Using density functional theory, we study the preferential ordering of rod-like guest particles immersed in a smectic host fluid. Within a model of perfectly aligned rods and assuming that the guest particles do not perturb the smectic host fluid, simple excluded-volume arguments explain that guest particles that are comparable in length to the host particles order in phase with the smectic host density layering, whereas guest particles that are considerably shorter or longer order in antiphase. The corresponding free-energy minima are separated by energetic barriers on the order of the thermal energy *k*_{B}*T*, suggesting that guest particles undergo hopping-type diffusion between adjacent smectic layers. Upon introducing a slight orientational mismatch between the guest particles and the perfectly aligned smectic host, an additional, smaller free-energy barrier emerges for a range of intermediate guest-to-host length ratios, which splits the free-energy minimum into two. Guest particles in this range occupy positions intermediate between in-phase ordering and in-antiphase ordering. Finally, we use Kramers’ theory to identify slow, fast, and intermediate diffusive regimes for the guest particles as a function of their length. Our model is in qualitative agreement with experiment and simulation and provides an alternative, complementary explanation in terms of a free-energy landscape for the intermediate diffusive regime, which was previously hypothesized to result from temporary caging effects [M. Chiappini, E. Grelet, and M. Dijkstra, Phys. Rev. Lett. **124**, 087801 (2020)]. We argue that our simple model of aligned rods captures the salient features of incommensurate-length guest particles in a smectic host if a slight orientational mismatch is introduced.

## I. INTRODUCTION

There is a large interest in the transport of particles in crowded, structured dispersions, from both a biological and a technological perspective.^{1–5} In this context, liquid-crystalline suspensions of colloidal particles have attracted significant attention as a model system due to their rich phase behavior.^{4,6,7} In particular, their nematic, smectic, and columnar phases all possess broken rotational or translational symmetries. Considerable work has been done to characterize the highly heterogeneous diffusive behavior that this gives rise to, as we describe below.

For example, in the nematic phase, liquid-crystalline particles collectively align along a preferential axis, which naturally creates an easy axis for transport of particles.^{8–16} In the smectic phase, on the other hand, liquid-crystalline molecules exhibit, in addition to orientational order, a lamellar structure in which the aligned particles are arranged in layers. In the lamellar structure, intra-layer diffusion is found to occur more or less continuously, whereas inter-layer diffusion occurs in discrete “hops.”^{17–20} It has been shown that only the former is affected by the flexibility of the molecules.^{21–23} The latter, on the other hand, is strongly influenced by neighboring particles forming temporary cages around each other, analogous to what is thought to happen in glassy systems.^{24,25} The caging effect competes with the permanent barriers resulting from the smectic layering, which gives rise to non-Gaussian diffusive motion.^{26} To facilitate inter-layer diffusion, the particles form string-like clusters that collectively move along the nematic director: particles cooperatively move into the spaces vacated by other particles in different layers.^{27}

Finally, the columnar phase is likewise characterized by a loss of translational invariance, though column-like structures with hexagonal or hexatic symmetries take the place of lamellar layers. Like in the smectic phase, the diffusive motion in the columnar phase is heterogeneous and non-Gaussian. Transient cages affect both intra-column diffusion and hopping-type inter-column diffusion.^{28} Similar to the smectic phase, the latter occurs through the cooperative motion of string-like clusters.^{29,30}

Building on these important insights, Alvarez *et al.* have recently studied, in vitro, the transport properties of rod-like fd virus particles in the smectic phase;^{31,32} this is the system we shall be concerned with in this paper. Specifically, they used two mutants of the fd virus to create a smectic phase composed of short “host” particles, in which a trace amount of long “guest” particles is immersed. Counter-intuitively, they found that the long guest particles diffuse faster than the short host particles do.

In an attempt to explain the fast diffusion of long and incommensurate guest particles, Chiappini *et al.* performed molecular dynamics computer simulations on dense suspensions of slender rods with hard-core repulsive interactions.^{33} They considered a binary mixture in which a trace amount of incommensurate-length guest rods are immersed in a smectic phase of host rods. Depending on the guest-to-host length ratio, they identified slow, intermediate, and fast regimes for the diffusion of the guest rods.

