Journal of Physics: Condensed Matter
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Self-diffusion of nonspherical particles fundamentally conflicts with
effective sphere models
To cite this article: Felix Roosen-Runge et al 2021 J. Phys.: Condens. Matter 33 154002
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J. Phys.: Condens. Matter 33 (2021) 154002 (8pp) https://doi.org/10.1088/1361-648X/abdff9
Self-diffusion of nonspherical particles
fundamentally conflicts with effective
sphere models
Felix Roosen-Runge
1,2,∗, Peter Schurtenberger
1and Anna Stradner
11 Division of Physical Chemistry, Lund University, Naturvetarv¨agen 14, 22100 Lund, Sweden
2 Department of Biomedical Sciences and Biofilms-Research Center for Biointerfaces (BRCB), Faculty of Health and Society, Malmö University, Sweden
E-mail:felix.roosen-runge@mau.se
Received 30 October 2020, revised 9 January 2021 Accepted for publication 26 January 2021 Published 18 February 2021
Abstract
Modeling diffusion of nonspherical particles presents an unsolved and considerable challenge, despite its importance for the understanding of crowding effects in biology, food technology and formulation science. A common approach in experiment and simulation is to map nonspherical objects on effective spheres to subsequently use the established predictions for spheres to approximate phenomena for nonspherical particles. Using numerical evaluation of the hydrodynamic mobility tensor, we show that this so-called effective sphere model fundamentally fails to represent the self-diffusion in solutions of ellipsoids as well as rod-like assemblies of spherical beads. The effective sphere model drastically overestimates the slowing down of self-diffusion down to volume fractions below 0.01. Furthermore, even the linear term relevant at lower volume fraction is inaccurate, linked to a fundamental
misconception of effective sphere models. To overcome the severe problems related with the use of effective sphere models, we suggest a protocol to predict the short-time self-diffusion of rod-like systems, based on simulations with hydrodynamic interactions that become feasible even for more complex molecules as the essential observable shows a negligible system-size effect.
Keywords: nonspherical particles, diffusion, hydrodynamic interactions, crowding effects, effective sphere models, particle mobility
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1. Introduction
While theoretical accounts often use model systems based on spherical particle shape, suspensions occurring in nature as well as bio- and nanotechnology usually contain more com-plex particle shapes. This fundamental difference undoubtedly challenges the accuracy of theoretical predictions. Extensions from spherical systems to nonspherical systems are far from ∗Author to whom any correspondence should be addressed.
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trivial, and usually imply that an elegant and tractable theoret-ical approach becomes complicated and numertheoret-ically costly. In particular the prediction of dynamics of nonspherical particles is challenging, as diffusion is not only governed by direct inter-actions between particles, but also hydrodynamic interinter-actions affecting the mobility of particles. Consequentially, theoreti-cal accounts of diffusional properties are thus often based on rough estimations or limiting cases, such as the well-known Edwards–Doi theory for rotational diffusion of long rods [1, 2], the well-known results for the hydrodynamic radius of individual ellipsoids [3] or one individual long rod [2,4].
In this context of simplified models of complex particles, many simulational and experimental accounts are based on
J. Phys.: Condens. Matter 33 (2021) 154002 F Roosen-Runge et al effective sphere models. The conventional effective sphere
model for self-diffusion is based on the condition of equal diffusion coefficients D0 in the dilute limit, i.e. the effective sphere radius equals the hydrodynamic radius Rh of the non-spherical particle [5]. Various approaches of data analysis for experimental techniques make use of this concept such as by accessing the hydrodynamic radius as an isotropic measure in dynamic light scattering or nanoparticle tracking analysis. When extending to real solutions with finite volume fraction
φ, the self-diffusion in a solution of nonspherical particles is
usually modeled as
D(eff)(φ) = D(HS)(φeff) with φeff= 4π 3 R3 h Vp φ (1)
with the effective volume fraction φeff resulting from the hydrodynamic radius Rh, and the bare particle volume Vp. For hard spheres, a semiempirical equation for the short-time self-diffusion as a function of volume fraction φ reads [6]
D(HS)(φ) = D
0(1− 1.56φ)(1 − 0.27φ) (2)
which combines exact results for the linear term with an assumed diffusional arrest for random close packing at
φrcp= 0.64, and agrees reasonably well with experimental and simulation data [6,7]. Given the fact that the effective sphere model is constructed using a related physical quantity in the dilute limit, the model has been expected to perform well at low volume fractions, and fail beyond a specific finite volume fraction. To our knowledge, this intuitive expectation has, however, not been validated, probably due to the chal-lenging problem to obtain reliable and accurate self-diffusion coefficients for nonspherical particles.
