Soft Matter View Article Online TUTORIAL REVIEW View Journal | View Issue 111 years of Brownian motion 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. www.rsc.org/softmatter 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 This journal is © The Royal Society of Chemistry 2016 Soft Matter, 2016, 12, 6331--6346 | 6331 View Article Online Tutorial Review Soft Matter 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)Áv(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)Áv(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) À 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 6332 | Soft Matter, 2016, 12, 6331--6346 This journal is © The Royal Society of Chemistry 2016 View Article Online Soft Matter Tutorial Review 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 eÀ4Dt: (3) may also be determined in eqn (8) by the slip length b.