They rationalized these regimes by computing the periodic effective potential or “molecular field” experienced by a guest rod making use of the average spatial distribution of guest rods. For the slow and fast diffusive regimes, this potential exhibits a single minimum per smectic layer, separated from the next by a free energy barrier. Conversely, for the intermediate diffusive regime, a double-barrier potential develops, which exhibits two minima per smectic layer that are of equal depth. In this regime, guest particles order neither in phase nor in antiphase with the smectic host fluid. Instead, the guest particles occupy intermediate positions, as illustrated in Fig. 1. Figure 1 shows for the various ordering scenarios both an indication of the guest particle density along the director axis and a schematic representation of how the guest particles (blue) arrange themselves in the smectic host fluid (purple).

The intermediate positioning of incommensurate-length guest particles in a layered environment has also recently been observed experimentally for short rods dispersed in a colloidal membrane of long rods.^{34} The long rods are twice as long as the short rods, which seem to adhere to the lamellar interface. In line with these experimental findings, Chiappini *et al.* stated that in the intermediate diffusive regime, their guest particles “anchor” to the lamellar interface.

We interpret their explanation for the “anchoring” behavior as a temporary caging effect, superimposed on the permanent barriers resulting from the lamellar structure of host rods. In this regime, guest rods that occupy intermediate positions create large voids in the adjacent layers into which they protrude. These voids can be filled via small orientational fluctuations of the host particles, which by doing so effectively form a cage around the guest particle. Conversely, guest particles that order in phase or in antiphase with the smectic host fluid create much smaller voids in the adjacent layers. To fill these voids, larger orientational fluctuations of the smectic host particles are required (see the supplementary material of Ref. 33 for a visualization of this). As a result, guest particles at intermediate positions are much more effectively caged than guest particles in phase or in antiphase with the smectic host fluid, resulting in their preferential ordering. The caging rationale is consistent with the Doi theory for rod diffusion, in which the reorientational motion of the rod is restricted to a tube-like region by its neighbors.^{35–37}

Here, we carry out a simple theoretical experiment to further investigate the preferential ordering of a guest particle immersed in a smectic phase of host particles. In Sec. II, we establish a model of perfectly aligned rods for the smectic host particles. We compute the density profile of the host particles by means of a bifurcation analysis.^{21,38–45} Then, in Sec. III, we calculate the molecular field experienced by a guest particle due to the presence of the smectic host particles as a function of position,^{46} neglecting the effect of the guest particle on the smectic host. In particular, we investigate the role of an *imposed* orientational deviation of the guest particle from the perfectly aligned host particles.

The analysis that follows in Sec. IV naturally identifies periodic equilibrium positions for the guest particle. Short guest particles tend to order in antiphase with the smectic host. This incidentally also holds for small spheres immersed in a smectic phase composed of rods^{47} and for platelets with a small diameter immersed in a smectic phase composed of platelets with a larger diameter.^{48} Conversely, guest particles comparable in length to the host particles tend to order in phase with the smectic host. We show that if the guest particles are perfectly parallel to the smectic host particles, there is a sharp transition between the two regimes. If the guest particles are at a slight angle with the smectic host particles, an intermediate regime emerges where guest particles anchor to the lamellar interface. We show that these regimes repeat for guest particles exceeding the host particles in length, with a periodicity of twice the smectic layer spacing, consistent with the work of Chiappini *et al.*^{33}

In our model, the intermediate regime arises because the mutually excluded volume between a guest particle and a host particle that are slightly misaligned gradually tapers off as they move further apart. This suggests that it is “cost-effective” to slightly penetrate a smectic layer near its interface than to fully penetrate it. For guest particles of intermediate length, this effect culminates in positioning intermediate between in phase and in antiphase with the smectic host fluid. We stress that this picture requires a slight orientational deviation between guest and host particles, similar to how the reorientation of host particles hinders the diffusion of guest particles in the explanation proposed by Chiappini *et al.*^{33} Our explanation, however, stems entirely from a free-energy landscape, rather than temporary caging effects.

Finally, in Sec. VI, we analyze the energy barriers separating adjacent equilibrium positions of guest particles. Using Kramers’ theory,^{49} we identify slow, fast, and intermediate diffusion regimes as a function of guest particle length, in line with what is found in experiment and simulation.^{31–34} We summarize our findings in Sec. VII.