This lack of theoretical validation is surprising given the strongly increased interest in nonspherical systems in the last 15 years due to its relevance for inter alia the assembly of com-plex structures [8], as well as for the understanding of macro-molecular crowding in biology [9], and its role for cellular reaction pathways [10]. Recent experimental colloidal exam-ples studied the self-diffusion of ellipsoids in channels [11], in quasi-2D confinement [12], through obstacle arrays [13], and random energy landscapes [14], as well as the anisotropic collective diffusion of magnetic ellipsoids in external fields [15, 16], and the influence of particle shape on flow in microchannels [17, 18]. Besides ellipsoids, dynamics of rod-like particles has been studied in recent years, including fast self-diffusion of long rods through smectic phases of short rods [19], thermodiffusion of rods [20,21] and strongly hin-dered diffusion in long stiff ribbon-like peptide aggregates [22].
A plethora of studies during the last decade addressed the diffusion of nonspherical proteins in crowded conditions [23, 24], in particular also studying phenomenologically the role of tracer and crowder shape for diffusion [25,26]. As the most prominent example of nonspherical proteins, solutions of Y-shaped antibodies were intensively studied due to their med-ical and pharmaceutmed-ical importance, characterizing long-time and short-time self-diffusion [25,27–29], as well as collective
diffusion and viscosity [30–34]. For globular proteins of differing compact shapes, various studies on short-time diffusion in the last decade addressed e.g. lysozyme [35–37], β-lactoglobulin [38], crystallin proteins [37,39–41], hemoglobin in red blood cells [42, 43] and bovine serum albumin [37,44].
The short-time diffusion is of key relevance for a com-prehensive understanding of multi-scale dynamics for several reasons. First, while long-time diffusion is governed by a com-plex interplay of hydrodynamic and direct potential interac-tion, the short-time limit of diffusion is defined by only the instantaneous mobility tensor, i.e. before potential interaction effectively affects the dynamics. Understanding this short-time limit is thus of central importance to disentangle effects of potential interactions such as transient molecular docking from slowing down due to hydrodynamic interactions [45]. As short-time diffusion represents the initial stage for a broad range of processes ranging from transport over escape from transient structures to dynamical arrest, the short-time limit acts as a general prefactor of long-time dynamics, and thus is essential for a quantitative understanding.
Second, a reliable description of the slowing down due to hydrodynamics at short time scales is also particularly important for Brownian and Langevin dynamics simulations of nonspherical particles, as these coarse-grained approaches rely on correct particle mobilities, i.e. correct short-time self-diffusivities, which technically set the link between time and length scales. This aspect is of special importance when con-sidering polydisperse systems, where the mobility of different shapes might be reduced by different factors, in a similar way as the inhomogeneous slowing down of spheres of different size in polydisperse solutions [29].
Third, several experimental techniques access the short-time regime, such as dynamic light scattering for large colloids or quasi-elastic neutron scattering for proteins [23]. A reliable knowledge on the slowing down depending on shape and vol-ume fractions thus opens opportunities to study conformation and also clustering and interactions in crowded solutions.
