Gaussian field theory for the Brownian motion of a solvated particle
GGaussian field theory for the Brownian motion of a solvated particle
Thomas Speck Institut f¨ur Theoretische Physik II, Heinrich-Heine-Universit¨at D¨usseldorf,Universit¨atsstraße 1, 40225 D¨usseldorf, Germany
An alternative derivation of Brownian motion is presented. Instead of supplementing the linearizedNavier-Stokes equation with a fluctuating force, we directly assume a Gaussian action functional forsolvent velocity fluctuations. Solvating a particle amounts to expelling the solvent and prescribinga boundary condition to the solvent on the interface that is shared with the solute. We study thedynamical effects of this boundary condition on the solvent and derive explicit expressions for thesolvent mean flow and velocity correlations. Moreover, we show that the probability to observesolvent velocity fluctuations that are compatible with the boundary condition reproduces randomBrownian motion of the solvated particle. We explicitly calculate the translational and rotationaldiffusion coefficients of a spherical particle using the presented formalism.
PACS numbers: 05.40.-a,05.40.Jc
I. INTRODUCTION
Our traditional understanding of Brownian motion isthat of a solute getting kicked randomly by surroundingsolvent molecules. Einstein’s seminal contribution hadbeen to derive the diffusion equation–and in particularan explicit expression for the diffusion coefficient–basedon this insight [1]. One route to derive Brownian mo-tion from more fundamental equations is fluctuating hy-drodynamics [2], which augments the linearized Navier-Stokes equations with a random stress tensor obeyingthe fluctuation-dissipation theorem. Integrating out thefluid velocity field, the stochastic equation of motion fora solvated particle can then be derived, which takes onthe form of a generalized Langevin equation [3]. Thisapproach is not without problems since the correlationsof the random stress tensor near the solute are assumedto be the same as those in the isotropic bulk fluid. Thereis, however, evidence that the presence of rigid bodiesinfluences the local stress fluctuations [4].The purpose of this Brief Report is to give a somewhatalternative view on the dynamics of solvated particles, aview in which Brownian motion arises from suppressingsolvent fluctuations. Instead of the stress, we considerdirectly the (memoryless) fluctuations of the solvent ve-locity. We employ Gaussian field theory [5] to study howa solute modifies these velocity fluctuations in its vicin-ity. Gaussian field theory has been applied successfullyto, e.g., dielectric relaxation dynamics [6] and the hy-drophobic effect [7]. In analogy to the modification ofdensity fluctuations due to the excluded volume of thesolute, we show that imposing a solvent velocity on thesolute-solvent interface leads to a flow and diminishessolvent velocity fluctuations in the vicinity of the solute.Loosely speaking, the solvent loses entropy, which is com-pensated by the random motion of the solute.
II. THE PURE SOLVENT
We consider an incompressible quiescent fluid withbulk viscosity η . We assume that in the absence of anysolvated object the probability P [ u ] = exp {−S [ u ] } toobserve a given history of velocity fluctuations u ( r , t )away from zero is Gaussian with action S [ u ] = 12 (cid:90) d t (cid:90) d r d r (cid:48) u ( r , t ) · χ − ( r , r (cid:48) ) · u ( r (cid:48) , t ) . (1)The functional inverse of χ is defined through (cid:90) d r (cid:48)(cid:48) χ − ( r , r (cid:48)(cid:48) ) χ ( r (cid:48)(cid:48) , r (cid:48) ) = δ ( r − r (cid:48) ) , (2)where denotes the identity matrix. Normalization of P is achieved through choosing the appropiate functionalmeasure [d u ]. Clearly, the matrix χ is related to thevelocity correlations, (cid:104) u ( r , t ) u T ( r (cid:48) , t (cid:48) ) (cid:105) = χ ( r , r (cid:48) ) δ ( t − t (cid:48) ) . (3)The brackets denote averages over different realizationsof the velocity field, whereas the subscript indicates thepure solvent. Eq. (3) implies temporally uncorrelated ve-locity fluctuations. We know that this is not strictly cor-rect as hydrodynamics predicts a power law tail for theparticle