## Abstract

The reactive flux formalism (Chandler 1978 *J. Chem. Phys.* **68**, 2959–2970. (doi:10.1063/1.436049)) and the subsequent development of methods such as transition path sampling have laid the foundation for explicitly quantifying the rate process in terms of microscopic simulations. However, explicit methods to account for how the hydrodynamic correlations impact the transient reaction rate are missing in the colloidal literature. We show that the composite generalized Langevin equation (Yu *et al.* 2015 *Phys. Rev. E* **91**, 052303. (doi:10.1103/PhysRevE.91.052303)) makes a significant step towards solving the coupled processes of molecular reactions and hydrodynamic relaxation by examining how the wall-mediated hydrodynamic memory impacts the two-stage temporal relaxation of the reaction rate for a nanoparticle transition between two bound states in the bulk, near-wall and lubrication regimes.

## 1. Introduction

Determination of the reaction kinetics is fundamental to characterizing many physico-chemical processes occurring in soft matter such as bond formation/breaking between ligands and receptors mediating particle–cell or cell–cell adhesion, and in molecular detection and sensing applications where the intermolecular or interparticle interactions are subject to conformational changes. While the characterization of the reaction rate for such activated processes and relaxation is often reported and analysed at the phenomenological level based on the mass action kinetics over certain macroscopic time scales, multiple dynamical processes could occur at distinct characteristic (shorter) time/length scales, thus leading to challenges in the interpretation of the result. The objective of this work is to present a theoretical approach that predicts the transition dynamics of a Brownian nanoparticle with a self-consistent incorporation of relevant forces and temporal correlations.

For controlled synthesis and formulation of microcapsules [1] or polymeric nanoparticles [2–4] in a microfluidic device, various reactions are modulated in the capillary flow. For particle–surface or surface–surface interactions mediated by the action of receptors and ligands, the adhesion can compete with rotational mobility of particles, intervening tether dynamics [5–7], and size or shape-dependent margination effects [8,9]. In cell adhesion, cell membrane curvature undulations regulate protein mobility [10–13] and thereby influence fluctuating membrane-mediated transient adhesive kinetics. In biomedical applications such as vascular targeted drug delivery involving advanced functional materials, engineering the reaction rate constant of functionalized nanoparticle binding to the target tissue such as the endothelium is essential to the characterization of the targeting efficacy [14]. Moreover, such drug-loaded nanoparticles have to overcome physical and biological barriers to effectively penetrate deep into the tissue [15,16]. Ordered nanomaterials of macroscopic dimensions synthesized using templating free approaches such as convective deposition [17] can be further configured using functionalized assembly techniques. Such techniques include DNA hybridization [18], and may realize new classes of advanced functional materials (e.g. photonic crystals or metamaterials [19]). The examples above are fundamentally unified via how the hydrodynamic memory in the inertial regime (i.e. at the time scale comparable with the fluid viscous relaxation time) and thermal fluctuations unavoidably influence the ‘reaction trajectory’ of the transition from one state to another, thus influencing the average transition rate as well as the distribution of rates observed in single molecule or single particle tracking experiments in a geometry or confinement-dependent fashion. While it is anticipated that hydrodynamic fluctuations may play a role in the reaction dynamics, a framework to quantify the effects is still not available in the literature. In order to take into account hydrodynamic interactions in the predicted reaction rate, it is necessary to apply a rigorous and computationally tractable framework that generates transition pathways with temporal memory encoding the hydrodynamic and thermal effects, and further unambiguously characterizes the corresponding microscopic time correlation functions.