In this study, we attempt a critical test whether effec-tive sphere models accurately reproduce the short-time self-diffusion of nonspherical systems. First, using hydro-dynamic calculations, we numerically determine short-time self-diffusion for rod-like rigid linear assemblies of n spher-ical beads, which we refer to as n-rods. In this context, rod-like rigid linear assemblies of n spherical beads present an ideal system, since these n-rods combine the methodological advantage of individual spherical objects for the calculation of mobility matrices with the nonspherical shape of the overall particle. Second, we compare the results to predictions using the effective sphere model, and explain the observations by effective radii linked to different particle properties. Third, we extend this approach by analytical results for ellipsoidal particles to obtain a more general view on nonspherical par-ticles. Finally, we suggest a protocol for reliable predictions of short-time self-diffusion for nonspherical particles at the full volume fraction range, based on computationally feasible hydrodynamic simulations.
2. Theory and computational methods
2.1. Self-mobility Miiwith first-order correction due to hydrodynamic interactions
Here, we revisit the method of reflections and Faxen’s theorem for nonspherical particles (for a more general discussion, see e.g. [46]). As we are interested in the first-order correction of self-diffusion, we disregard rotational diffusion for this theo-retical part. In particular, we focus on the first-order correc-tion for the mobility matrixM, which relates the velocities v of all particles to the forces F of all particles on the solvent: v =MF. The starting point is a single particle i in a quiescent solvent, which by definition yields
v(0)i = μiFi (3)
with the velocity v(0)i of particle i, the individual-particle mobility matrix μi, and the force F
(0)
i on particle i. The motion v(0)i of particle i at position ri affects particle j at position rj through an induced flow field, which in the far field is approximately
u(0)(rj) = T(r− ri)Fi (4) with the Oseen tensor T(r− ri). The force F(0)j caused on particle j by the flow field u(0)(r
j) is given by Faxen’s law F(0)j =Fju(0)(r) = ξju(0)(rj) +Dju(0)(r)|r=rj (5)
with the individual-particle friction matrix ξj= μ−1j . For spherical systems, Dj=16ξjR2∇2, whereas for nonspherical particles Dj is a sum of differential operators acting on the flow field around rjand depending on the particle shape and orientation [47].
Combining equations (4) and (5), the velocity of particle j is now given by v(1)j = μj Fj+FjTi jFi (6) with Ti j= Tji= T(rj− ri). The reflected flow field u(1)(r) is determined from stick-boundary conditions on the surface of particle j, yielding in the far field
u(1)(r) = Ti jξi v(1)j − u(0)(rj) (7) = T(r− rj) Fj+DjTi jFi . (8)
Finally, the force by u(1)(r) on particle i is given by Faxen’s theorem
F(1)i =Fiu(1)(r) = ξiu(1)(ri) +Diu(1)(r)|r=ri. (9) Thus, the total velocity of particle i reads
vi= v(0)i + μiF(1)i (10) = v(0)i + (I + μiDi) Ti j Fj+DjTi jFi (11) =μi+ Ti jDjTi j Fi+ (I + μiDi) Ti jFj+O(∂4), (12)
where we have dropped terms of higher order in the last line, since terms of these order would require also a higher-order expression for the induced flow fields. From the last line, the self-mobility matrix Miiof the ith particle with first-order cor-rection due to the presence of other particles is given by the square bracket in equation (12) as
Mii= μi+ Ti jDjTi j. (13) This matrix is the basic quantity from which the short-time self-diffusion coefficient can be calculated by the average of the eigenvalues.
2.2. Mobility matrices for n-rod solutions
To obtain the self-diffusion coefficient, we numerically cal-culated mobility matrices for a configurational ensemble of
n-rods obtained using Monte Carlo simulations.
The n-rods were formed as linear and rigid assemblies of
n touching spherical beads. The system configurations X were
sampled by NVT Monte Carlo simulations for numbers of hard spherical beads N = 96, 144, 192, 240, 288 in a cubic box with periodic boundary conditions, i.e. with particle numbers
M = N/n, respectively. We performed these simulations for n = 1, 2, 3, 4, 5, 6, and for 27 values for the volume fraction φ
between 0.000 01 and 0.4.