## II. BIFURCATION ANALYSIS

We start by computing the density profile of the smectic host particles. We consider a collection of hard, rod-like particles with a diameter *d* and a length *L*. The particles also have hemispherical end-caps. Although an orientational-positional coupling is known to influence the nematic-to-smectic transition,^{21,50,51} we shall for simplicity presume the rod-like particles to be perfectly parallel to each other. This seems justified, as we are primarily interested in the form of the density profile, and the nematic phase becomes anyway highly aligned in the vicinity of the nematic-to-smectic transition.^{39,52–56} Various authors have also studied the effect of introducing an orientational distribution, which enables them to map out the phase diagram of the isotropic, nematic, and smectic phases using the same model system.^{41,43,57–60}

*qualitatively*correct density profile,

^{41–43,57,59–61}even though its predictions for what density that the nematic–smectic transition occurs and the periodicity of the smectic density wave are not

*quantitatively*correct. Corrections upon this approximation are possible, but for reasons of simplicity, we choose to ignore this. We return to this issue below. Within the second virial approximation, the free-energy functional is

*β*= 1/

*k*

_{B}

*T*being the reciprocal thermal energy, where

*k*

_{B}is the Boltzmann constant and

*T*is the temperature. In addition,

*ν*is an unimportant volume scale, and $\rho r$ is the number density of particles with their centers of mass located in an infinitesimal volume d

**r**centered at position

**r**. The first term represents the ideal part of the free energy, and the second term represents the excess part of the free energy. The Mayer function, $fMr,r\u2032$, is a proxy for the inter-particle interactions. It yields negative unity if two particles with their centers of mass located at

**r**and

**r**′, respectively, overlap and zero otherwise,

^{21,38–45}

_{j}being dimensionless amplitudes and

*q*being the wave number of smectic ordering. The corresponding dimensionless wave number is

*Q*=

*qL*. Next, we insert (3) into (1) and demand stationarity with respect to Δ

_{j}and

*Q*,

*ϵ*, where Δ

_{j}=

*ϵ*

^{j}

*a*

_{j}. Here,

*ϵ*is an indication of the distance from the bifurcation point, and

*a*

_{j}are expansion coefficients. We do the same for the dimensionless wave number and the packing fraction

*η*=

*ρπd*

^{2}

*L*/4, which gives

*ϵ*enables an order by order solution to stationarity equations (4), where $\eta 0,Q0=0.576,4.420$ denotes the bifurcation point; this is where the smectic phase splits off from the nematic phase. The remaining coefficients

*a*

_{j},

*η*

_{j}, and

*Q*

_{j}characterize the structure of the dispersion deeper into the smectic phase. On account of the somewhat unwieldy expressions that we find for these quantities, we list them in the supplementary material. Taken together, they predict a density profile as a function of the expansion parameter, according to

*ϵ*is a measure for the distance to the nematic-to-smectic transition. As

*ϵ*increases, we move deeper into the smectic phase. Since we find that

*η*

_{1}= 0, we can make this explicit by expressing

*ϵ*from Eqs. (6) and (5) in favor of

*η*. Our expansion reproduces Mulder’s results if we remove the hemispherical end-caps

^{39}and suggests a second-order nematic–smectic phase transition consistent with previous work on the Onsager model at the level of the second virial approximation.

^{58–60}Further modification of the theory, such as taking into account the third virial contribution

^{43,58}or expanding the free energy with the chemical potential or pressure as the thermodynamic control parameter,

^{62}is required to capture the density jump at the nematic-to-smectic transition point.

^{63}

Figure 3 shows the density profile for various values of the packing fraction *η*. The curves are in increasingly light green with the increasing packing fraction, from the transition point of *η* = 0.576 up to *η* = 0.720. Close to the nematic-to-smectic transition (dark green curve), the density profile is approximately harmonic and has a small amplitude. As the packing fraction increases, the amplitude increases and the profile becomes more sharply peaked. In addition, the smectic wavelength decreases slightly with the increasing packing fraction. The limitations of Eq. (6) become evident for *η* > 0.720, where the equation locally predicts negative values for the number density. Hence, we restrict ourselves to the regime *η*_{0} ≤ *η* ≤ 0.720.

