas for the microscopic diffusion coefficients
By applying the bead-shell method, we can calculate the microscopic diffusion coefficients D ∥ tp , D tp , and D r tp of trimer particles for different q. Note that this particle geometry can only possess an aspect ratio in the interval of 1.00 ≤ q ≤ 3.00. The resulting values are given in Table I in units of Ds and τ −1 D . Note that each of these values is determined via multiple shells and by extrapolating to the limit of infinite beads with zero diameters. On average, we use between seven and eight different σ B for the bead-shell calculations corresponding to a trimer with a specific q. See Refs. 3, 34, 39, and 40 for more information on the extrapolation process.
In Fig. 3, the microscopic diffusion coefficients calculated via the bead-shell method are depicted (in units of Ds and D r s ) with points. By approximating the end-effect corrections with quadratic functions of 1/q, we can close the gaps between the bead-shell results  and obtain empirical formulas. The corresponding fits yield the following relations: They match the bead-shell results remarkably well [see Fig. 3, where the lines depict Eqs. (11)- (13)]. Note that we completed the dataset of Table I with the theoretical values of a sphere for q = 1.00 before calculating the fits. For information on the goodness of fit, see Sec. SI of the supplementary material.

B. Comparison with established results
We can now compare the determined formulas for the microscopic diffusion coefficients of a trimer with established results. We start by looking at the total three-dimensional diffusion coefficient of translation, and the rotational diffusion coefficient of a linear string of three (non-overlapping) spheres (i.e., trimers with q = 3.00) obtained with bead-shell calculations in Ref. 39. Through their computations, the authors find D 3D = 0.58Ds and D r = 0.11 D r s . Our empirical formulas yield D 3D = 0.57Ds and D r = 0.11 D r s . Thus, we find an excellent agreement between the established values of Ref. 39 and our results.
In simulations, the equations of Tirado and García de la Torre 11,12,17 for cylinder-shaped rods, are often the go-to approximation 29 for arbitrary elongated particles and are shown to match experimental data 17 for rod-shaped particles with an aspect ratio in the range of 2.00 ≤ q ≤ 30.0. Figure 4 shows a comparison between the formulas for trimers (solid lines) and the cylinder-shaped rods (dashed lines). While we find the same overall behavior, the diffusion coefficients of the trimers are larger than their cylinder counterparts. In the interval of 1.00 ≤ q ≤ 3.00 (2.00 ≤ q ≤ 3.00), the maximum relative differences for D ∥ , D , and D r are 7.8% (7.7%), 7.5% (7.1%), and 65.9% (47.0%), respectively. By comparing the microscopic diffusion coefficients for small aspect ratios, one should bear in mind that the equations of Tirado and García de la Torre were calculated 11,12 and tested 17 for q > 2.00, which does not allow for strong assessments regarding the accuracy of corresponding values for q < 2.00. Nevertheless, the general differences of the microscopic diffusion coefficients should be completely geometric in nature and are the results of the deviations in the particle shape. This can, in particular, be seen while shrinking down  the aspect ratios of the particles: While the trimer slowly becomes a simple sphere, the cylinder approaches the shape of a small disk (with equal height and diameter). The obvious differences between these two geometric shapes motivate the dissimilarities in the diffusion coefficients, which should be measurable in experiments. 34 Note that large relative differences in the diffusion coefficients mean that using the equations of Tirado and García de la Torre in simulations modeling the motion of short trimers could lead to distinct discrepancies in the particle dynamics.
C. Application example: Long-time diffusion coefficients BD simulations are a simple but powerful tool to corroborate equilibrium and non-equilibrium studies. The corresponding integrators for rod-like particles given by Eqs. (8) and (9) contain the microscopic diffusion coefficients and, thus, the calculated empirical formulas open up the BD algorithm for studying trimers.