Using hydrolib [48], we calculated the full mobility
matri-ces M(X) = M∞(X) +R−1lubr(X) [49, 50] for an ensemble
of individual system configurations, accounting for periodic boundary conditions and rigid constraints within the n-rods. To avoid overlapping beads due to numerical inaccuracies, we rescaled all coordinates by a factor 1.000 01. In the calculation of the hydrodynamic interactions, short-range lubrication cor-rectionsR−1lubr(X) as well as long-range contributionM∞(X) up to multipole expansion order 3 are included [48]. We cal-culated the dilute diffusion coefficient D0independently from the mobility matrix μi using hydrolib for a single particle in infinite space.
The 6M× 6M mobility matrix M can be decomposed into
6× 6 submatrices Mi j expressing the hydrodynamic action
of the jth on the ith particle [46, 48]. The isotropic self-diffusion coefficient for the ith particle is given by the average of the eigenvalues of the 3× 3 translational submatrix of Mii [46]. We used the value and standard deviation of the aver-age over all particles and multiple system configurations as the self-diffusion coefficient DN for a given system size, and an estimation of the related error, respectively.
To correct for system-size effects, we obtained the transla-tional self-diffusion coefficient D from the established scaling formula DN(φ) = D(φ)− 1.7601D0(η0/η)(φ/N)1/3[7,51] via an error-weighted fit of D(φ) and the rescaled inverse viscosity
η0/η(φ) for each individual volume fraction φ.
3. Results and discussion
3.1. Short-time self-diffusion of n-rods
Figure 1 displays the short-time self-diffusion coefficients
J. Phys.: Condens. Matter 33 (2021) 154002 F Roosen-Runge et al
Figure 1. Normalized self-diffusion coefficients decrease with increasing aspect ratio of n-rods. Volume fraction-wise extrapolation (symbols) agrees well for aspect ratios below 4 with results using the diffusion ratio in equation (14) (see section4.1) (solid line). The effective sphere model (dashed line) clearly deviates from the simulations already at lower volume fractions.
self-diffusivity shows a systematic decrease with increasing aspect ratio n over the full volume fraction range. The self-diffusion of these nonspherical particles is thus slowed down significantly more than that of spherical systems at similar volume fraction. To our knowledge, this is the first system-atic data set simulating the short-time self-diffusion including hydrodynamic interactions for these rod-like systems.
For spheres (n = 1), our results are in very good agreement with previous results using accelerated Stokesian dynamics [7]. We remark that a qualitatively similar trend of a decreased short-time self-diffusion for ellipsoids compared to spheres at similar volume fraction was obtained for ellipsoids using multi-particle collision dynamics [52], supporting that the observed decrease at fixed volume fraction with aspect ratio is not specific to rods.
3.2. Evaluation of the effective sphere model
Strikingly, the predictions D(eff ) from the effective sphere model using equation (1) (dashed lines) significantly deviate from D(φ) at already low volume fractions. To obtain a clearer picture on the disagreement, we plot the diffusion ratio
α(n)(φ) = D (n) N (φ)/D (n) 0 − 1 D(HS)N (φ)/D(HS)0 − 1 (14) between rescaled normalized self-diffusion of nonspherical and spherical particles for the same system size (symbols in figure2). While not a physical quantity per se, this ratio allows to extract two very relevant pieces of information from D(φ): first, the low-φ limit α(n)(φ = 0) corresponds to the initial slope of D(φ), characterized by two-body hydrodynamic inter-actions. Second, the volume fraction at which α(n)(φ) starts to decrease indicates the volume fraction where crowding effects render higher-order hydrodynamic interactions increasingly influential.
Figure 2. Ratio α(n)(φ) from equation (14) for all aspect ratios (purple: n = 1; blue: n = 2; red: n = 3; green: n = 4) and system sizes (: N = 96; : N = 144; : N = 192; : N = 240;
◦: N = 288).
The diffusion ratio α(φ) can be easily evaluated for the effective sphere model (dashed lines in figure 2), with the dilute limit α(eff)0 = αeff. The comparison to simulation data reveals that the effective sphere model fails in two ways.