Finally, a note on the value of the packing fraction at the nematic-to-smectic transition, *η*_{0} = 0.575, is in order. This packing fraction is considerably larger than that reported for computer simulations, namely, *η*_{0} = 0.36.^{64} A similar discrepancy applies to the wavelength of the density profile at the transition, which the theory (*λ* = 1.40*L*) overestimates in comparison with results of computer simulations (*λ* = 1.27*L*).^{64} Experimentally, the smectic wavelength is smaller still (*λ* ≈ 1.05*L*)^{65} albeit that this might also be due to the impact of a finite bending flexibility.^{21,60} The discrepancies are a known shortcoming of the second virial approximation for lyotropic liquid crystals^{40} and are often addressed by multiplying the excess free energy with a phenomenological correction term $1\u221234\eta /1\u2212\eta 2$.^{66–68} The correction moves the transition point to *η*_{0} = 0.338, although it leaves the smectic wavelength unaffected. A similar correction is possible by application of scaled particle theory.^{69,70} We stress, however, that adopting such a correction would also limit the range of validity of Eq. (6) to even lower packing fractions, without appreciably altering the form of the density profile. Finally, we note that although more accurate free energy functionals can, in principle, also be obtained through fundamental measure theory,^{71} this renders the analysis significantly more involved and it is not *a priori* obvious how the approach we present below ought to be formulated in that context. We therefore forego carrying out a (phenomenological) correction. Since we are interested in describing densely packed suspensions,^{31–33} we use *η* = 0.720 for the remainder of this work.

## III. MOLECULAR FIELD CALCULATION

**r**

_{g}. The Mayer function $fMr,r\u2032,\theta $ describes hard-core repulsion, with

*θ*being the angle the guest particle makes with the smectic director. As discussed above (see Fig. 2), the Mayer function essentially restricts the integral to the overlap volume of the guest particle with the host particles.

In particular, we highlight the explicit dependence of Eq. (8) on *θ*: in contrast to the host particles, for which we demand perfect alignment, we allow the guest particles to exhibit a slight orientational deviation. This allows us to investigate—by way of a simple theoretical experiment—the qualitative effect of misalignment on the ordering behavior of the guest particle.

We evaluate Eq. (8) for a guest particle with length *L*_{g} and diameter *d* that is oriented at an angle *θ* with respect to the smectic host particles. To this end, Fig. 4 schematically shows the volume that is excluded to the guest particle due to the presence of a single host particle. Here, the *z* axis is chosen along the director, and the *x* axis is chosen such that the particles lie in the *x*–*z* plane. Figure 4(a) illustrates that the host particle generally excludes the guest particle from a parallelogram-shaped region of the *x*–*z* plane (black dashed line). The width of the excluded region is greatest if the guest particle is at the center of a smectic layer and gradually decreases as the guest particle moves further away along the director. The anisotropic shape of the excluded region suggests that it is (in a sense) “cost-effective” to slightly penetrate a smectic layer at the interface rather than to penetrate all the way to the center of the layer. This effect results directly from the orientational mismatch between guest and host particles and vanishes in the limit *θ* = 0° [Fig. 4(b)].

*x*coordinate into the function $FMz\u2212zg,y,\theta $, which obeys

*L*

_{g}cos

*θ*≤

*L*, and

*L*

_{g}cos

*θ*≥

*L*+ 2

*σ*sin

*θ*. Here, $L\xb1\u2261L\xb1Lg\u2061cos\u2061\theta 2$, and the three-dimensional character of the excluded region is captured in terms of the parameter $\sigma =d2\u2212y2$. We remark that Eqs. (10) and (11) take on a form that is slightly more complicated still in the regime

*L*<

*L*

_{g}cos

*θ*<

*L*+ 2

*σ*sin

*θ*. However, since we restrict ourselves to thin rods and small angles (

*d*≪

*L*,

*θ*≪ 1), the range of guest particle lengths this regime applies to is exceedingly small. As it turns out, none of the results we report on in this paper fall within the aforementioned range, and so we do not cover it here.

## IV. MOLECULAR FIELD ANALYSIS

Figure 5 shows the result of numerically evaluating Eq. (9) for various guest particle lengths. Plotted are the molecular fields scaled to the thermal energy, Δ*βU*_{sm}, as a function of the distance along the director axis scaled to the smectic wavelength, *z*/*λ*. All molecular fields are translated such that their minima coincide with Δ*βU*_{sm} = 0, and we show results for an aspect ratio of $dL=140$ and an average packing fraction of *η* = 0.720. At lower packing fractions, the qualitative features are unchanged, although the magnitude of the molecular fields decreases. To facilitate a comparison with experiment and simulation, we scale the guest particle length to the smectic layer spacing of the host: *r* ≡ *L*_{g}/*λ*. This ensures that the discrepancy in the predicted smectic wavelength between theory, simulation, and experiment, as discussed above, does not obscure our results.^{33}