For a spherical particle of mass *m* in an incompressible fluid, the Langevin equation predicts its velocity relaxation to follow an exponential decay (overdamped limit) with a dissipative friction coefficient (*ζ*), *t*^{−3/2} scaling for a particle in an unbounded fluid [21]. When a reaction or adhesion occurs near a surface, not only the friction coefficient is augmented, but also the velocity temporal correlation becomes more complex. Several analyses have been reported for a particle close to a planar bounding wall, where the gap between the particle surface and the wall is greater than the particle size. For example, Gotoh & Kaneda [22] have theoretically investigated the temporal velocity correlation for a colloidal particle at a distance much greater than its radius from a no-slip planar surface and found that the algebraic decay switches from a *t*^{−3/2} scaling to a *t*^{−7/2} scaling when the particle moves perpendicular to the wall. In a computational study, Pagonabarraga *et al.* [23] have analysed the transient motion of a colloidal particle close to a planar wall for both slip and no-slip boundary conditions, and confirmed the findings of Gotoh and Kaneda. Extending these studies, Felderhof [24] has presented an analytical study of wall-mediated VACF of a Brownian particle at long times from the perspective of frequency-dependent admittance. Specifically, he found that for the perpendicular motion, a *t*^{−7/2} scaling over intermediate times is in fact followed by a long-time tail that exhibits a *t*^{−5/2} scaling with a negative amplitude caused by the wall reflection of the fluid vortex. Along the same line, Franosch & Jeney [25] included the effect of a confining potential such as an optical trap and investigated the dynamics of a spherical particle in the near-wall regime. In the situation where the gap between the particle surface and the wall is much smaller than the particle size, the classical result of Taylor [26,27] shows that the strong lubrication force enhances the friction coefficient by a factor of particle radius over gap thickness. Moreover, in a recent direct numerical simulation (DNS) study of Vitoshkin *et al.* the transient velocity relaxation is comprehensively investigated (including stochastic non-equilibrium and wall hydrodynamic interactions, see [28–32] for a discussion), and the long-time scaling exhibits the same power-law index as the near-wall result [33].

While these studies have provided important information regarding the hydrodynamic memory effect on the particle velocity relaxation over various time scales in different hydrodynamic regimes (bulk, near-wall, lubrication), it is only recently that, an attempt at incorporating these temporal correlations in a unified framework is made. Yu *et al.* proposed the composite generalized Langevin equations (GLEs), a non-Markovian stochastic equation of motion that resembles the hydrodynamic effect of the linearized Navier–Stokes equations, to describe the instantaneous particle trajectory subject to hydrodynamic, Brownian and binding force interactions [30]. The transition from the bulk to the near-wall and lubrication regimes related to a nanoparticle approaching a planar wall is taken into account by a memory function that is constructed in a composite fashion. In their initial analyses, it is demonstrated that when the nanoparticle is subject to the correct hydrodynamic memory function in the unbound state, the direct inclusion of a harmonic potential may allow one to recover the correct temporal correlations of the nanoparticle in its bound state. Moreover, in the presence of a strong binding force, it is shown that the fluid viscous relaxation can compete with the binding force relaxation, thus emphasizing the importance of the incorporation of the memory function. On this basis, in order to investigate the transition rate of the Brownian nanoparticle in the examples mentioned above, and unambiguously quantify the influences of couple hydrodynamic fluctuations on the reaction dynamics, we generalize the framework to consider the Brownian dynamics of a nanoparticle within a bistable potential that defines the free-energy landscape for two bound states and the transition between them.

According to the fluctuation–dissipation theorem or Onsager’s regression hypothesis [28], the macroscopic time-dependent disturbance in the probability for particle in a given state is equivalent to the time correlation of the microscopic fluctuations of the probability for particle in that state. Given a well-defined transition state in a potential energy surface, the time derivative of the correlation function of the microscopic fluctuations along the reaction coordinate at the transition state is by definition the reaction rate. This serves as the important foundation of this work when we characterize the reaction dynamics. Once the particle trajectories within the potential energy surface are obtained, we determine the reaction rate by analysing the time correlation function of the flux of those particle trajectories that initially start from one state, pass over the barrier and commit to the other state, namely, the reactive flux correlation function [29].

In the following sections, we present our methodology of theoretically investigating the reaction kinetics of a Brownian nanoparticle in the presence of wall-mediated hydrodynamic fluctuations (§2), delineate the results across various hydrodynamic regimes and binding strengths (§3) and conclude our findings (§4).

