111 Years of Brownian Motion

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a b a Cite this: Soft Matter, 2016, Xin Bian,* Changho Kim and George Em Karniadakis* 12, 6331 We consider the Brownian motion of a particle and present a tutorial review over the last 111 years since Einstein’s paper in 1905. We describe Einstein’s model, Langevin’s model and the hydrodynamic models, with increasing sophistication on the hydrodynamic interactions between the particle and the fluid. In Received 18th May 2016, recent years, the eﬀects of interfaces on the nearby Brownian motion have been the focus of several Accepted 29th June 2016 investigations. We summarize various results and discuss some of the controversies associated with new DOI: 10.1039/c6sm01153e findings about the changes in Brownian motion induced by the interface.

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1 Introduction dust in a fluid. However today, its theory can be also applied to describe the fluctuating behavior of a general system interact- Soon after the invention of the microscope, the incessant and ing with the surroundings, e.g., stock prices. irregular motion of small grains suspended in a fluid had been It was not until 1905 that physicists such as Albert Einstein,3 observed. It was believed for a while that such jiggling motion William Sutherland,4 and Marian von Smoluchowski5 started to was due to living organisms. In 1827, the botanist Robert gain deep understanding about Brownian motion. While the Brown systematically demonstrated that any small particle existence of atoms and molecules was still open to objection, suspended in a fluid has such characteristics, even an Einstein explained the phenomenon through a microscopic inorganic grain.1 Therefore, the explanation for such motion picture. If heat is due to kinetic fluctuations of atoms, the should resort to the realm of physics rather than biology. Since particle of interest, that is, a Brownian particle, should undergo then this phenomenon has been named after the botanist as an enormous number of random bombardments by the sur- ‘‘Brownian motion’’.2 In the classical sense, the phenomenon rounding fluid particles and its diﬀusive motion should be Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. refers to the random movement of a particle in a medium, e.g., observable. The experimental validation of Einstein’s theory by Jean Baptiste Perrin unambiguously verified the atomic nature of matter,6 which was awarded the Nobel Prize in Physics in 1926. a Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Since the seminal works in the 1900s, this subject has fostered E-mail: [email protected], [email protected], [email protected] many fundamental developments on equilibrium and non- 7,8 b Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA equilibrium statistical physics, and enriched the applications

Xin Bian received his PhD in Changho Kim received his PhD in 2015 from Technische Universita¨t 2015 from Brown University and Mu¨nchen and is a postdoctoral is a postdoctoral researcher in the research associate in the Division Computational Research Division of Applied Mathematics at Brown at the Lawrence Berkeley National University. His research focuses on Laboratory. His research areas multiscale modeling and simula- include molecular theory and tion of soft matter physics. simulation, and mesoscopic modeling of soft materials.

Xin Bian Changho Kim

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of fluid mechanics such as the rheology of suspensions.9–11 scales may not be clearly differentiated in the MSD, but easily It also motivated mathematically rigorous developments of distinguished in the VACF, as will be shown in Fig. 3 of probability theory and stochastic differential equations,12–14 Section 4. which in turn boosted the stochastic modeling of finance. For In an order of progressively more accurate hydrodynamic example, one of its remarkable achievements is the Black– interactions between the particle and the fluid, we organize Scholes–Merton model for the pricing of options,15 which was various theoretical models as follows. At first in Section 2 we awarded the Nobel Memorial Prize in Economical Sciences in introduce the pure diffusion model corresponding to Einstein’s 1997. More recently, Brownian motion has been playing a central microscopic picture. Subsequently, we describe the Langevin and fundamental role in the studies of soft matter and bio- model in Section 3, which considers explicitly the inertia of the physics,16,17 shifting the subject back to the realm of biology. Brownian particle. We describe the hydrodynamic model in Other areas of intensive research driven by Brownian motion Section 4, which further includes the inertia of the fluid and include the microrheology of viscoelastic materials,18–21 artificial takes into account the transient hydrodynamic interactions Brownian motors22 and self-propelling of active matter,23,24 between the particle and the fluid. The persistent VACF fluctuation theorems for states far from equilibrium,25–27 and from this model has far-reaching consequences for physics. quantum fluctuations.28,29 In Section 5, we explore the hydrodynamic model in confine- In this work, we focus on the classical aspect of Brownian ment, with its subtle hydrodynamic interactions among the motion based on selective references from 1905 until 2016, particle, the fluid and the confining environment. The results which spans the last 111 years. More specifically, we attempt to of the confined Brownian motion are significant, since the interpret previous theories from a hydrodynamic perspective. passive microrheology using a Brownian particle to determine To this end, we mainly consider a spherical particle of sub- interfacial properties has become more and more popular due micrometer size suspended in a fluid and the particle is subject to its non-intrusive properties. Along the presentation, we shall to free and constrained Brownian motion. Special focus will be focus mainly on the analytical results of the theoretical models given to the velocity autocorrelation function (VACF) of the and make short excursions to experimental observations and particle, denoted by C(t)=hv(0)v(t)i with the equilibrium numerical studies. Controversial results will be highlighted. ensemble average hi. It measures how similar the velocity v Finally, we conclude this work with some perspectives in Section 6. after time t is to the initial velocity.30 In general, due to its interaction with the surrounding fluid, the particle’s velocity 2 Pure diffusion becomes randomized and the magnitude of hv(0)v(t)i diminishes as t increases. Compared to the well-known mean-squared In this section, we summarize Einstein’s seminal work in 2 displacement (MSD), which is denoted by hDr (t)i with the 1905,†3 which has two innovative aspects. The first part displacement Dr(t)=r(t) r(0), the VACF contains equivalent formulates the diffusion equation to relate the mass diffusion dynamical information. This can be clearly seen by the follow- to the MSD, which is a measurable quantity. This relation was 31,32 ing relation: also discovered by von Smoluchowski,5 but with a slightly ð t different factor. The second part is to connect two transport

Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. d Dr2ðtÞ ¼ 2 CðtÞdt; (1) processes: the mass diffusion of the particle and the momentum dt 0 diffusion of the fluid. Hence, the diffusion coefficient can also be expressed in terms of the fluid properties. The connection which suggests that the VACF can be calculated from the between the two transport processes was also obtained by second derivative of the MSD. Nevertheless, the VACF reveals Sutherland independently.4 In the end of this section, we discuss the dynamics in a more direct way; over several time scales of the validity of the model. By considering the VACF, we demon- different orders involved, characteristic behaviors of disparate strate the limitations of the model and clarify its underlying assumptions.