Below, we discuss the behavior of the molecular field as a function of the ratio of the lengths of the guest and host particles, *r*. For each value of *r*, three different curves are plotted in Fig. 5, indicating various values of the orientational deviation of the guest particle from the smectic director. The dotted curve serves as a reference for perfect alignment, *θ* = 0°. The solid curve corresponds to an orientational deviation of *θ* = 2.5°, which one can show is the mean orientational deviation in a nematic phase at the same packing fraction.^{52} The dashed curve illustrates the effect of further increasing the orientational deviation, *θ* = 5°. These values of *θ* cover orientations up to two standard deviations away from the mean of the host rod orientation distribution function and thus include almost all expected orientations.

For short guest particles (*r* = 0.25), the molecular field has maxima at the center of each smectic layer (*z*/*λ* = −1, 0, 1) and minima in between smectic layers (*z*/*λ* = ±0.5). The locations of the minima indicate equilibrium positions of the guest particles, suggesting that they order in antiphase with the smectic host particles. Dimensionally, they “fit” in between smectic layers. The maxima, on the other hand, represent energy barriers separating adjacent equilibrium positions. As the orientational mismatch between guest and host particles increases (dotted to solid to dashed curve), the height of the energy barriers also increases. This is because the excluded volume—and, by proxy, the free energy barrier—is proportional to sin *θ*, with *θ* being the angle between guest and host particles.^{46} The free energy barriers are highest at *θ* = 90°, in which case guest particles order in antiphase with the smectic host regardless of their length. This, of course, is to be expected.

For guest particles comparable in length to the smectic layer spacing (*r* = 0.91), the minima of the molecular field are located at the center of each smectic layer, and the maxima are located in between layers. Such guest particles order in phase with the smectic host, rather than in antiphase. This ordering ensures that guest particles only interact with the layer they are embedded in, rather than multiple adjacent layers. Here, increasing the orientational mismatch between guest and host particles slightly decreases the free energy barriers, as guest particles that are at an angle with respect to the smectic host particles do not extend as far into adjacent layers. Note that this holds only for comparatively small angles; for large angles, the trend described further above is recovered again.

Strikingly, guest particles of intermediate length (*r* = 0.47) experience a molecular field with maxima both at the center of each smectic layer and in between the smectic layers. The minima are located at intermediate positions between the centers of the smectic layers and the gaps separating adjacent layers. The relative height of the different free energy barriers varies with the angle *θ*, which, as we will show further below, determines the range of guest particle lengths for which double-barrier potentials are recovered.

It seems that we reproduce the same propensity for guest particles of intermediate length to “anchor” to the lamellar interface as was reported in experiments and computer simulations.^{33,34} In our model, this propensity follows directly from the anisotropic shape of the excluded volume, as illustrated in Fig. 4. By introducing a slight orientational mismatch between the guest particle and the host particles, the volume excluded to the guest particle gradually decreases as it moves further away from a host particle. This means that, in contrast to a perfectly aligned guest particle, a slightly misaligned guest particle is denied less free volume if it slightly penetrates a smectic layer than if it fully penetrates a smectic layer.

This effect is subtle and gets overruled in the case of short guest particles and guest particles comparable in length to the smectic layer spacing. In those cases, there are strong incentives to order in antiphase and in phase with the smectic host particles, respectively. For guest particles of intermediate length, however, the excluded volume is minimized by slightly penetrating into an adjacent layer: the guest particles occupy positions intermediate between the centers of the smectic layers and the gaps separating adjacent layers grant the guest particle the greatest free volume.

Our model suggests that the free-energy landscape drives guest particles of intermediate length to occupy positions intermediate between the centers of the smectic layers and the gaps separating adjacent layers. This is purely based on geometric excluded-volume considerations. A commonality of our explanation with that of Chiappini *et al.* is that both require an orientational mismatch between the guest and the host particles.

Their explanation, however, appeals to temporary caging effects, rather than a free-energy landscape. They hypothesize that guest particles of intermediate length are caged much more effectively if they occupy positions intermediate between the centers of the lamellar layers and the gaps in between the lamellar layers, than if they order in phase or in antiphase. This is because the protrusion of the guest particle into adjacent layers generates big voids in the former case, which are easily filled through slight reorientations of nearby host particles. In the latter case, the voids are much smaller and thus require larger tilt angles of the host particles to fill. Hence, it seems that both explanations require an orientational mismatch between the guest and the host particles.