## 2. Theory and method

We generate stochastic trajectories of a Brownian nanoparticle subject to a bistable energy landscape defining the stabilities of and transition between two states. We employ the composite GLEs [30], which recapitulate hydrodynamic interactions and thermal fluctuations in three specific regimes (bulk, near wall and lubrication), as well as capture temporal correlations correctly in the presence of a binding spring force. Applying the technique, we explore the effect of wall-mediated hydrodynamic memory on the dynamics of rate processes of a nanoparticle close to a bounding wall. This problem represents a minimal yet non-trivial system for studying the coupling of hydrodynamic memory with rate processes discussed above. Here, we consider a neutrally buoyant Brownian particle of mass *m* and radius *a*=250 nm immersed in a water-like fluid with a viscosity *η*=10^{−3} kg ms^{−1} and density *ρ*=1 kg m^{−3} in one dimension. If the particle is at an average separation *h* from a planar wall with an instantaneous relative position *x*(*t*), and is moving perpendicularly to the wall with a velocity *U* subject to a locally harmonic potential with *k* being the local spring constant (figure 1*a*), the translational equation of motion is described by the composite GLE as:
*M*=(3*m*/2)(1−*a*^{3}/8*h*^{3})^{−1} is the particle effective mass and *U*(*x*,*t*)=d*x*/d*t*. On the right-hand side, the first term denotes the Stokes friction with *β*=(1−9*a*/8*h*)^{−1} being the *O*(*a*/*h*) enhanced resistivity to the particle motion due to the planar wall (consistent with the order in [24]), the second term represents the fluid momentum diffusion around the particle in the absence of the bounding wall surface with *τ*_{w}=*h*^{2}*ρ*/*η* being the time for fluid momentum to diffuse over *h*, the third term corrects the wall effect with *F* being the potential of mean force defined later in equation (2.2) and the last term denotes the random force expressed as *R*(*t*)=*R*_{w}(*t*)+e^{−t/τw}*R*_{c1}(*t*)+(1−e^{−t/τw})*R*_{c2}(*t*) with 〈*R*_{w}(*t*)*R*_{c1}(*t*′)〉=〈*R*_{w}(*t*)*R*_{c2}(*t*′)〉=〈*R*_{c1}(*t*)*R*_{c2}(*t*′)〉=0. Utilizing the fluctuation–dissipation theorem [31] and the superposition principle for linearized systems, we obtain: 〈*R*_{w}(*t*)*R*_{w}(*t*′)〉=12*πηaβk*_{B}*Tδ*(*t*−*t*′), *t*≪*τ*_{ν}=*a*^{2}*ρ*/*η*; the time for fluid momentum to diffuse over *a*) exponential correlation with the friction coefficient of 6*πηaβ* and an effective mass *M*, an intermediate-time (*τ*_{ν}<*t*≪*τ*_{w}) algebraic decay of *t*^{−3/2} scaling, and a long-time (*t*≫*τ*_{w}) negative tail of *t*^{−5/2} scaling. When the particle is far away from the wall, *M*=3*m*/2, *M*=3*m*/2 as if the particle drags its surrounding fluid in the bulk with a much enhanced instantaneous Stokes resistivity such that to leading order the first three terms on the right-hand side of equation (2.1) are replaced by −[6*πηa*^{2}/(*h*(*t*)−*a*)]*U*(*x*,*t*). Details of the position and VACFs in these limits are available in [30], which are consistent with the behaviour of the particle in the presence of a planar boundary as known from experiments [25,32], theories [22,24,25] and detailed numerical simulations of the fluid equations of motion (e.g. [33] and references within). Moreover, independent of hydrodynamic regimes, the mean-squared displacement of the particle during the course of our simulations does not exceed 2% of *a* [30], thus justifying the assumption of a nearly constant *h* in equation (2.1).