George Em Karniadakis received his 2.1 Diffusion equation and mean-squared displacement PhD in 1987 from Massachusetts The probability density function (PDF) f (x,t) of a Brownian Institute of Technology and is the particle satisfies the following diffusion equation in the one- Charles Pitts Robinson and John dimensional case: Palmer Barstow Professor of Applied Mathematics at Brown @f ðx; tÞ @2f ðx; tÞ ¼ D ; (2) University. His research interests @t @x2 include diverse topics in computa- where D is the diffusion coefficient of the Brownian particle. tional science both on algorithms This equation is derived under Einstein’s microscopic picture and applications. A main current by assuming that the difference between f (x,t + Dt) and f (x,t) thrust is stochastic simulations results from the position change Dx of the particle due to and multiscale modeling of

George Em Karniadakis physical and biological systems. † Einstein’s works on Brownian motion are collected and translated.33

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random bombardments. D may be expressed in terms of the where Z is the dynamic viscosity of the fluid, a is the radius of the second moment of Dx and higher moments are dropped off. particle, and a is the friction coefficient at the solid–fluid interface. For a Brownian particle initially located at the origin, the Note that the Navier slip length is defined as b = Z/a.35 For a =0,it formal solution to eqn (2) is a Gaussian distribution with mean corresponds to a perfect slip interface, whereas a = N corresponds zero and variance 2Dt: to the no-slip boundary condition originally adopted by George Gabriel Stokes in 1851.36 The mobility of a sphere with partial slip 1 x2 f ðx; tÞ¼pffiffiffiffiffiffiffiffiffiffiffi e4Dt: (3) may also be determined in eqn (8) by the slip length b. 4pDt By combining eqn (7) and (8), we arrive at the celebrated Stokes–Einstein–Sutherland formula3,4 Eqn (3) represents that the PDF of the particle evolves from a 8 Dirac delta function d(x)att = 0 to a Gaussian distribution with > kBT <> ; b ¼1; an increasing variance for t 4 0. Accordingly, the MSD of the 4pZa D ¼ > (9) particle, which is the second moment of the PDF, increases > k T : B ; b ¼ 0: linearly with time: 6pZa D 2 h x (t)i =2Dt. (4) This equation establishes the connection between the mass transport of the particle and momentum transport of the fluid. Here, Dx(t)=x(t) x(0) and the brackets denote the Therefore, one can attain one unknown quantity from the other ensemble average over the equilibrium distribution. For the available quantities via eqn (9). For example, given the known three-dimensional case, we have hDx2i = hDy2i = hDz2i and, values of k T and Z, and further D from eqn (5), one may therefore, for r ={x,y,z}, B determine the radius a of the Brownian particle.3 2 hDr (t)i =6Dt. (5) Alternatively, if a is known, Avogadro’s number NA can be determined by using the fact kB = Rg/NA, where Rg is the gas For a random walk like Brownian motion, both the velocity constant.3 Jean Baptiste Perrin actually followed this proposal 23 1 and displacement of the particle are averaged to be zero. and determined Avogadro’s number (NA = 6.022 10 mol ) Therefore, the simplest but still meaningful measurement is within 6.3% error,6 which settled the dispute about the theory the MSD, which determines the diffusion coefficient via eqn (4). on the atomic nature of matter.

2.2 Stokes–Einstein–Sutherland equation 2.3 Limitations and underlying assumptions In a dilute suspension of Brownian particles, the osmotic The main criticism of the diﬀusion model, as Einstein himself pressure force acting on individual particles is rV, where V realized later,38,39 is that the inertia of the particle is neglected. is a thermodynamic potential. Hence, the steady flux of particles This implies that an infinite force is required to change the driven by this force is fm1rV,wheref is the particle volume velocity of the particle to achieve a random walk at each step. concentration and m is the mobility coeﬃcient of individual Therefore, its velocity cannot be defined and its trajectories are

Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. particles. At equilibrium, the flux due to the potential force must fractal, as illustrated on the right in Fig. 1. Since an apparent be balanced by a diffusional flux as: velocity is deduced by two consecutive positions, it really depends on the time-resolution of the observations.40,41 If the fm1rV = Drf. (6) observations are separated by a diffusive time scale as in Einstein’s model, the particle appears to walk randomly. From Moreover, the concentration should have the form of V/kBT the MSD of the diffusion, we may determine. an effective mean f p e at equilibrium, where kB is Boltzmann’s constant pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffi and T is the temperature. By substituting the expression of f velocity over a time interval as v ¼ Dx2 Dt ¼ 2D Dt.As into eqn (6), we obtain Einstein’s relation: Dt - 0, this effective velocity diverges and cannot represent the real velocity of the particle. This also explains the early controver- D = mk T. (7) B sial measurements on the actual velocity of the particle.42,43 The mobility coefficient m is the reciprocal of the friction This unphysical feature can also be seen by calculating the coefficient x. Here, the definitions of m and x arise from a VACF from eqn (1) and (5): hv(0)v(t)i =3Dd(t), where d(t) is the Dirac delta function. This means that even after an infinitesimal situation where the particle moves at terminal drift velocity vd 1 time, the velocity becomes completely uncorrelated with the in a fluid under a weak external force Fext: m = x = vd/Fext. According to Stokes’ law,‡34 the mobility of a sphere in an previous one. A mathematical model corresponding to this case incompressible fluid at steady state is is a Gaussian white noise process for the velocity. Then, x(t) corresponds to a Wiener process, which is continuous but 1 1 þ 3Z=aa nowhere differentiable in time.13 m ¼ x1 ¼ ; (8) 6pZa 1 þ 2Z=aa Physically, however, we should be able to find a time scale 39 t o tb for the ballistic regime,§ where the velocity does not ‡ Stokes’ law is valid for the Knudsen number Kn = l/a { 1, where l is the mean 37 free path of fluid particles. § In general, the ballistic time scale tb is proportional to the Knudsen number.

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takes into account the inertia of the particle.¶ In Langevin’s formulation, which was thought to be ‘‘infinitely simpler’’ according to himself, the equation of motion for the Brownian particle is formally based on Newton’s second law of motion as

d2x dx m ¼x þ F~ðtÞ; (11) dt2 dt

where m is the mass of the particle, x is the friction coefficient defined earlier, and F˜(t) is a random force on the particle. In : this mode, the velocity of the particle v(t)=x(t) is well-defined and it is subject to two different types of forces exerted by the surrounding fluid: a friction force and a random force. It is further assumed that the random force is an independent Fig. 1 Fractal trajectory of Brownian motion according to Einstein’s Gaussian white noise process. Hence, F˜(t) satisfies diffusion model in two dimensions. On the left is the actual trajectory of a particle. On the right are the observed locations of the particle on hF˜(t)i =0,hF˜(t)F˜(t0)i = Gd(t t0), (12) diffusive time scales. Arrows indicate the apparent velocities of the particle. hF˜(t)x(t0)i =0,hF˜(t)v(t0)i = 0, (13)