In this section, we give an application example of the microscopic diffusion coefficients of trimers by utilizing said coefficients and the BD algorithm to obtain long-time diffusion constants. For each parameter set, we perform at least 2000 BD simulations of 150 spheres and 1 trimer in a quadratic simulation box with periodic boundary conditions. To change the sphere area fraction ϕ of the system, we adjust the size of the box length. The simulations are performed in units of the diameter σ of the spheres and the trimer segments, the energy kBT, the diffusion coefficient of the spheres Ds, and the Brownian time τD. We utilize a time step with the length Δt = 10 −5 τD. Initially, the particles are randomly placed in the simulation box before they equilibrate for 10τD. Afterward, the data are gathered in a time interval of 1000τD.
We start with the translational long-time diffusion coefficient D eff . For all analyzed sphere area fractions, the trimer performs a diffusive motion on large time scales. Hence, its mean squared displacement can be written as in the long time limit. By calculating ⟨Δ⃗ r 2 (t)⟩ with our BD simulations and using a linear fit for 200 τD < t < 650 τD (after equilibration) to approximate the long-time behavior, we can use this relation to obtain the translational long-time diffusion coefficient D eff . See also Sec. SII of the supplementary material for additional discussions regarding the calculation of the mean squared displacement and the calculation of the translational long-time diffusion coefficient. Figure 5(a) shows D eff of the trimer for different sphere area fractions and aspect ratios divided by the corresponding total twodimensional diffusion constants, We find that D eff is only weakly dependent on q (i.e., we find only a small decreasing trend, especially for smaller sphere area fractions) but strongly affected by the sphere area fraction ϕ. These findings are in line with previous studies on quasi-two-dimensional 20 and fully three-dimensional 32 dumbbells. Next, we can analyze the rotational long-time diffusion coefficient D r eff . To calculate D r eff , we assume that the autocorrelation function of the orientation vector follows the relation in the long-time limit. By using an exponential fit for 1τD < t < 550τD (after equilibration), we can extract D r eff . See also Sec. SII of the supplementary material.
We depict the rotational long-time diffusion coefficients of trimers for different sphere area fractions and aspect ratios divided by the corresponding microscopic diffusion coefficient of rotation D r in Fig. 5(b). We find that D r eff strongly depends on the sphere area fraction and the aspect ratio.
By utilizing linear and quadratic fit functions, we can again find empirical formulas for the long-time diffusion coefficients corresponding to trimers in monolayers surrounded by colloidal spheres. We find where at(q) = 0.600 q + 0.545, (23) bt(q) = −0.311 q − 1.596, (24) ct(q) = −0.006 q + 1.003 (25) The Journal of Chemical Physics ARTICLE pubs.aip.org/aip/jcp hold for the translational long-time diffusion coefficients, and ar(q) = 0.844 q 2 − 3.863 q + 3.399, (26) br(q) = −0.256 q 2 + 0.633 q − 0.368, (27) cr(q) = 0.002 q 2 − 0.009 q + 1.007 (28) determine the relation for the rotational long-time diffusion coefficient. The resulting formulas are depicted by the lines in Figs. 5(a) and 5(b). These relations can, for example, be used in future studies to roughly approximate the long-time diffusion coefficients of trimers in dependence on the aspect ratio and the sphere area fraction in the range of 0% ≤ ϕ ≤ 50%. For more information on the performed fits, see Sec. SIII of the supplementary material.

D. Comparison between trimers and spherocylinders
The spherocylinder shape is a particle geometry that is quite similar to the trimer shape in many respects. Hence, the question comes up whether it is possible to corroborate experiments based on trimers with simple spherocylinder simulations. On the other hand, it is interesting if complex rod-based systems can accurately be modeled via three-segmented Shish-Kebab models. Here, we compare the diffusive behavior of these two geometries to answer these questions. For this, we use the empirical formulas for spherocylinders that we calculated previously in Ref. 20.