We find similar behavior for guest particles that exceed the smectic layer spacing in length (*r* > 1) albeit with the trend reversed. Guest particles slightly longer than the smectic layer spacing (*r* = 1.26) order in phase with the smectic host, whereas guest particles comparable in length to twice the smectic layer spacing (*r* = 1.90) order in antiphase with the smectic host. Guest particles of an intermediate length (*r* = 1.45) experience a double-barrier potential. Due to the periodicity of our system, the molecular fields shown in Fig. 5 repeat for guest particle lengths larger than twice the smectic layer spacing, with a period of *r* = 2. This is consistent with the simulations of Chiappini *et al.*^{33}

To further characterize the molecular fields, Fig. 6 shows the equilibrium positions of the guest particles as a function of their length for various values of the angle *θ*. For *θ* = 0 (dotted line), the guest particles order either in phase (*z*_{*}/*λ* = 0) or in antiphase (*z*_{*}/*λ* = ±0.5) with the smectic host fluid. The former occurs for 0.27 < *r* < 1.27, and the latter for *r* < 0.27 and *r* > 1.27. There is a sharp transition between the two regimes. If an orientational deviation between guest and host particles is introduced (solid and dashed lines), a third regime emerges. This regime coincides with the emergence of a double-barrier potential and sees that the guest particles occupy intermediate equilibrium positions $0<z*/\lambda <0.5$ rather than ordering in phase or in antiphase with the smectic host fluid. The range of guest particle lengths to which this behavior applies is determined by the angle *θ*.

## V. COMPARISON TO SIMULATIONS

Next, we compare Figs. 5 and 6 with the molecular fields Chiappini *et al.* obtained from their computer simulations.^{33} Although the molecular fields we report on have smoother and more rounded edges than those extracted from computer simulations, the salient features we find are remarkably similar. Our simple model reproduces in phase ordering with the smectic host fluid for guest particles close in length to *r* = 2*n*, $n\u2208Z*$, and in antiphase ordering for guest particle closes in length to $r=n+1$, $n\u2208Z*$. In addition, we reproduce the “anchoring” to the lamellar interface of guest particles with an intermediate length, across a range of guest particle lengths similar to that found in simulations.^{33} We achieve this based on geometric excluded-volume arguments. Our explanation derives from a free-energy landscape, rather than a temporary caging effect.

Quantitatively, however, there is a difference in the magnitude of the molecular fields shown in Fig. 5 and those obtained from computer simulations.^{33} Our model predicts energy barriers of the order of 1–4*k*_{B}*T*. In contrast, computer simulations predict slightly larger energy barriers in the range of 2–6*k*_{B}*T*. Especially on the lower end of this spectrum, for relatively weak ordering potentials, the difference of 1*k*_{B}*T* can have a significant effect on the guest particle dynamics. For free energy barriers clearly in excess of *k*_{B}*T*, the guest particle is expected to display hopping-type diffusion, as observed in experiments and simulations.^{31–33} Conversely, if the energy barriers are (much) smaller than *k*_{B}*T*, no such hopping-type motion is to be expected. Note that even in this case, the diffusion may be non-Gaussian.^{24,26}

In order to understand this apparent *quantitative* disagreement between theory, simulations, and experiments, we remark that the height of the energy barriers generally scales with how pronounced the smectic host layers are. For sharply peaked layers, deep in the smectic phase, the host particles are highly concentrated at the center of each layer. Accordingly, certain regions of space are much less accessible to guest particles than others. For a host fluid that is only weakly smectic, the differences between “accessible” and “inaccessible” regions of space are much smaller. In light of this, we argue that the theory underestimates the energy barriers due to being limited in how deeply into the smectic phase it can probe. This is because the starting point of our analysis is an expansion around the nematic-to-smectic phase transition.

To test this hypothesis, we again compute the molecular field experienced by a guest particle due to the presence of a smectic phase of host particles, according to Eq. (9). However, instead of inserting Eq. (6) for the density profile, we extract this information directly from published computer simulations.^{33} For this, we fit to a Gaussian distribution the simulated molecular field experienced by a host particle and deduce the corresponding density profile of host particles according to $\rho z\u221dexp\u2212\Delta \beta Usm$. Inserting this density profile into Eq. (9) and carrying out the integration yield the molecular fields shown in Fig. 7.