0 change significantly, that is, Dx(t) E v(0)t, as illustrated on the where t Z t and the noise strength G is to be determined below. left in Fig. 1. In Einstein’s model, tb can be chosen from the time scale for the duration of successive random bombardments. From a mathematical point of view, eqn (11) is a stochastic 2 differential equation. Compared to Einstein’s model, x(t) has From the equipartition theorem, we have hv i = kBT/m,where m is the mass of the particle. Hence, we obtain the MSD better regularity; x(t) is now differentiable. However, v(t)is expression in the ballistic regime: continuous but not differentiable just as x(t) in Einstein’s model. In general, special care needs to be taken to handle a stochastic k T Dx2ðtÞ ¼ B t2: (10) differential equation, as the ordinary calculus may not hold. m However, since eqn (11) is subject to an additive independent ˜ In Einstein’s model, the time scale tb is neglected (i.e., assuming noise F(t), we can still legitimately apply the ordinary calculus to

tb - 0) and the MSD is a completely linear function in time. calculate the MSD and the VACF from eqn (11). A century ago, Einstein also did not expect that it would be possible to observe the ballistic regime in practice due to the 3.1 Two regimes of mean-squared displacement limitation of experimental facilities. Remarkably, such measure- We first derive an expression for the MSD and obtain from it 44 ments have recently become realistic in rarefied gas, normal two asymptotic limits at both short-time and long-time scales. 45 46,47 Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. gas and liquid, with increasing difficulty for fluids with Without loss of generality, we take x(0) = 0. After multiplying elevated density due to the diminishing of tb. However, the dx2 dx eqn (11) by x and using the fact that ¼ 2x and experiment on Brownian particles in a liquid is subtle, as it is dt dt 41 currently still difficult to resolve time below the sonic scale. d2x2 dx 2 d2x ¼ 2 þ2x , we have Therefore, the equipartition theorem can only be verified for the dt2 dt dt2 total mass of the particle and entrained liquid, but not at the 46,47 single particle level. We shall further discuss the effect of md2x2 xdx2 mv2 ¼ þ xF~ðtÞ: (14) the added mass in Section 4. 2 dt2 2 dt In summary, Einstein’s pure-diffusion model considers only the independent random bombardments on the particle, but By taking the average and using eqn (13), we obtain a differ- nothing else. Although the resulting MSD expression of eqn (4) d ential equation for z ¼ x2 : or (5) is always valid at a large time, the model has the single dt time scale of the mass-diffusion process t = a2/D, which is D mdz x denoted as the diffusive or Smoluchowski time scale.48 More- þ z ¼ k T; (15) 2 dt 2 B over, the model disallows a definition of velocity, possesses no ballistic regime, and its VACF does not contain any dynamical where the equipartition theorem, mhv2i = k T, was applied. information. These issues will be resolved in Langevin’s model. B Since hz(0)i =2hx(0)v(0)i = 0, the solution to eqn (15) is

2k TÀÁ 3 Langevin equation zðtÞ¼ B 1 ext=m : (16) x A remedy for the unphysical feature of Einstein’s model at the 49 ballistic time scale was proposed by Paul Langevin, which ¶ Langevin’s work is translated.50

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By integrating eqn (16), we obtain an expression for the MSD must hold. This represents a fundamental relation named as over the entire time range as:51–53 the fluctuation-dissipation theorem (FDT).55–57 Roughly speak- ing, the magnitude of the fluctuation G must be balanced by 2kBT m m Dx2ðtÞ ¼ t þ ext=m : (17) the strength of the dissipation x so that temperature is well x x x defined in Langevin’s model. Therefore, the pair of friction and

On the one hand, for t c tB = m/x, the exponential term random forces acts as a thermostat for a Langevin system. It becomes negligible, and we retrieve Einstein’s result eqn (4) should not come as a surprise that the frictional force and the from eqn (17): random force have such a relation, since they both come from the same origin of interactions between the particle and the d 2kBT x2ðtÞ ¼ ¼ 2D: (18) surrounding fluid molecules. dt x From the solution of velocity in eqn (21), we can also This may also be directly obtained by dropping off the expo- calculate the VACF of the particle. After multiplying eqn (21) nential term in eqn (16). by v(0), and further taking the average, we obtain

On the other hand, for t { tB or t - 0, by using the power 2 xt=m kBT xt=m t2 ÀÁ CðtÞhivð0ÞvðtÞ ¼ v ð0Þ e ¼ e : (25) series et ¼ 1 t þ þ Ot3 we obtain from eqn (17) m 2! Here, the random force term vanished due to eqn (13) and the k T Dx2ðtÞ ¼ B t2; (19) equipartition theorem was also used. It is simple to see that C(t) m decays exponentially and the relevant time scale is the Brownian

which is identical to the ballistic regime of eqn (10) discussed relaxation time, tB = m/x. in Section 2.3. Hence, we clearly see that Langevin’s model can If we take the time integral of the VACF, we find ð ð explain the ballistic regime as well as Einstein’s long-time 1 1 kBT xt=m kBT result of the MSD. The new relevant time scale is the relaxation hivð0ÞvðtÞ dt ¼ e dt ¼ ¼ D; (26) 0 m 0 x time of Brownian motion, tB = m/x. which is just the diffusion coefficient obtained by Einstein. 3.2 Fluctuation-dissipation theorem, velocity autocorrelation The relation in eqn (26) is not fortuitous, but known as the function and diffusion coefficient simplest example of the fundamental Green–Kubo relations.58–61 Now we turn to the velocity of the Brownian particle, which is These relate the macroscopic transport coefficients to the correla- the new element in Langevin’s model. Furthermore, we may tion functions of the variables fluctuating due to microscopic 62 characterize the full dynamics of the particle by the VACF. processes. Such relations were also postulated by the regression 63,64 Let us rewrite the Langevin equation in terms of velocity: hypothesis of Lars Onsager, which states that the decay of the correlations between fluctuating variables follows the dv m ¼xv þ F~ðtÞ; (20) macroscopic law of relaxation due to small nonequilibrium dt disturbances.8 The 1968 Nobel Prize in Chemistry was awarded to Onsager to glorify his reciprocal relations in the irreversible

Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. which is a first-order inhomogeneous differential equation and has the formal solution:53,54 process, which also formed the basis for further development ð of nonequilibrium thermodynamics by Ilya Prigogine and 1 t vðtÞ¼vð0Þext=m þ dt exðttÞ=mF~ðtÞ: (21) others.54,65–67 m 0 Similarly to the diffusion in the long-time limit, we may From this solution, we observe that the average of squared velocity define the time-dependent diffusion coefficient as 2 2 2xt/m ð hv (t)i has three contributions: the first one is hv (0)ie and the t 1 d k TÀÁ Ð DðtÞ hivð0ÞvðtÞ dt ¼ Dx2ðtÞ ¼ B 1 ext=m : (27) 2 xt=m t xðttÞ=m second one is the cross term e dt e vð0ÞF~ðtÞ , 0 2dt x m 0 which becomes zero due to eqn (13). The third contribution is of Note that the equivalence of the two definitions in terms of the second order in F˜(t) and, by making use of eqn (13), we have VACF and the MSD also follows from eqn (1). For Langevin’s model, ð ð t t ÀÁthis equality can be explicitly verified by using eqn (17) and (25). 1 xðttÞ=m 0 xðtt0Þ=m 0 G 2xt=m 2 dt e dt e GdðÞ¼ t t 1 e : m 0 0 2xm 3.3 Limitations and underlying assumptions (22) The Langevin model not only recovers the long-time result of Therefore, the mean-squared velocity is Einstein’s model, but also produces the correct ballistic regime at a short-time limit. An essential ingredient in the model is G ÀÁ v2ðtÞ ¼ v2ð0Þ e2xt=m þ 1 e2xt=m : (23) that the Brownian particle has an inertia, that is, mass m.Asa 2xm result, the velocity and the VACF become well-defined and At the long-time limit, we expect the equipartition theorem, 2 8 hv (t)i = kBT/m,tobevalid.Hence,theequality Coincidentally, the work of Onsager on Brownian motion and linear response laws was conducted when he was teaching at Brown University, although the G =2xkBT (24) latter Brown refers to the businessman and philanthropist Nicholas Brown, Jr.