As stated in Ref. 20, the microscopic diffusion coefficients of short spherocylinders can be computed via the following relations: Some resulting values for different aspect ratios are depicted by the points in Fig. 4. Strikingly, we find that the microscopic diffusion coefficients of the two shapes are nearly identical. The maximum relative discrepancies for D ∥ , D , and D r are only 0.8%, 1.6%, and 0.7%, respectively. The coincidence of the results can be interpreted in two different ways: Either our strategy cannot resolve the differences between the diffusion behavior of these shapes or the discrepancies between diffusion coefficients of short uniaxial particles can, indeed, be neglected. The latter option is favored by the fact that trimers and spherocylinders are spheres for q → 1.00. This means that the diffusion coefficients for small aspect rations near q = 1.00 are identical. In addition, for larger aspect ratios of 1.00 ≤ q < 2.00, the shapes are really similar as the curvatures of the surface created by the trimer segment overlaps are small. Hence, here, the diffusion coefficients should be close as well. However, for larger q → 3.00, more noticeable differences are expected as the particle geometries are distinctly different.
Note also that we could reproduce established results (for q = 3.00) with the empirical formulas for trimers. In addition, the relations for the dumbbells and spherocylinders that we calculated with the same strategy in Ref. 20 were successful in reproducing predictions and experimental measurements as well. This is in favor of the accuracy of the applied strategy.
The linear trimer shape can, in theory, be considered a friction bead model for spherocylinders in itself. Using simple strings of spherical beads to approximate cylindrical shapes is an often-used method to perform hydrodynamic calculations. See Ref. 48 where this approach is applied to model worm-like macromolecules with continuous contours and circular cross sections. In such calculations, it is often the case that additional "corrections" or adjustments are needed to obtain the best results. For example, in the mentioned work, the diameter of the beads modeling the worm-like chains is scaled to maintain the original volume of the macromolecule. By comparing this with our system of linear trimers and spherocylinders, we find that the influence of the difference in the volume of the two particle shapes (and, in particular, the corresponding impact on the diffusion coefficients) seems to be negligible in the whole analyzed aspect ratio range. This is quite intuitive for small q, where the volume difference vanishes.
By utilizing the same methods as before, we also computed the long-time diffusion coefficients for a Kihara spherocylinder in a monolayer surrounded by colloidal spheres. The results are depicted by the squares in Figs. 5(a) and 5(b). Interestingly, the results are also approximately identical to the ones obtained for trimers. Note that we found a distinct discrepancy in the rotational long-time diffusion coefficient while comparing pairwise interacting dumbbells and spherocylinders in Ref. 20. Strikingly, this means that (at least) three segments are necessary to approximate Kihara spherocylinders via a Shish-Kebab model regarding their dynamics, and simple dimes are insufficient.
Overall, we found impressive similarities between the diffusive behavior of spherocylinders and trimers. Therefore, we conclude that for both dilute three-dimensional systems and dense binary monolayers, pairwise interacting trimers should be sufficient to approximate short spherocylinders, and vice versa. This is quite interesting because we found deviations in the rotational long-time diffusion coefficients of spherocylinders and dumbbells, which seem to vanish by introducing the third segment in the middle of the particle.

E. Shaped-overarching approximations for the microscopic diffusion coefficients of short elongated particles
Due to the strong similarities in the long-time diffusion coefficients of trimers and spherocylinders (see Fig. 5), our empirical formulas [Eqs. (21) and (22)] for the long-time diffusion coefficients are valid for both particle shapes. As stated before, we also found that their microscopic diffusion coefficients are nearly identical (see Fig. 4). To take advantage of the similar results for the microscopic diffusion coefficients of spherocylinders and trimer particles, we propose simplified (and shape-independent) relations for approximating the microscopic diffusion coefficients of arbitrary rod-like particles with axis symmetry that are spheres for q = 1.00. Sometimes, one is not interested in the exact dynamics of a system. Hence, if the diffusion coefficients of a sample system can deviate by ∼10% while modeling it, it is unnecessary to precisely match the formulas for microscopic diffusion coefficients to the specific particle shapes. Here, the following equations can be utilized to roughly estimate D ∥ , D , and D r .