The shape of the potential energy employed is motivated by the simulated effective one-dimensional (perpendicular to the cell surface) potential of mean force for a ligand-functionalized nanoparticle multivalently bound to a receptor-expressing endothelial cell [14]. For clarity and simplicity, when the nanoparticle is at the potential minimum of the bound state *A*, *x*=0, i.e. the nanoparticle is at its average position. As seen in figure 1*b*, the potential energy landscapes are bistable (continuous and everywhere differentiable) defined by the one-dimensional reaction coordinate, *x*, in the direction perpendicular to the wall along the receptor–ligand binding pair with a transition state *x**. Parameter values explored correspond to physical systems and scenarios detailed previously [14,30,34]. For example, the value of *k*=1 N m^{−1} mimics the potential of mean force for the aforementioned ligand-functionalized nanoparticle multivalently bound to a receptor-expressing endothelial cell [14] (figure 1*a*), where the deeper and shallower (fixed as 1 *k*_{B}*T*) wells denote the stable (*A*) and metastable (*B*) bound states, respectively. Changing receptor expression on the cell can vary the energy landscapes [35], as figure 1*b* depicts. The five different shapes of the potential shown in figure 1*b* are defined by:
*F** being the transition state energy. The parameters for the five energy landscapes are summarized in table 1. Potential 1 (solid black) is the reference potential, potential 2 (solid blue) and 3 (solid red) have the same transition state landscape as potential 1 but lower energy barriers, and potential 4 (dashed blue) and 5 (dashed red) correspond to more diffuse transition state landscapes (with *k**=2*kF**/(*kx*^{*2}−2*F**)) and lower energy barriers than potential 1.

The integral-differential equation (equation (2.1)) is solved with finite-difference discretization in *x* and a random number generator for power-law correlated noise subject to initial conditions *x*(0)=*x*_{0} and *U*(0)=*U*_{0}; for details, see [30]. For the transition from the stable state to the metastable state (i.e. *x*(0)=0 and *h*−*a*)/*a*∼*O*(1), the non-stationary nature of the trajectory introduced due to the bridging functions e^{−t/τw} and 1−e^{−t/τw} requires selections of the initial conditions from various particle configurations across the energy landscape. Therefore, for each trajectory, we randomly choose the particle initial position and velocity from the (Maxwell-) Boltzmann distribution, *F*.

We realize 400−600 (10^{17} steps; Δ*t*≤10^{−10} s), 10000 (10^{14} steps; Δ*t*≤10^{−9} s) and 200 (10^{5} steps; Δ*t*≤10^{−10} s) stochastic trajectories for the bulk, near-wall and lubrication regimes, respectively. According to the fluctuation–dissipation theorem, given appropriate functions identifying the state of the particle, the time-dependent rate constant for a transition between two states is calculated through ensemble averaging the particle trajectories using the reactive flux formalism [28,29] with the characteristic state functions *h*_{A} and *h*_{B},
*A* at *t*=0 and passes over the barrier at *t* with a given speed, i.e. the average flux of all such trajectories *A* to *B*. Here, the dot denotes the time derivative. The correlation function for trajectories *A* and *B* and yields *k*=1 N m^{−1} and potential 1 are shown in figure 1*c*,*d*. Other selective reactive flux correlation functions for different force constants, potential landscapes and hydrodynamic regimes are presented in the electronic supplementary material. As seen in figure 1*c*, the time integral gradually increases to a plateau value, indicating that the reaction rate decreases to zero as the system attains equilibrium. According to the mass action kinetics for *τ*_{rxn} that denotes the characteristic time for reaction to occur and is defined by *k*_{−} and *k*_{+} denote the *d*, *k*_{−} over a relaxation time scale of *τ*_{mol} that indicates the time scale for barrier crossing. After this transient relaxation, the system loses memory of its initial condition, and a long-time exponential decay with the characteristic time scale *τ*_{rxn}≫*τ*_{mol} sets in. For molecular or atomic reactions, the dynamics near the barrier involves bond vibrations and *A*; at long times, *A* a while ago before barrier crossing. In the inset of figure 1*d*, an exponential distribution of the life time (denoted as the residence time) of *A* or *B* is recovered. This is consistent with the argument that when barrier crossing is a rare event with subsequent crossings being statistically independent, the distribution of the event should be Poissonian and the event waiting time should exhibit an exponential decay.