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continuous in time. By considering a very small relaxation time (and also of the Einstein model) is probably that the friction m/x - 0 in eqn (20), the Langevin dynamics degenerates to be coeﬃcient x is taken as the solution of the steady Stokes flow, the overdamped Brownian dynamics of Einstein’s model. whereas a Brownian particle undergoes erratic movements The limitations of the Langevin model can be revealed by constantly. Therefore, the steady friction may be valid only if considering a corresponding microscopic model, that is, the the surrounding fluid becomes quasi-steady immediately after Rayleigh gas,68 which contains ideal gas particles and a massive each movement, or less strictly, before the relaxation time

particle. Several attempts were made to derive the Langevin tB = m/x of the Brownian particle. This deficiency was already equation from this microscopic model in the early 1960s.69,70 It pointed out in the early lectures of Hendrik Lorentz:**78 was realized that the derivation is possible if the interaction x =6pZa is a good approximation only when the mass density

between the Brownian particle and any gas particle takes place ratio r/rB of the fluid and the Brownian particle is so small that the only for a short microscopic time.68,70 This condition can be fluid inertia is negligible. We shall discuss later why this is true. rigorously verified under the ideal gas assumption and the Since the seminal work of Alder and Wainwright, it was very infinite mass limit of the Brownian particle (i.e., m - N), and soon widely acknowledged that unsteady hydrodynamics plays thus the microscopic justification of the Langevin equation can a significant role in the dynamics of the Brownian particle. This be provided through the Rayleigh gas model. For Brownian motivated many theoretical physicists to work on this subject motion in a real gas or a liquid, a mathematically rigorous from various perspectives, and so the algebraic decay was justification is intractable. One of the reasons is that if the fluid understood by several approaches: a purely hydrodynamic particles interact among themselves, a collective motion (e.g., approach based on the linearized Navier–Stokes equations,80,81 a correlated collisions) of the fluid particles can occur, which generalized Langevin equation approach based on the fluctuating implies that the aforementioned condition may not hold. hydrodynamics,82–84 the mode-coupling theory,85–87 and the We will see in Section 4 that the Langevin description is kinetic theory.88 Although these methodologies have different valid only if the Brownian particle is sufficiently denser than perspectives and mathematical sophistication, all of them respect the surrounding fluid, where the inertia of the fluid may be the inertia of the surrounding fluid and corroborated the same neglected. This fact was exploited in a recent experiment,45 scaling of the asymptotic decay on the VACF.89 where a silica bead is trapped by a harmonic potential53 in air The bold assumption of quasi-steady state in the Langevin and the experimental VACF corroborates well the results of the model can be examined only if we consider the unsteady solution Langevin model.71 For a general case of arbitrary density, the of the hydrodynamics, which has been available for more than a collective motion of the fluid particles and their inertia should century from the independent works of Basset and Boussinesq. be reconsidered carefully. 4.2 Boussinesq–Basset force For a spherical particle undergoing unsteady motion influenced 4 Hydrodynamic model by the inertia of the surrounding fluid, its resistant force was known to Boussinesq and Basset:90–93 Although the VACF of a Brownian particle was never explicitly ð ffiffiffiffiffiffiffiffi t Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. measured in the first half of the twentieth century due to M p v_ðtÞ FðtÞ¼6pZavðtÞ v_ðtÞ6a2 pZr pffiffiffiffiffiffiffiffiffiffidt; (28) experimental limitation, it was widely believed to decay expo- 2 0 t t nentially. When a new era of computational science began in 4 where M ¼ pa3r is the mass of the fluid displaced by the the 1950s, this belief was put to the test and it marked the 3 failure of the molecular chaos assumption.72 particle. Note that eqn (28) is obtained by linearizing (dropping the vrv term) the incompressible Navier–Stokes equations together 4.1 Observation of algebraic decay in VACFs with the no-slip boundary condition on the particle. For a stationary : motion v(t) = 0, only the first term on the right-hand side remains, Using molecular dynamics (MD) simulations, some pioneers which is just the Stokes friction in eqn (11). The second term is due started to realize that the VACF of molecules does not follow to the added mass of an inviscid incompressible origin, while the strictly an exponential decay, but has a slowly decreasing third term is the memory effect of the viscous force from the characteristic. This long persistence was found in fluids retarding fluid, which is referred to as the Bousinesq–Basset force. described by both the Lennard-Jones potential73,74 and the Now let us discuss when the Bousinesq–Basset force hard-core potential.75,76 A milestone took place in 1970 when becomes as important as the Stokes friction. Since the former Alder and Wainwright77 delivered a definite answer for the is expressed as a convolution integral, we may understand it long persistence of the VACF as an algebraic decay, that is, better in the frequency domain. By taking the Laplace trans- C(t) B td/2 for t - N. Here d is the dimension of the problem. Ð form of eqn (28), that is, FðoÞ¼ 1eotFðtÞdt, we obtain Meanwhile this scaling was confirmed by independent numerical 0 82 simulations of Navier–Stokes equations, which indicate that a F(o)=x(o)v(o) with 76,77 (transient) vortex flow pattern forms around a tagged particle. M pffiffiffiffiffiffiffiffiffi xðoÞ¼6pZa þ o þ 6pa2 Zro: (29) These observations from computer simulations led to many 2 intriguing questions as to what is missing in the Langevin model. The most suspicious assumption of the Langevin model ** Lorentz’s lectures are translated,79 see page 93 of the translation.

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From the transformation, we note that any model with only the 4.4 Heuristic derivations of the algebraic decay steady friction should be considered to be a zero-frequency Here, we discuss how the algebraic decay appears in the 80 theory. If we compare the first and third terms on the right- generalized Langevin eqn (30), and how it can be explained hand side of eqn (29), the latter becomes larger than the former from a hydrodynamic perspective. The first question can be 2 2 for frequency o 4 Z/ra , or equivalently for time t o ra /Z. answered by deriving a differential equation that the VACF Since the relaxation time in Langevin eqn (11) is tB = m/x = C(t)=hv(0)v(t)i satisfies. After multiplying eqn (30) by v(0) and 2 2rBa /9Z, the fluid inertia has non-negligible effects on the taking averages, we obtain the Volterra equation (also known as dynamics of the Brownian particle for t o (9r/2rB)tB. Hence, if the memory function equation103) 9r/2r 1, the fluid inertia is negligible, which also confirms B { ð the insightful remark made earlier by Lorentz. t mC_ðtÞ¼ xðt tÞCðtÞdt: (32) Alternatively, we may realize the significance of the fluid 0 inertia more directly by considering the vorticity o = ru, which satisfies the diffusion equation qo/qt = nr2o,94 where It is known that if either C(t)orx(t) decays algebraically, then the kinematic viscosity n = Z/r. The time scale for the vorticity to the other also decays algebraically with the same power law and pffiffiffiffi travel a distance of the radius of the Brownian particle is the opposite sign.104 From the o term of x(o) in eqn (29), we 2 3/2 tn = a /n. For the Langevin model to be valid, it must be know that x(t) decays like t with negative values at large time 3/2 tn { tB or 9r/2rB { 1 so that the transient behavior of the t. Therefore, it is expected that C(t) also decays like t but fluid plays a negligible role in the particle dynamics. This with positive values at large time t. This mathematical argu-

hydrodynamic argument is also in agreement with the analysis ment shows that no matter how small r/rB is, the asymptotic of the molecular theory.70 decay of the VACF is always algebraic rather than exponential.