From Fig. 7, it is clear that using a more sharply peaked density distribution as input for our calculations has an appreciable effect. First, the shapes of the molecular fields have sharper, more defined features as compared to Fig. 5, bringing them more closely in line with the simulation data. This also makes the “anchoring” effect showed in the center two panels more evident. In addition, the magnitude of the energy barriers is increased to the range 2–5*k*_{B}*T*, suggesting hopping-type diffusion. We expect the correspondence with simulations to further improve upon relaxing the assumption of perfect alignment for the smectic host particles. This restriction slightly underestimates the mutually excluded volume between guest and host particles and thus the molecular field. The above makes us confident that the approach we present here captures the essential physics of the system and that our qualitative results should also hold deeper into the smectic phase.

## VI. DIFFUSIVITY

Finally, we briefly discuss the dynamics of a guest particle immersed in a smectic host fluid. The motion of the guest particle can be modeled as a diffusive process in an external field, where the external field is given by the molecular fields shown in Fig. 5. We assume that, in this potential, diffusion occurs primarily in the form of hops between (adjacent) equilibrium positions of the guest particle. We justify the use of hopping-type diffusion by reiterating that the molecular fields we report capture the salient physics of Fig. 7 and Ref. 33. We thus expect that the qualitative features of Fig. 5 persist for more pronounced smectic phases, which are known to exhibit energy barriers clearly exceeding the thermal energy.

*D*

_{0}denotes the (self-)diffusivity of the particle along the director axis in a flat potential. In general, this self-diffusivity consists of a contribution along the long axis of the particle,

*D*

_{‖,0}, and a contribution perpendicular to it,

*D*

_{⊥,0}; in an ideal solution,

*D*

_{‖,0}= 2

*D*

_{⊥,0}.

^{72}Since we consider only small orientational deviations of the guest particle with the director axis, we neglect the latter and approximate

*D*

_{0}≈

*D*

_{‖,0}. We will interpret the self-diffusivity

*D*

_{0}as corresponding to guest particle diffusion in a nematic host fluid at the same packing fraction, where we assume that in the long-time limit, caging no longer plays a dominant role.

^{24,25}

Figure 8 shows the diffusivity computed from Eq. (12) based on the energy barriers shown in Fig. 5. We find that, regardless of the orientational mismatch between guest and host particles (different curves), the diffusivity exhibits sharp peaks around specific guest particle lengths. Upon decreasing the packing fraction, the qualitative behavior stays the same, but the valleys in Fig. 8 become more shallow. The behavior we find is in line with computer simulation results and rationalizes the experimental observation of long guest particles diffusing faster than short host particles.^{31,33}

For a guest particle that is perfectly aligned with the smectic host particles (*θ* = 0°, dotted curve), the diffusivity peaks at *D* ≈ *D*_{0}. This suggests that there exist values of the guest particle length for which the molecular field is (almost) completely flat. The guest particle no longer “senses” the smectic layering and diffuses as if in a nematic phase. If we for the moment neglect the hemispherical end-caps of the particles, a straightforward calculation shows that this is the case for *L* + *L*_{g} = *nλ*, $n\u2208Z+$, i.e., if the combined length of the guest and host particles equals the length of one, or multiple, smectic layers. In that case, the volume excluded to a guest particle due to the presence of a host particle, *πd*^{2}(*L* + *L*_{g}), is exactly commensurate with the periodicity of the smectic host fluid. In the plot, the corresponding peak locations are located at *r* ≈ 0.27 and *r* ≈ 1.27.

If the guest particle is at a slight orientational deviation from the smectic host particles (*θ* = 2.5°, solid line), the peaks in diffusivity both become lower and move to larger guest particle lengths. The former implies that there no longer exists a guest particle length for which the molecular field becomes flat. Instead, a double-barrier potential develops in which the one barrier grows, while the other shrinks. To understand the latter, we have to realize that the diffusivity peaks if the guest particle “senses” the smectic layering the least. This occurs if just as the guest particle loses contact with one smectic layer, it gains contact with another. Since the distance between smectic layers is fixed, a longer guest particle is required to cover the said distance if it is oriented at an angle with respect to the smectic director. The above trend continues if the orientational mismatch between guest and host particles increases further (*θ* = 5°, dashed line). Note that the qualitative features we report here remain unchanged if we use the molecular fields of Fig. 7 instead of those of Fig. 5.