By applying the Taylor series to the logarithm and the endeffect corrections of the relations (11)- (13), and rounding the results, we derive for arbitrary short rod-like particles with an aspect ratio 1.00 ≤ q < 2.00 that are spheres for q = 1.00. The first terms of these equations, Ds|L = kBT/(3πηL) and D r s |L = 3Ds|L/L 2 , correspond to the diffusion coefficients that are expected for perfect spheres with a diameter equals to the rod length L. The second and third terms adjust for finite elongations. Therefore, these last two terms are first-order corrections that can be applied to the diffusion coefficients Ds|L and D r s |L of an enlarged sphere with diameter L in the case of short rod-like particles.
Compared to the diffusion coefficients of trimers (spherocylinders) with aspect ratios in the interval 1.00 ≤ q < 2.00, our simplified equations give the maximum relative deviations of 5.0% (5.3%), 1.8% (2.3%), and 4.6% (3.6%) for D ∥ , D , and D r , respectively. They can even be used to approximate dumbbells 20 and prolate spheroids: 18,19 By using the corresponding formulas for the microscopic diffusion coefficients of both (in their form as written in Ref. 20), we find the maximum relative discrepancies of 3.5%, 0.9%, and 2.9% for dumbbells, and 4.6%, 5.8%, and 17.3% for spheroids (in the aspect ratio interval 1.00 ≤ q < 2.00). Thus, the simplified equations can be used for various rod-like particles (that are spheres for q = 1.00) and accompany the usual approximation D ∥ = 2D used for the other extreme-really long rod-like particles (see Refs. 13,26,and 27). A comparison between the simplified equations (dashed FIG. 6. Comparison between the total three-dimensional and the rotational diffusion coefficients according to the empirical formulas for a linear trimer (solid lines) and the simplified equations (dashed lines) for arbitrary short rods that are spheres for an aspect ratio q = 1.00. For 1.00 ≤ q < 2.00, the simplified equations excellently approximate the empirical formulas. They are even viable for larger q. lines) and the empirical formulas for trimers (solid lines) is depicted in Fig. 6 (we compare the total three-dimensional and the rotational diffusion coefficients D 3D and D r ). Note that the simplified equations can also be viable for larger aspect ratios of q ≥ 2.00 depending on the setup-dependent, acceptable deviations.

IV. CONCLUSION
With the application of bead-shell calculations, empirical fits, and Brownian dynamics simulations, we were able to achieve a thorough investigation of the diffusion behavior of linear trimer particles. In detail, we computed continuous equations for the microscopic diffusion coefficients in three dimensions, and for the longtime diffusion coefficients in colloidal monolayers. These relations can be applied to various future studies reaching from theoretical calculations, numerical studies to experiments.
To obtain the empirical formulas for the translational microscopic diffusion coefficients parallel and perpendicular to the elongated axes of the particles, as well as the rotational microscopic diffusion coefficient, we performed bead-shell calculations for different aspect ratios. The results were then utilized to perform fits (where the corresponding end-effect corrections are approximated by quadratic functions of the inverse aspect ratio) to extract continuous relations. We found that these equations reproduce predicted results for an aspect ratio of q = 3.00 and differ distinctly from the well-established results for cylinder-shaped rods.
We first suggested this straightforward approach of combining the well-established bead-shell method with fit functions inspired by relations for cylinder-shaped particles 11,12,17 in our previous work, 20 to which the presented study is a direct continuation. With the successful application of this methodology to the linear trimer shape, we were able to prove the excellent robustness of this approach. In particular, we would like to highlight that the relations (1)-(3) in combination with a series expansion of the end-effect corrections regarding the inverse aspect ratio approximates the bead-shell results remarkably well. This empirically indicates that this specific form of equations (that is established for cylinder-shaped particles) might be a universal property of arbitrary elongated particles.