In summary, while the Langevin equation provides a fair However, for smaller r/rB, the exponential decay yields to approximation for 9r/2rB { 1, e.g., a dense particle in gas, it algebraic decay later in time and the Langevin model becomes B does not apply well to the case of 9r/2rB 1, for example, a a better approximation. pollen particle in water, that is the historic observation The persistent scaling of the VACF can also be easily under- recorded by Robert Brown. stood by a heuristic hydrodynamic argument. Suppose a particle has initial velocity v , due to viscous diffusion, after time t,a 0 pffiffiffiffiffi 4.3 Generalized Langevin equation vortex ring (d = 2) or shell (d = 3) with radius r nt develops. Now that the importance of the fluid inertia is recognized, we ThetotalmasswithintheinfluencedzoneisM* B rrd.Ifthe may discuss the equation of motion for the Brownian particle. surrounding fluid is entrained and moves with the particle at time mv mv For a rigid particle suspended in a continuum fluid described t, by momentum conservation we have vðtÞ¼ 0 0ðntÞd=2. by the fluctuating hydrodynamics,93 the following generalized M r 83,84 B d/2 Langevin equation can be formulated: Then, it is simple to see that C(t) (nt) . The argument above ð assumes that the particle does not move when the vortex forms. If dv t m ¼ xðt tÞvðtÞdt þ F~ðtÞ: (30) the particle moves evidently as the vortex develops, we may still Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. dt 0 extend this hydrodynamic argument by adding in the self- diffusion constant D of the tagged particle into the scaling so Compared to the original Langevin eqn (20), eqn (30) is non- that we have C(t) B [(n + D)t]d/2. In fact, by introducing the Markovian as the friction force is history-dependent. The memory evolution of the probability distribution function of the tagged kernel x(t) is the inverse Laplace transform of eqn (29). In particle, the following expression was derived rigorously (one- ˜ addition, the random force F(t) is non-white or colored, which dimensional case):87 can be observed via the fluctuation-dissipation relation57

2kBT 3=2 hF˜(0)F˜(t)i =3kBTx(t). (31) lim CðtÞ¼ ½4pðD þ nÞt : (33) t!1 3r At first glance, eqn (30) seems to be simple. We note, This power law scaling is demonstrated by dissipative particle however, that the form is quite general and all the complicated dynamics simulations in Fig. 2. information is hidden in the memory kernel x(t) or in the If the momentum diffusion is much stronger than the mass statistics of the random force F˜(t). diffusion or if the Schmidt number Sc = n/D is very large (e.g.,a Although theoretically well known, the colored power spectral solid particle suspended in a liquid), we can ignore the con- density of the thermal noise, which is the Fourier transform of tribution of D. Under this condition, which is favored by the eqn(31),hasbeenconfirmedbyexperimentsonlyrecently.95,96 We linearized hydrodynamics, the full expression of C(t) was also note that the same form of equation as eqn (30) can be derived from the fluctuating hydrodynamics of an incompres- obtained from microscopic equations of motion for a Hamiltonian sible fluid for a neutrally buoyant particle:83,107 fluid through the Mori–Zwanzig formalism.97–102 In fact, the "# emergence of a non-Markovian process is a typical scenario when ð pffiffiffi 2 2k T 1 1 xexnt=a insignificant variables (fast fluid variables in our case) are elimi- CðtÞ¼ B dx : (34) 3m 3p 1 þ x=3 þ x2=9 nated in a Markovian process under coarse-graining.54 0

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Fig. 2 Asymptotic limit of the velocity autocorrelation function for a diffusive particle. Eqn (33) with or without diffusion coefficient D is compared with the results of tagged fluid particles in dissipative particle dynamics (DPD) simulations. The inset shows the long-time limit in the logarithmic scale. Input parameters of DPD are taken from a previous 105,106 work, which correspond to a fluid with kBT =1,r =3,n = 0.54, and D = 0.15 in DPD units.

Other than the integral form of eqn (34), an alternative closed form of C(t) is also available.82,108,109 We compare the VACF from the hydrodynamics theory with that of Langevin’s model in Fig. 3(a). We observe that the Langevin model underestimates the decay rate of the VACF at

short time (t t tn) while overestimates it at long time (t \ tn).

4.5 Diffusion coefficient and mean-squared displacement The time-dependent diffusion coefficient D(t) of a Brownian particle can be obtained directly by integrating eqn (34) as shown in eqn (27). Furthermore, the MSD may also be obtained by further integrating D(t) or directly from the VACF as110,111

Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. ð t Dr2ðtÞ ¼ 2 ðt tÞCðtÞdt: (35) 0

The non-diffusive signatures of the MSD and the time- dependent diffusion coefficient due to hydrodynamic memory have been validated for Brownian particles in a suspension probed by dynamic light scattering.††107,111 More recently, to avoid any (weak) hydrodynamic interactions between particles, optical trapping interferometry has been applied to a single micrometer particle112 which is trapped in a weakly harmonic potential.113 Consequently, the hydrodynamic theory for the non-diffusive regime has been explicitly confirmed with excel- lent accuracy.112 We compare the time-dependent diffusion coefficients and MSDs from different theoretical models in Fig. 3(b) and (c). We observe that the D(t) from Langevin’s Fig. 3 C(t), D(t), and hDx2(t)i of a Brownian particle (1D) according to the model approaches exponentially fast to Einstein’s diffusion Langevin model, incompressible viscous hydrodynamics, and its correc- coefficient, whereas it takes a substantially longer time for tion due to compressible effects at the short time scale. Relevant time scales are sonic time t = a/c , viscous time t = a2/n, Brownian relaxation cs s n the hydrodynamic model to reach a plateau value. 2 time tB = m/x, and diffusive time tD = a /DN. The definitions of variables

are in the text. For a demonstrative purpose their values are a =1,cs = 50, †† An analytical work on the non-diﬀusive MSD from the physics community of r = rB =1,n = 1, and kBT = 1 in reduced units. Hence tc = 0.02, tB = 0.22, the former Soviet Union seems to predate other relevant works,114 and it has been s tn = 1.0, and tD = 18.85. recently translated.115