## VII. CONCLUSION AND DISCUSSION

In summary, we perform an idealized theoretical experiment to further clarify the preferential ordering and dynamics of trace amounts of guest particles immersed in a smectic host fluid. Using a model of perfectly parallel rods for the smectic host particles, we calculate the periodic potential experienced by a guest particle on the basis of excluded volume arguments. We pay particular attention to the effect of an imposed orientational deviation of the guest particle with respect to the smectic director.

We find that if the guest particle is perfectly aligned with the host particles, it orders either in phase or in antiphase with the smectic host depending on its length. There is a sharp transition between these two cases. If the guest particle is at a slight angle with respect to the host particles, a third regime emerges. For a range of intermediate guest particle lengths, the periodic potential exhibits two distinct energy barriers, and the guest particle occupies positions intermediate between the centers of the smectic layers and the gaps in between layers. We explain this finding, which has been verified experimentally and in computer simulations,^{33,34} using geometric excluded-volume arguments. Our explanation derives from a free-energy landscape, whereas Chiappini *et al.* proposed that guest particles are caged most effectively by the host particles if they occupy these intermediate positions, i.e., if they “anchor” to the lamellar interface. We argue that, in reality, a combination of the two effects dictate the distribution of guest particles in the smectic host. Incidentally, we note that while “caging” is typically interpreted kinetically, it could also be interpreted thermodynamically, as the host particles engaging in the effective caging of a guest particle are able to make larger use of their orientational freedom.

Finally, to make connection with kinetics, we study the guest particle dynamics using Kramers’ theory. In accordance with computer simulations, we find that the diffusivity of the guest particle is sharply peaked around specific values of the guest particle length. These values can be predicted for perfectly aligned guest particles and rationalized why long guest particles diffuse faster than short host particles in experiments.^{31} Upon introducing a slight orientational mismatch between guest and host particles, the peaks in diffusivity shrink and move to larger guest particle lengths.

Our findings show that a simple, geometric model is able to reproduce and explain the salient features of experiment and simulation. Consequently, our work highlights the possible mechanisms driving the preferential ordering and dynamics of elongated particles in an ordered environment.

Obviously, that is not to say that there is no room to improve and expand upon our current work. For example, our current model incorporates collective effects of the host particles through the second virial approximation, albeit that they are integrated out. The coupling to the guest particles is one way; however, the guest particles do not affect the structure of the smectic host fluid. This means that we neglect collective effects involving both particle species, which is reasonable as long as the fluctuations of host particle density do not become comparable to the host particle density. The model could be extended to include such effects by treating both particle species simultaneously in, for example, the second virial framework.

In addition, our decision to focus on the effect of an imposed orientational mismatch between guest and host particles may seem somewhat artificial. In principle, it is possible to allow the guest particle to freely choose the magnitude of this orientational mismatch. This can be done by computing the molecular field across a range of values for *θ* and then performing a Boltzmann-weighted average. We have opted not to pursue this route, as our assumption of a perfectly aligned smectic host biases the Boltzmann average to disproportionately small angles. For the theory to be internally consistent, both the host and guest particles would need to be allowed orientational freedom. This would obviously make the theory very much more complicated. It would be logical then to also go beyond the second virial theory and, for instance, apply fundamental measure theory.^{71}

## SUPPLEMENTARY MATERIAL

The supplementary material lists the expansion coefficients following from the bifurcation analysis in Sec. II. Together with Eqs. (6) and (7), these describe the density profile of the smectic host fluid.

## ACKNOWLEDGMENTS

This research received funding from the Dutch Research Council (NWO) in the framework of the ENW PPP Fund for the top sectors and from the Ministry of Economic Affairs in the framework of the “PPS-Toeslagregeling.”

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Guido L. A. Kusters**: Conceptualization (equal); Formal analysis (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (equal). **Martijn Barella**: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting). **Paul van der Schoot**: Conceptualization (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.

## REFERENCES

*New Approaches to Problems in Liquid State Theory: Inhomogeneities and Phase Separation in Simple, Complex and Quantum Fluids*

*Theory of Simple Liquids: With Applications to Soft Matter*

*The Theory of Polymer Dynamics*