Finding analytic formulas for the diffusion coefficients of different particles shapes is generally a highly relevant topic as these quantities are often crucial to describe a system's dynamics. Our work adds a set of equations for linear trimer particles to the literature and, thus, improves the general knowledge of the diffusion of complex-shaped particles. Having continuous relations for the microscopic diffusion coefficients has a large number of advantages. For instance, through the empirical formulas, we now precisely know the dependence of the trimer diffusion on the particle dimensions. This can come in handy for analytical calculations, where the empirical formulas can simply be inserted into the computations to obtain quantities depending on the diffusion coefficients. Furthermore, in an experimental setup, one usually does not have a perfectly monodisperse size distribution. In fact, one might find a finite polydispersity that influences the results of the studies. If one tries to model such systems with simulations, one can simply define the particle lengths randomly according to the size distribution of the experiment and calculate the corresponding diffusion coefficients with our continuous relations. A similar model would be quite If one considers a program that should apply to trimer particles of arbitrary aspect ratio, it is highly beneficial to have continuous equations of the microscopic diffusion coefficients at disposal. Finally, we would like to point out that the computed empirical formulas are also suitable for systems with non-static aspect ratios. If the aspect ratio varies with time, one can simply insert the corresponding timedependent particle dimensions in the empirical formulas to extract the time evolution of the diffusion coefficients. This might be important in biological systems, where one might find micro-organisms that change their shape depending on external stimuli or to propel themselves forward. By using the computed empirical formulas, we performed Brownian dynamics simulations of colloidal monolayers consisting of a single linear trimer surrounded by spheres as an application example. With this, we calculated empirical formulas for the longtime diffusion coefficients via linear and quadratic fits, and we found that the translational long-time diffusion coefficient strongly depends on the sphere area fraction, but it shows only a small downward trend regarding the aspect ratio. On the other hand, the rotational long-time diffusion coefficient strongly depends on both the sphere area fraction and the aspect ratio.
While relations for the long-time (or self-diffusion) coefficients of some particle shapes are known, 20,29,32 to our knowledge, such equations specifically tailored to trimer particles were missing. The computed approximations for the long-time diffusion coefficients can now be used to roughly estimate the dynamics of linear trimers at the interface between two fluids. Possible setups could be done similarly to Langmuir-Blodgett studies performed in Ref. 30. One should also mention the recent work in Ref. 49, where (linear) trimer particles that can be used in monolayer experiments were manufactured. Our calculated empirical formulas could be applied to approximate the dynamics in corresponding experimental studies.
In a direct comparison of the diffusive behavior of linear trimers and spherocylinders (where we used the microscopic diffusion coefficients for spherocylinders that we calculated previously 20 ), we found that the microscopic and long-time diffusion coefficients are almost identical. This shows that Shish-Kebab models with three segments can be used to model the dynamics of Kihara spherocylinders, and vice versa.
The similarity between the dynamics of linear trimers and spherocylinders is an important fact and in contrast to previous results where dumbbell particles and spherocylinders were found to have different rotational long-time diffusion coefficients. 20 The results presented in our manuscript indicate that both particle shapes (i.e., linear trimers and spherocylinders) might be used interchangeably in complex problems of statistical and computational physics. For example, in experiments, one can use colloids with trimer or spherocylinder geometry to study properties that are universal for both shapes. In the same way, one can apply the most optimal implementation for a specific problem in simulations to approximate the universal dynamics of both particle types.
Based on the similarities between spherocylinders and trimers, we introduced simplified versions of the formulas for the microscopic diffusion coefficients at small aspect ratios, which can be applied to particle geometries such as dumbbells, linear trimers, prolate spheroids, and spherocylinders.
The empirical formulas for the microscopic diffusion coefficients open up the possibility for various (Brownian dynamics) simulations containing linear trimer particles. With this, the phase behavior of linear trimer particles could be analyzed in future studies. Furthermore, experimental measurements of the diffusion coefficients of linear trimers would be interesting to confirm the empirical formulas and prove the similarities in the diffusive behavior of short rod-like particles.

SUPPLEMENTARY MATERIAL
See supplementary material for additional information (such as a goodness of fit analysis) regarding the performed fits.