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It is worth noting that even when the fluid inertia is particle.120,121 From the ratio of the added mass and the important for the dynamics such as the asymptotic decay of M r particle mass ¼ , it is simple to see that for a lighter C(t) of the BrownianÐ particle, the equation for the diffusion 2m 2rB 1 fluid the compressibility becomes less important for the parti- coefficient D1 ¼ 0 CðtÞdt ¼ kBT=x always holds. This means that the steady motion or the zero-frequency mobility compo- cle dynamics. nent provides the largest displacement and dominates the Similarly any viscoelasticity effects may be incorporated into diffusive process.83,109 Therefore, the Stokes–Einstein–Sutherland the generalized friction at a different frequency after introdu- 80 formula in eqn (9) is still correct for a diffusive process, which is cing a new relaxation time scale. Moreover, one would need to universally captured by Einstein’s model, Langevin’s model and select a suitable viscoelastic model and also determine its the hydrodynamic model. relaxation time by other means. The problem is that viscoelasticity includes a vast range of time scales, but most models 4.6 Limitations and underlying assumptions do not. The heuristic approach above assumes that the long-time decay The hydrodynamic theory is based on continuum-fluid of the VACF for the particle is solely affected by the dynamics of mechanics, which necessarily cannot resolve the ballistic d 4 vortex formation driven by the transversal component of the motion over t 0 accurately. This fact is indicated in the hydrodynamic equations.89,116 The longitudinal component inset of Fig. 3(c), where the Langevin model shows a finite drives compressibility effects, which vanish in a sonic time period for the ballistic regime, whereas the hydrodynamic scale, and therefore, they do not contribute to the long-time model deviates from it quickly. In the hydrodynamic model behavior of the dynamics.87 If the short-time dynamics is of (also in Langevin and Einstein models), we consider only the interest, the compressibility should be reconsidered. continuous friction such as the Stokes or Bousinesq–Basset drag When the fluid is considered mathematically to be incom- on the particle, but ignore the Enskog friction on the Brownian 122,123 pressible, the particle mass m is augmented by an induced particle due to molecular collisions with the solvent. mass M/2, where M is the mass of the fluid displaced by the Here we focused on the translational motion of a single particle.93 Due to this mathematical treatment, for any infini- spherical particle with the no-slip boundary condition. There are various extensions based on this simple scenario. For tesimal time dt, C(dt)=kBT/(m + M/2). However, the equiparti- example, for a sphere with the slip or partial-slip boundary tion theorem requires that C(t) starts with C(0) = kBT/m. Therefore, the incompressible assumption generates a discon- condition, the magnitude but not the scaling of the asymptotic 80 tinuity of C(t) at short time and violates the equipartition decay changes. The dynamics for a particle with an arbitrary 83,108,124 theorem of statistical physics.‡‡117,118 A similar paradox was shape can be formulated as a similar problem. recognized when inverse-transforming eqn (29) to get x(t), The VACF of the angular velocity for a rotating particle may R p which is singular at t = 0 and leads to a substantial difference also be calculated with an asymptotic behavior as C (t) 5/2 83,125,126 between v(0) and v(dt) in the case of impulsive particle t ,§§ and the non-spherical shape alters only its 127 motion.89,93 The unphysical consequences at short time may magnitude but not the power law. For a test particle immersed be alleviated by realizing that every fluid is (slightly) compres- in a suspension of particles, the asymptotic power law does not Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. sible. Therefore, we may find a reconciliation of the dynamics change and its magnitude is obtained by replacing the fluid 128,129 from short to long time by considering the propagation of viscosity with the suspension viscosity. The unsteady equa- sound waves and incorporating a frequency dependent friction tion of motion for a sphere in a nonuniform flow is also 130 at a frequency similar to the inverse of the sound speed available. For a Brownian particle of molecular size, the value c .80,117,118 For a neutrally buoyant particle, the sound wave of its radius or slip length on the surface is always conceptually s 131 dissipates 1/3 of the total energy and the contribution on the subtle in a continuum description and needs extra care. VACF from the compressibility effects reads109,117,118 "# pffiffiffi ! pffiffiffi ! c t ffiffiffi 5 Effects of confinement kBT 3 s 3cst p 3cst CsðtÞ¼ e 2a cos 3 sin : (36) 3m 2a 2a In the past few decades, the effects of an interface on a nearby Brownian particle have been attracting a lot of attention. On the We may see in Fig. 3(a) that adding the compressible correction one hand, it is physically interesting to study the dynamics of of eqn (36) to the incompressible VACF of eqn (34) indeed the Brownian particle in a confined environment, where the respects the equipartition theorem at short time. The effects of momentum relaxation of the fluid is influenced by the inter- the compressibility are not so apparent for the diffusion face. On the other hand, it is practically beneficial to deduce coefficient or MSD, as indicated in Fig. 3(b) and (c). the interfacial properties from the observed dynamics of the Another interesting phenomenon at the short-time scale due Brownian particle, which is analogous to the passive micro- to sound propagation is the ‘‘backtracking’’, which may con- rheology technique for unbounded viscoelastic characterization.18 tribute negatively to the overall friction experienced by the Different from the unbounded case, the motion of a Brownian

‡‡ Another contemporary work by Giterman and Gertsenshteiˇn119 was recently §§ A slightly earlier work132 on the rotating motion from the physics community brought to attention.115 of the former Soviet Union has been recently translated.115

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particle near an interface is stronglyinfluencedbyitshydrody- For the perpendicular motion, an exact solution can be namic interactions with the interface, and its studies date back as obtained using the bi-polar coordinates142,143 133,134 early as Hendrik Lorentz’s reciprocal theorem. X1 a 1 4 nðn þ 1Þ From the unbounded motion of a Brownian particle, we m1 ¼ sinh a ? h m 3 ð2n 1Þð2n þ 3Þ learnt that the diffusive process is dominated by the steady or 1 n¼1 2 3 zero-frequency mobility. This is still true in the confined case. Therefore, at first we may ignore the thermal agitations of the 6 2 sinhð2n þ 1Þa þð2n þ 1Þ sinh 2a 7 6 17; fluid and describe the mobility of a spherical particle immersed 4 1 5 4 sinh 2 n þ a ð2n þ 1Þ2 sinh 2a in Stokes flow bounded by a plane wall in Sections 5.1 and 5.2. 2 Due to the linearity of Stokes flow, the particle’s parallel and (40) perpendicular motions to the wall can be decomposed and handled separately. Subsequently, we will discuss the diffusion where a = cosh1(h/a). Although eqn (40) was immediately and VACFs of a Brownian particle near a wall in Section 5.3, validated by experiments,144 it is expressed as an infinite series, followed by other more sophisticated scenarios revealed in which is inconvenient as a reference solution. An appropriate Section 5.4. form as a good approximation to eqn (40) may be obtained by the regression method145,146 5.1 Mobility with no-slip interface a 6 10ða=hÞþ4ða=hÞ2 When no-slip boundary conditions are assumed on the surfaces m? : (41) h 6 3ða=hÞða=hÞ2 of both the particle and the wall, Hiding Faxeń derived an expression for the mobility coefficient mJ of the parallel motion We summarize different approximations for the mobility 135–137 using the method of reflection in his PhD dissertation hampered by a no-slip plane wall in Fig. 4. The results from different methods agree with each other at the intermediate a 9 a 1 a 3 45 a 4 1 a 5 m m 1 þ ; (37) and large distance, that is, mJ with h/a \ 1.5 and m> with h/a \ 3. k h 1 16 h 8 h 256 h 16 h Differences appear only at the short distance; Lorentz’s image technique is not accurate for either mJ or m>. The method of which includes the effects of a second reflection; mN is the reflection improves the accuracy of mJ, but fails at the lubrication Stokes mobility coefficient (denoted above as m) and h is the regime (h/a o 1.1), which is covered by eqn (39) of Perkins and distance from the center of the sphere to the wall surface. Jones. The method of reflection in eqn (38) overcompensates the Following the method of reflection applied by Sho¯ichi Wakiya,138 deviation on m> from Lorentz’s image technique. Adding only a m> the mobility coefficient of the perpendicular motion can also few terms of the series in eqn (40) already provides a convergent be obtained as139 value for m>, which is readily represented by the regression form of eqn (41). a 9 a 1 a 3 m m 1 þ : (38) ? h 1 8 h 2 h Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. Eqn (37) and (38) represent a hindered motion due to the presence of the wall compared to the mobility coefficient mN in the unbounded case. If we truncate eqn (37) and (38) at the first order of a/h, we recover the earlier approximations obtained by the Lorentz’s image technique.134 From these first-order approximations, it is simple to deduce that the perpendicular motion is impeded more strongly than the parallel one. Both the image technique and the method of reflection are only accurate for a { h. For the parallel motion, there is no closed form for the solution of mobility over the entire range of h. Instead, Perkin and Jones started out with the Green tensor for a semi-infinite fluid and matched a series result (at large h) with an asymptotic one derived from lubrication theory (at small h) to get the 140,141 mobility valid for a wide range of h Fig. 4 Mobility of a sphere near a no-slip wall. The results from Lorentz’s image technique are taken up to the first order of a/h in eqn (37) and (38); 1 a 8 a a the results from the method of reflection are the complete expressions in mk 1 ln 1 þ 0:029 h 15 h h eqn (37) and (38). The prediction of mJ in eqn (39) from Perkins and Jones is (39) shown to be more accurate at the short distance, as indicated in the inset. a 2 a 3 þ 0:04973 0:1249 ; The prediction of m> with series solution is taken from eqn (40) up to h h n = 10 and including higher n does not change the sum of series significantly. The regression approximation for m> in eqn (41) is almost which is more accurate than eqn (37) for small h. identical to the series solution.