First, while α(eff )(φ) remains constant up to φ≈ 0.05,
α(sim)(φ) starts deviating from the dilute limit already at vol-ume fractions φ≈ 0.001. This difference indicates that the effective sphere model represents the self-diffusion coefficient with an incorrect functional form on the full volume fraction range. This incorrect functional form is most likely caused by the effects due to anisotropic shape on hydrodynamic interac-tions on length scales of few particle diameters. The significant deviations of the self-diffusion coefficients at finite concen-trations is of high relevance for a quantitative interpretation of self-diffusion e.g. in situations of macromolecular crowd-ing. In particular, our results imply that the reduction factor of particle mobility due to increased volume fraction depends on particle shape, and should thus not be applied uniformly for all particles, with implications for accurate setups for e.g. coarse-grained simulations.
Second, the dilute limit α(φ = 0) is not correctly repro-duced in the effective sphere model, which implies that even the initial slope of D(φ) is inaccurate when using predictions from the effective sphere model. Thus, already the leading order of hydrodynamic interactions at large interparticle dis-tances is not correctly taken into account, suggesting a fun-damental failure of effective sphere models. We remark that the deviation becomes increasingly important for aspects ratios beyond 4, while it might be in practice below the detection limit for short n-rods.
3.3. Differing effective radii of n-rods as tracer and crowder
To understand the fundamental origin of this misconception, we revisit the different roles a particle has in the context of diffusion and hydrodynamic interactions, by investigating the leading order of the self-diffusion coefficient in the method of reflections (see Theory section). The general idea of the 4
method of reflection is to first calculate the slowing down of the mobility of a given particle by the reflected hydrodynamic flow at another particle, and then average over all particles. Each particle thus acts in different fundamental roles, as a tracer and as a hydrodynamic as well as a steric crowder, and each of these roles can be used to motivate an effective radius of anisotropic particles.
(a) Volumetric radius: as a particle with finite volume V, a simple volumetric measure is given by
Rvol= 3 V 3 4π = 3 √ nR (15)
which can be considered as the first-order estimation of steric effects. R denotes the radius of the spherical bead. (b) Hydrodynamic radius: as a diffusing tracer, the particle
exerts random forces on the fluid, whose strength can be quantified by the hydrodynamic radius
Rh= kBT 6πηD0 = 1 6πη 3 f1−1+ f2−1+ f3−1 (16) with friction coefficients f1,2,3along the main inertial axes of the particle. The hydrodynamic radius of an arbitrary particle is the effective radius of a sphere with the same isotropic diffusion coefficient D0in the dilute limit. From the mobility matrix of a single n-rod in an infinite system, we thus directly obtain Rh.
(c) Reflective radius: as a crowder, the particle reflects the hydrodynamic flow exerted by a tracer back to the tracer. This hydrodynamic interaction between only two parti-cles causes a slowing down of the self-diffusion Dtwo of the tracer [53]: Dtwo= D0 1−5 4 RhR3refl r4 +O(r −6). (17)
Here, Rrefl of an arbitrary particle corresponds to the effective radius of a sphere with similar scalar reflection cross-section, averaged over all orientations. To determine
Rrefl, we use a system of two n-rods with varying inter-particle distance (respective center-of-mass) and varying relative and absolute orientations to calculate the mobil-ity matrix. Figure3 displays the obtained slowing down ΔD = (D0− D)/D0(symbols) for the sphere, along with fits of equation (17). The good agreement on a rescaled axis r/4
RR3
refl (figure3 inset) shows that the obtained
Rreflvalues indeed provide a good rescaling at large dis-tances with the expected r−4dependence. The small devi-ations at shorter distances growing with rod length n is expected due to near-field flow fields of n-rods distinctly different from spheres.
The results for the effective radii are shown in figure4, and discussed after the next section on prolate ellipsoids.