## 3. Results and discussion

The effects of the energy landscape and hydrodynamic memory on *τ*_{mol}, *k*_{−} and *τ*_{rxn} across various potential energies, force constants and hydrodynamic regimes. It is evident that the order of magnitude values of *a* and *k*_{−} and *τ*_{rxn} in figure 3*a*,*b* depend primarily on the force constant *k* in the energy well. However, the hydrodynamic memory has the effect of broadening the distributions of the values of these properties in the near-wall and bulk regimes. Later, we discuss this effect in terms of the broadening of the distribution of *τ*_{rxn}. In figure 2*a*, we observe that for a given *k* or *h*/*a*, *F*=*F**−*F*_{A}, independent of the transition state landscape. This behaviour is most notable in the lubrication and bulk regimes, which have relatively smaller statistical noise. In figure 2*b*, it is shown that the order of magnitude values of *τ*_{mol} for different cases are similar to the fluid viscous relaxation time, *τ*_{ν}, implying that the transient relaxation for barrier crossing couples with the fluid momentum diffusion in the vicinity of the particle surface. At *A*, and may be directly compared with the rate defined by the transition state theory (TST) [28]. The TST relates the rate constant to the energy barrier by *c*) with a stronger force constant and negligible hydrodynamic memory, a linear dependence *α*≈3.5 is observed and *τ*_{mol}. However, in figure 2*d*, for particle in the bulk the much longer hydrodynamic memory makes *k* were high and in the Langevin limit.

In figure 3*a*, however, we observe that for a given *k* or *h*/*a*, the plateau rate constant *k*_{−} is apparently impacted by both the energy barrier and the landscape of the transition state in the presence of hydrodynamic interactions. Specifically, given the same Δ*F*, sharp transition states (potentials 2 and 3) lead to higher reaction rates than those with broader energy barrier landscapes (potentials 4 and 5). At short times, the particle dynamics at the transition state is characterized by *τ*_{mol} in figure 2*b* and couples with the local fluid viscous relaxation at the time scale of *τ*_{ν}. Over long times, for the choices of the parameters explored here, the energy barrier for *k*_{B}*T*) is smaller than that for *b* *τ*^{−1}_{rxn}≈*k*_{+}. Moreover, for a fixed Δ*F*, we observe longer *τ*_{rxn} for broader landscapes of transition-free energy barriers (cf. potential 2 to 4 and potential 3 to 5; see table 1 for the force constant over the barrier), implying that the rattling time over the barrier also affects the overall transition rate between two states, as predicted by the reactive flux formalism [29]. The long-time hydrodynamic memory leads to substantial disturbances to the reaction relaxation. Consequently, although the probability distribution of *τ*_{rxn} is consistent with the residence time distribution characterizing a Poisson process for a particle in the lubrication regime (figure 3*c*), it exhibits a broad distribution in the bulk regime (figure 3*d*), where the hydrodynamic memory is much greater. These results indicate that transition rates measured in single molecule experiments will show distribution in rates across multiple realizations of the trajectory that are dictated by the underlying hydrodynamic interactions. Moreover, in figure 4, the comparison between the reaction relaxation time predicted from *τ*_{rxn,−}) and from *τ*_{rxn,+}) shows that for a particle in the lubrication regime with negligible hydrodynamic memory (solid symbols), *τ*_{rxn} closely obeys the mass action kinetics and *τ*_{rxn,+}≈*τ*_{rxn,−}. By contrast, notable deviations between the estimates for *τ*_{rxn} when derived from