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5.2 Mobility with slip interface Although the no-slip boundary condition on the fluid–solid interface cannot be justified from first principles, classical experiments over several decades indeed support its validity, and the no-slip boundary condition has become a cornerstone of continuum-fluid mechanics.31,37,147,148 However, many recent experiments indicate violations of the no-slip boundary condition in micro-channels even of the micrometer scale.149–152 Since the slip length of the interface may depend on the shear rate153 and dynamic response of gas bubbles,154 any external perturba- tion from measurements, such as shear flow, could affect the intrinsic properties of the interface. A passive Brownian particle may be an effective probe to sense the interfacial properties locally, as it only leads to a minimal intrusion to the natural environment near the interface. We again start with a spherical particle immersed in Stokes flow bounded by a single plane wall. The no-slip boundary condition still applies to the particle surface, whereas for the plane wall we define the slip-length b from its boundary condition as35,148 @u u ¼ 0; u ¼ b k; (42) ? k @n where n is the normal direction to the wall. This is the same definition as for the slip length of a particle in eqn (8); the normal component of the velocity vanishes at the interface, whereas the tangential component extrapolates linearly to vanish at distance b inside the solid. For a small slip length b { h, Lauga and Squires applied the image technique (a { h)toobtain155 Fig. 5 Mobility of a sphere near a slip wall. a b 9 a b m ; m 1 1 ; (43) k h h 1 16 h h larger the slip length of the wall is, the stronger mobility a nearby particle has. It is worthwhile to note that a large slip length a b 9 a b m ; m 1 1 ; (44) b/h 4 1(e.g., b/h =50orN) may cause the parallel mobility ? h h 1 8 h h Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. coefficient to be even greater than that of the unbounded case as where the mobility coefficients are now functions of both a/h and shown in Fig. 5(a), whereas it does not affect the perpendicular b/h.Forano-slipwallb = 0, eqn (43) and (44) reduce to eqn (37) mobility significantly as indicated in Fig. 5(b). Therefore, we and (38) to the first order of a/h. suggest that the parallel motion of the particle should be probed For a large slip length b c h (and a { h), another preferably to determine the interfacial properties, as it is more asymptotic limit is obtained155 sensitive to the slip length of the interface and provides a much wider range of mobility coefficients for measurements. a b 3 a 5h h m ; m 1 þ 1 þ ln ; (45) So far, we have assumed that the particle surface has a no- k h h 1 8 h b b slip boundary condition. Even if the particle–fluid interface a b 3 a h also possesses a slip length, the Stokeslet (Green’s function) in m ; m 1 1 þ : (46) the image technique does not change.155 Therefore, the mobi- ? h h 1 4 h 4b lity modifications due to the slip wall in eqn (43)–(46) still hold. For b - N, the terms of h/b disappear in eqn (45) and (46) and In this case, we only need to replace mN in these equations by the mobility for a perfect slip wall is recovered. In this case, the one presented in eqn (8), which takes into account the eqn (46) corroborates the pioneering work of Brenner.142 modifications induced by the slip length at the particle surface. The higher-order terms are not included in these solutions and the results are accurate only to the first order of a/h and b/h 5.3 Diffusion coefficient and asymptotic decay of VACFs (small slip length) or h/b (large slip length). We summarize the From the mobility coefficients we may write down the diffusion first-order modifications for the mobility of a particle near a coefficients for a particle in the vicinity of a plane wall as wall with a slip boundary condition in Fig. 5. Due to the image technique, the further away from the wall the particle is located a a D ¼ m kBT; (47) (larger h/a), the more accurate are the solutions. In general, the k h k h