3.4. Differing effective radii of ellipsoids as tracer and crowder
As a second case of nonspherical particles, we summarize and derive analytical results for prolate ellipsoids with semiaxes (R, R, nR) and the aspect ratio n.
Figure 3. ΔD = (D0− D)/D0=54RhR3refl/r4for n-rods as a function of distance r and length n. The calculated values (symbols) can be fitted using equation (17) (lines) to obtain values for the reflective radius Rrefl. Inset: when rescaling r, all curves fall on top of each other at larger r with the expected r−4behavior.
Figure 4. Effective radii of rod-like aggregates of n spherical beads with radius R (symbols), and prolate ellipsoids with semiaxes (R, R, nR) (dashed lines): the hydrodynamic radius Rh, the reflective radius Rrefland the volumetric radius Rvolcharacterize effective spheres with similar self-diffusion coefficient, similar reflective behavior regarding hydrodynamic flows, and same volume, respectively. The different dependencies of the effective radii on aspect ratio n render effective sphere models intrinsically problematic.
(a) Volumetric radius: from the well-known volume of a prolate ellipsoid, we directly obtain
Rvol= 3 V 3 4π = 3 √ nR (18)
which is the identical result to the n-rod case.
(b) Hydrodynamic radius: the hydrodynamic radius can be obtained by Perrin’s seminal result [3] on the frictions fi along the major axes,
fi= 16πη
S + a2 iPi
(19) with the solvent viscosity η and the elliptical integrals
Pi= ∞ 0 ds (s + r2 i) (s + r2 1)(s + r22)(s + r23) . (20)
Using the isotropically averaged mobility, one obtains [54]
J. Phys.: Condens. Matter 33 (2021) 154002 F Roosen-Runge et al Rh= kBT 6πηD0 = 1 6πη 3 f1−1+ f2−1+ f3−1 = 2 3S (21) with the elliptical integral
S = ∞ 0 ds (s + r2 1)(s + r22)(s + r23) (22)
for the semiaxes (ra, rb, rc). The integral was evaluated numerically for half-axes ra= rb = R and rc= nR with quasi-continuously varying aspect ratio n.
(c) Reflective radius: our analytical derivation of the reflec-tive radius of a prolate ellipsoid starts from Faxen’s law as in equation (5). The differential operatorDjin Faxen’s law is in principle based on a surface integral, leading toward the well-known isotropic formula
D(sph) j = 1 6ξjR 2∇T∇ = πηR3∇T∇ (23) for spheres.
For nonspherical particles, Dj contains a potentially infinite series of differential operators with nonisotropic contributions. While no analytic expression is known for rigid assemblies of spherical beads, the case of ellip-soids can be used to explore the effects of nonsphericity. The first term of an infinite series of increasing orders of differential operators reads [47]
D(ell) j = 1 6R T ⎛ ⎝f01 0f2 00 0 0 f3 ⎞ ⎠ R∇TRT ⎛ ⎝r 2 1 0 0 0 r22 0 0 0 r32 ⎞ ⎠ R∇. (24) The friction coefficients f1, f2 and f3 are well-known functions depending on the semiaxes r1, r2, and r3 (see previous section) [3]. The rotation matrix R transforms from the lab frame to the particle frame along the semiaxes.
For large separations, the particle orientations are uncorrelated, and an isotropic average of equation (24) yields
Dj(ell)= πηR3refl∇T∇ + πηQ3refl∇ ⊗ ∇ + O(∂4) (25)
R(ell)refl = 1 5f r 2− 1 30f r 2 1/3 /(πη) (26) Q(ell)refl = − 1 10f r 2+ 1 10f r 2 1/3 /(πη) (27)
with the averages
f = ( f1+ f2+ f3)/3 (28) r2 = (r2 1+ r22+ r23)/3 (29) f r2 = ( f 1r21+ f2r22+ f3r23)/3. (30) We note that ∇ ⊗ ∇T(r) = 0, which cancels out the second term with Qrefl. The comparison between equation (23) and the remaining first term in equation (25) reveals the physical meaning of Rrefl: the effective reflec-tive radius Rrefl is a scalar quantity representing an effective sphere with equal reflection strength of
hydrod-ynamic flow fields as the orientation-averaged nonsphe-rical particle.