Replotting figure 3*a* for potential 1 in figure 5*a* further provides an alternative perspective of how the hydrodynamic memory impacts the rate constant across different hydrodynamic regimes. The limiting scaling for particle dynamics are first noted: (1) inertia-controlled motion with a characteristic reaction-attempting frequency *t*_{c}=6*πηa*/*k* as the spring force −*kx*=−d*F*/d*x* balances the drag force ∼6*πηa*. We observe that changes in the hydrodynamic regime and/or force constant generally lead to variations in the rate constant that are bounded by the limiting scaling, indicating that the hydrodynamic disturbances always shift the order of magnitude value of the kinetic parameter from *k*^{1/2} scaling toward the *k* scaling. This result also suggests that as the particle moves relative to the boundary, the reaction kinetics is coupled to the local hydrodynamic memory as the particle transitions between different hydrodynamic regimes. Finally, in various technological applications involving adhesive nanoparticles, the size of the particle is a key design parameter as both reaction kinetics and hydrodynamic relaxation strongly depend on the particle size. In figure 5*b*, not surprisingly, based on the same argument as in figure 5*a*, we observe that the dependence of *k*_{−} on *a* is bounded by the scaling of the inertia-driven effect (as *τ*_{k}∝*a*^{3/2}) and the viscous-force-balanced effect (as *t*_{c}∝*a*).

## 4. Conclusion

In conclusion, using the composite GLEs with the reactive flux formalism, we have systematically investigated the transient reaction kinetics of a Brownian nanoparticle subject to a locally harmonic, bistable potential with different force constants across various hydrodynamic regimes. Although the mass action kinetics gives approximate estimates of the reaction rate parameters, hydrodynamic memory perturbs the simple dependence and yields variations in the actual values of the parameters. Even for Newtonian fluids, hydrodynamic interactions with memory functions naturally lead to a two-stage temporal decay of the reaction rate. This decay occurs at a time scale (approx. nanoseconds) much longer than the time scale (approx. picoseconds) of a molecular-relaxation-driven two-stage decay. We may view this hydrodynamic-driven reaction relaxation fundamentally different from the relaxation of molecular reactions. This hydrodynamic effect is important in many biological and physical processes (discussed in our introduction) not only due to solvent interactions, but also due to the presence of wall boundaries. It is also evident from our analyses that the ‘reaction memory’ time *τ*_{mol} is strongly coupled with the ‘fluid memory’ time *τ*_{ν}, a characteristic that is not realizable through the Langevin equation alone without the correct memory function. Our results suggest that this context-dependent hydrodynamic effect can change the ensemble-averaged rate constant by two orders of magnitude. We also showed that the dependence of the rate constant versus the binding force stiffness constant and versus the particle size are both bounded between the two limiting scaling behaviours: namely, that of an under-damped harmonic oscillator, and that of drag-force-balanced motion in the high-friction limit. Moreover, the hydrodynamic coupling also influences the reaction time distribution in the individual stochastic trajectories. Based on the observed wide distributions of the rate constant and reaction time in stochastic trajectories, we recommend that hydrodynamic effects be considered in interpreting such distributions in high-precision single molecule or single particle-tracking experiments. We envision that our methodology can be generalized to physiologically relevant conditions by including other (athermal) temporal correlations such as those due to the matrix in which the nanoparticle is transported, or due to cytoskeletal motion in order to describe the dynamics of internalized nanoparticles.

## Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

## Authors' contributions

Formulated the theory and designed the simulations: H.-Y.Y. and R.R.; performed the simulations: H.-Y.Y.; analysed data: H.-Y.Y. and R.R.; wrote the manuscript: H.-Y.Y., D.M.E., P.S.A. and R.R. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The composite GLE aspect of this work was supported in part by the National Institutes of Health (NIH) grant no. R01 EB006818 and the reactive flux formalism aspect of this work was supported by NIH U01 EB016027. Computational resources were provided in part by the Extreme Science and Engineering Discovery Environment (XSEDE) grant no. MCB060006 and by the National Science Foundation grant NSF DMR-1120901.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3590399.

- Received May 24, 2016.
- Accepted November 21, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.