Tutorial Review Soft Matter a a D ¼ m k T: (48) capability of a Brownian particle as a probe for the wettability ? h ? h B at a liquid–solid interface. For the diffusion coefficients of a spherical particle near a plane It is still quite challenging to obtain the confined VACFs wall with the no-slip boundary condition, these analytical with a great accuracy from experiments. Available experimental expressions have been corroborated by experiments,141,156,157 results157,166 exhibit non-negligible noises, from which neither and fluctuating-hydrodynamics simulations.158 For a partial- the scaling nor the magnitude of the VACFs could be conclu- slip wall, the analytical results on parallel mobility are also sive. Therefore, this dispute is yet to be settled. verified by deterministic continuum simulations.159 In Section 4, we have seen that the friction due to the 5.4 Limitations and underlying assumptions transient dynamics of the fluid plays a significant role in the We focused on the mobility of a particle due to a single nearby VACF of the Brownian motion. This is still true in the confined wall. Effects due to two-wall confinements or two-particle inter- case but more involved. For the unsteady motion of a sphere in actions are more involved, but can be tackled.139,156,159,167–170 We viscous flow bounded by a plane wall, where the no-slip have assumed an incompressible fluid and ignored any com- boundary condition applies to both solid interfaces, Wakiya pressible behavior of the fluid. For the short-time dynamics, 160,161 calculated the parallel mobility and Gotoh and Kaneda however, sound propagation also plays a decisive role for the 162 worked out the mobility perpendicular to the wall. Further Brownian motion in a confined environment,171–175 just as in the 83 extending the work of Hauge and Martin-Lo¨f based on unbounded case. fluctuating hydrodynamics of the unbounded case, Gotoh and The random force on a Brownian particle in confinement is Kaneda obtained the asymptotic VACFs in the confined case also non-white as in the unbounded case. Moreover, the 162 with dominant terms as intensity of the power spectral density on position fluctua- 2 kBTh 5=2 tion or thermal noise is shifted by the wall, as measured C ðtÞ pffiffiffiðntÞ ; (49) 96 k 8rp p experimentally. However, a recent analytical calculation from Felderhof176 on the spectrum of position fluctuations, where a k Th4 static wall-slip length is assumed, does not agree with the C ðtÞ B pffiffiffiðntÞ7=2: (50) ? 32rp p experimental results. This disagreement suggests that the slip length on the wall is dynamic and introducing a frequency- c 2 These solutions are valid for t th = h /n, which is the time for dependent slip length could potentially improve the modeling the vorticity propagation between the sphere and the wall. based on the continuum fluid mechanics.153,154,177–181 Never- 5/2 7/2 The power laws of t and t for the confined VACFs theless, it is not certain that this hypothesized continuum 163 were verified by lattice Boltzmann simulations. However, boundary condition may faithfully reflect the Brownian motion Felderhof recently claimed that these analytical results are in a confined fluid at molecular length-time scales, where lock- erroneous and the simulations are also too short to achieve ing and delayed relaxation caused by the epitaxial ordering of 164 an asymptotic limit. Instead, Felderhof performed the calcu- the fluid structure near the interface may be significant.165,182 lation himself and found that VACFs behave asymptotically at Furthermore, the mobility of a Brownian particle due to the

Published on 04 July 2016. Downloaded by Brown University 27/03/2018 20:05:04. 164 large t as 183 ÀÁ atomistic collisions confined in a microscale channel may not 2 2 kBT 3h a always be described by the linearized hydrodynamic equations. C ðtÞ pffiffiffi ðntÞ5=2; (51) k 24rp p

k Ta2 k Th4 6 Summary and perspectives C ðtÞ B pffiffiffiðntÞ5=2 þ B pffiffiffiðntÞ7=2: (52) ? 24rp p 32rp p We summarized three theoretical models for a Brownian par- For the parallel motion, the magnitude is slightly different from ticle suspended in a fluid: Einstein’s model, Langevin’s model, that of eqn (49). For the perpendicular motion, however, it is and the hydrodynamic model and its extensions near a con- even qualitatively different; the long-tail is dominated by a fined interface. From the perspective of hydrodynamic inter- scaling of t5/2 with negative values as in eqn (52) rather than actions between the particle and the fluid, each model is more t7/2 with positive values as in eqn (50). elaborate than its preceding one. In a recent ms-long molecular dynamics (MD) simulation It is simple to differentiate the capability of different models with Lennard-Jones interactions, the asymptotic scaling of the by taking into account the disparate time scales involved. parallel motion is again confirmed to be t5/2.165 Furthermore, Einstein’s model considers only the diffusive time scale 2 Huang and Szlufarska utilized a more general result than tD = a /D, when the particle undergoes a random walk, and eqn (51) for a denser particle to validate the magnitude of the thus a statistical description of its displacements is feasible asymptotic decay.165 However, there was still no direct evidence without involving the momentum coordinates of either the to confirm whether the magnitude in eqn (49) or eqn (51) is particle or the fluid. Langevin’s model introduces the inertia more accurate. The Brownian motion was also employed by of the particle, and therefore an extra time scale is introduced,

Huang and Szlufarska to detect a breakdown of the no-slip tB = m/x, where x is the friction coeﬃcient due to the viscous boundary condition at short time, which demonstrates the fluid at steady state. Hence, the model separates two asymptotic

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15 regimes, that is, the ballistic regime t { tB and the diﬀusive a numerical time step must be about 10 s for stability.

regime t c tB. Moreover, due to the particle inertia its velocity Therefore, it is still impractical to simulate such a simple is well-defined and the velocity autocorrelation function (VACF) scenario with a full atomistic description. The most realistic encodes the full dynamics of the particle with an exponential class of numerical methods to study the Brownian motion and 2 decay. If the relaxation time of the viscous fluid tn = a /n is its relevant areas seems to be the mesoscopic methods, which

comparable to or larger than tB, that is 9r/2rB \ 1, which is a may cover a wide range of spatial-temporal scales. This category typical scenario for a colloidal suspension (e.g., a pollen particle includes the mesh-based methods, such as finite difference,192 in water), the inertia of the fluid must be explicitly taken into finite volume,193–195 and lattice Boltzmann,196,197 and also the account. The hydrodynamic model is based on the solution of particle-based methods, such as dissipative particle dynamics,198 the linearized Navier–Stokes or unsteady Stokes equations, smoothed particle hydrodynamics,158,199 and stochastic rotation which is employed to calculate the full dynamics (signified by dynamics/multiple-particle collision dynamics.200–202 the VACF) of a Brownian particle. The coupling between the inertias of the particle and the fluid is mediated by their transient hydrodynamic interactions, and this leads to an algebraic decay of Acknowledgements the particle’s VACF. The power law scaling indicates significant implications, such as the failure of the molecular chaos assump- We would like to thank Prof. Bruce Caswell for a critical reading tion, which is expected from Langevin’s model. of the manuscript. This work was primarily supported by the When a Brownian particle jiggles near an interface, the Computational Mathematics Program within the Department relaxation of the fluid due to vortex development is aﬀected of Energy oﬃce of Advanced Scientific Computing Research as by its encounter with the interface. Naturally, this introduces a part of the Collaboratory on Mathematics for Mesoscopic 2 new time scale th = h /n, which indicates the time of vorticity Modeling of Materials (CM4). This work was also sponsored propagation between the particle and the wall. For t c th, the bytheU.S.ArmyResearchLaboratoryandwasaccomplished asymptotic limit of the VACFs (including parallel and perpendi- under Cooperative Agreement Number W911NF-12-2-0023. This cular components) for the particle may be calculated and they research was also partially supported by the grants U01HL114476 still follow the power law scalings. However, the actual magni- and U01HL116323 from the National Institutes of Health. tude and power law from analytical approaches remain controversial. Existing results from experiments are also imperfect for a consensus. Perhaps new experimental techniques illu- References strated by Raizen’s and Florin’s groups46,47 may provide a definite answer for the asymptotic limit in the near future. 1R.Brown,The miscellaneous botanical works of Robert Brown, Furthermore, if the sonic time scales in a liquid such as The Ray Society, London, 1866, vol. I. t t0 cs = a/cs and cs = h/cs are to be considered for the dynamics 2 M. D. Haw, J. Phys.: Condens. Matter, 2002, 14, 7769. of a Brownian particle, perhaps extra innovations in experi- 3 A. Einstein, Ann. Phys., 1905, 322, 549–560. mental facilities are yet to be developed. 4 W. Sutherland, Philos. Mag., 1905, 9, 781–785.

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