The results for the effective radii are shown in figure4, and discussed in the next section.
3.5. Comparison of effective radii of n-rods and ellipsoids
Figure 4 shows a comparison between the effective radii related to the role of tracer and crowder of both n-rods and pro-late ellipsoids. Here, the hydrodynamic radius Rh, the reflec-tive radius Rrefl and the volumetric radius Rvol are shown in units of basic radius R, and several conclusions can be drawn from the comparison.
First, for each nonspherical particle type, the different tive radii differ. This disagreement implies that a simple effec-tive sphere model with one radius cannot accurately describe diffusion in non-dilute conditions.
Second, for each nonspherical particle type, we obtain the order Rh> Rrefl> Rvol, which suggests that the friction in gen-eral shows a stronger dependence on nonspherical shape than the reflection of hydrodynamic flow fields. Furthermore, the fact that Rvol is smallest reflects the well-known result that nonspherical particles usually appear larger in effective sphere models. We remark that this result is more general than only for Rhand Rrefl, but also true for e.g. an effective hard sphere model of hard ellipsoids with similar second virial coefficient or similar intrinsic viscosity [55].
Third, from the comparison between the two nonspherical particle types, we find that the effective radii are specific to the respective particle shape. While the reflective radius Rreflis overall very similar for n-rods and ellipsoids with similar n, the hydrodynamic radius Rhshows a clear deviation. This finding stresses that an accurate effective representation of nonspher-ical particles cannot be made based on simple overall mea-sures such as aspect ratio, but has to include also more detailed information on the specific particle shape.
4. Methodological discussion
Given the failure of effective sphere models, we discuss in this section a protocol, how predictions of the self-diffusion pro-file as a function of volume fraction φ can be generated for the specific nonspherical particle. We stress that the outlined methodological approach is valid for short-time self-diffusion only, while long-time diffusion as well as collective diffusion require independent studies using more advanced simulation setups [56,57].
4.1. Diffusion ratio α as system-size independent quantity to predict self-diffusion
Importantly, the ratio α(φ) for n-rods obtained from the sim-ulations (figure 2) shows no observable system-size depen-dence. This observation is of high general methodological interest, since it allows for reliable predictions for the rescaled self-diffusion from simulations with smaller particle numbers, which is usually employed in hydrodynamic simulations to keep computational costs in a reasonable range. Using Pad´e 6
Table 1. Effective radii and parameters for the Pad´e approximant of α(φ). n α(n)0 a(n)1 a(n)2 b1(n) b(n)2 Rvol/R Rh/R Rrefl/R 1 1.000 0.00 0.00 0.00 0.00 1.000 1.000 1.000 2 1.388 41.60 −62.17 63.68 92.95 1.260 1.392 1.348 3 1.713 66.38 −102.86 90.16 150.11 1.442 1.728 1.621 4 2.028 111.20 −182.19 139.80 278.59 1.587 2.036 1.863 5 2.315 60.99 −112.50 78.54 150.35 1.710 2.324 2.122 6 2.572 −1.19 −2.43 6.48 −11.43 1.817 2.598 2.344 approximants α(n)(φ)≈ α(n)0 (1 + a(n)1 φ + a2(n)φ2)/(1 + b(n)1 φ + b(n)1 φ2) (31) (solid lines in figure2), we obtain an excellent parametrization for the ratio (see table1for parameter values).
Combining this parametrization with a known dependence for hard spheres D(HS)(φ) (see e.g. equation (2)), predic-tions for the self-diffusion of n-rods can be calculated for the full volume fraction range independent of system-size scaling relations. Given the systematic deviation of simula-tions for hard spheres from equation (2), we parameterize the short-time self-diffusion for hard spheres by the polynomial
D(HS)sim (φ)/D0= 1− 1.865φ − 0.347φ2+ 1.644φ3 (solid pur-ple line in figure 1), and obtain excellent agreement of the prediction for the n-rod diffusion based on the Pade approx-imant (solid lines) with the extrapolated data from system-size scaling (symbols).
This finding is highly relevant, as the system-size scaling relation has so far only been established for spheres. This inde-pendent method presents an important consistency check, and suggests that indeed the usual system-size scaling can be used. We remark that we observe a deviation for n 4 at high vol-ume fractions, while for n = 2 and 3 both methods agree. We speculate that this deviation indicates a possible problem with the standard extrapolation in dense systems, where the box size is smaller than 2 rod lengths. In this context, we remark that it is generally preferable to use larger box sizes to minimize any finite-size effects [58] and enhance sampling statistics, as long as the larger box is not prohibitive regarding computational costs.
4.2. Protocol for a robust prediction of short-time self-diffusivities
The simulation results can be used as semiempirical pre-dictions, by reformulating equation (14) and inserting equation (2), yielding
D(n)N (φ)/D(n)0 = 1− α (n)
(φ)(1.83− 0.42φ)φ. (32) Here, α(n)(φ) can be replaced by the Pad´e expression in equation (31), with the coefficients reported in table 1. It is important to notice that the estimation of the parameters has to be repeated for each specific particle shape.
In this context, the observation of a negligible system-size effect of α(n)is of great importance, as simulations with hydrodynamic interactions are exceedingly costly for systems with many particles. The proposed protocol for other particle geometries thus includes four basic steps:
(a) estimate the mobility tensor for nonspherical particles, and of hard spheres at the same system size, for the desired range of volume fractions. The system size can be cho-sen comparably small, e.g. with a cubic box size of twice the particle diameter. Potential numerical tools include hydrolib [48] as well as computational implementations of Stokesian dynamics [59–61]. Ideally as many orders of hydrodynamic interactions as possible should be taken into account.
(b) Calculate the system-size dependent diffusion coefficient
DN(φ) from the trace of the mobility submatrices, and determine the dilute limit D0either by extrapolation or by a separate calculation ideally of a single particle in infinite space.
(c) calculate the diffusion ratio α(φ) as in equation (14) indi-vidually for each particle geometry, and obtain a reason-able parametrization e.g. using Pade approximants as in equation (31).
(d) use the semiempirical expression in equation (32) to gen-erate the (system-size independent) prediction for the diffusion coefficient.
5. Conclusion
We have shown that analytical effective sphere models fail both fundamentally and practically to represent the short-time self-diffusion of non-spherical particles in solutions. Using numerical results for rod-like assemblies of spherical beads, and analytical results for ellipsoids, we explain the failure of effective sphere models by the mismatch of relevant effective radii representing volume, hydrodynamic friction and reflec-tion of hydrodynamic flows on nonspherical particles. We pro-pose a protocol to overcome this lack of predictive theory by using numerical calculations of the mobility matrix for solu-tions of nonspherical particles. Importantly, the proposed pro-tocol allows to use comparable small system sizes in sim-ulation, which are accessible in hydrodynamic simulations. The finding opens opportunities for future investigations to validate and exploit this predictive framework against experi-mental results on nonspherical systems which are increasingly becoming available in nanoscience, and have high relevance in nature.
Acknowledgments
We acknowledge helpful advice from, and inspiring discus-sions with Johan Bergenholtz (Göteborg, Sweden), Jan Dhont
J. Phys.: Condens. Matter 33 (2021) 154002 F Roosen-Runge et al (Jülich, Germany), Joakim Stenhammar (Lund, Sweden), Jin
Suk Myung (Daejeon, South Korea) and Niels Boon (Rot-terdam, Netherlands). We gratefully acknowledge financial support from the Swedish Research Council (VR Grant No. 2016-03301), and the Knut and Alice Wallenberg Foundation (project Grant No. KAW 2014.0052) and the Royal Physio-graphic Society in Lund.
Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.
ORCID iDs
Felix Roosen-Runge https://orcid.org/0000-0001-5106-4360
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