velocity autocorrelation function [8, 9]. How-ever, for the sake of simplicity, here we aim for a sim-ple Markovian description which is appropriate for suffi-ciently coarse-grained time. Brownian motion in confinedgeometries such as nanopores [10] might require to takeinto account memory.Now image that an external force density f ( r , t ) actson the solvent. Up to linear order, the fluctuation-dissipation theorem [11] relates this force to an instanta-neous mean velocity profile (cid:104) u ( r , t ) (cid:105) = 12 T (cid:90) d r (cid:48) χ ( r , r (cid:48) ) · f ( r (cid:48) , t ) (4)through the correlations Eq. (3). The factor 1 / δ -function. Of a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l course, Eq. (4) is nothing else than Faxen’s theorem, fromwhich we can deduce the correlations χ ( r ) = T πηr (cid:18) + rr T r (cid:19) (5)using the Oseen tensor [12]. However, we will not needan explicit expression for χ in the following. III. SOLVATING A PARTICLE
We solvate a particle expelling the fluid from the vol-ume V the particle occupies. For the clarity of presen-tation, we consider a spherical particle. Then rotationaland translational diffusion decouple and in the followingwe focus on the translational diffusion. The particle isdescribed by its instantaneous velocity v . By choosingthe velocity correlations Eq. (3) we assume a time scaleseparation between the relaxation time of the fluid andthe time scale on which the solvated particle diffuses.From the perspective of the solute it thus appears as ifthe solvent responds instantaneously with a mean flowprofile (cid:104) u ( r ) (cid:105) v to a change of v .To calculate this flow profile, we exploit the time scale separation through fixing v and consider the generatingfunction Q [ φ ] ≡ (cid:90) [d u ] (cid:90) [d ψ ] e −S +i C + (cid:82) d t (cid:82) d r φ ( r ,t ) · u ( r ,t ) (6)with conjugate field φ ( r , t ). We implement the “no-slip”or “stick” boundary condition such that the fluid layeron the particle surface has the same velocity as the mov-ing particle. The boundary condition gives rise to theconstraint (cid:89) t δ ( u ( r , t ) − v ( t )) = (cid:90) [d ψ ] e i C [ ψ ] in Eq. (6) with functional C [ ψ , u | v ] ≡ (cid:90) d t (cid:90) d r [ u ( r , t ) − v ( t )] · ψ ( r , t ) . (7)The auxiliary function ψ ( r , t ) is non-zero only on theparticle surface δV and zero elsewhere.We evaluate the path integrals in Eq. (6) in two steps.First, integration over the velocity fluctuations results in Q [ φ ] = (cid:90) [d ψ ] exp (cid:26) − (cid:90) d t ψ ◦ χ s ◦ ψ + i (cid:90) d t ψ ◦ b + 12 (cid:90) d t (cid:90) d r d r (cid:48) φ ( r , t ) · χ ( r , r (cid:48) ) · φ ( r (cid:48) , t ) (cid:27) . (8)To ease the notational burden, we define the operation g ◦ h ≡ (cid:90) δV d r g T ( r ) h ( r ) (9)as the integral over all positions that are elements of theparticle surface. The matrix χ s quantifies fluid velocitycorrelations in the absence of, but on the interface thefluid would share with, the solvated particle. The linearterm in Eq. (8) couples to the vector b ( r , t ) ≡ (cid:90) d r (cid:48) χ ( r , r (cid:48) ) · φ ( r (cid:48) , t ) − v ( t ) , (10)where r ∈ δV . Performing the integration over the auxil-iary vector field ψ we finally obtain the generating func-tion Q [ φ ] = N exp (cid:26) − (cid:90) d t b ◦ χ − ◦ b + 12 (cid:90) d t (cid:90) d r d r (cid:48) φ ( r , t ) · χ ( r , r (cid:48) ) · φ ( r (cid:48) , t ) (cid:27) (11)with (irrelevant) prefactor N due to the functional deter-minant of χ s . The inverse matrix is determined through χ s ( r , r (cid:48)(cid:48) ) ◦ χ − ( r (cid:48)(cid:48) , r (cid:48) ) = δ ( r − r (cid:48) ) , r , r (cid:48) ∈ δV, (12) i.e., the inversion is carried out on the subspace definedby the particle surface.From the generating function Eq. (11) it is straightfor-ward to calculate the mean flow profile as the functionalderivative (cid:104) u i ( r , t ) (cid:105) v = δ ln Qδφ i ( r , t ) (cid:12)(cid:12)(cid:12)(cid:12) φ =0 (13)with respect to the conjugate field. Using δb i ( r (cid:48) , t (cid:48) ) δφ j ( r , t ) = [ χ ( r (cid:48) , r )] ij δ ( t − t (cid:48) ) (14)together with the symmetry property χ ( r (cid:48) , r ) = χ ( r , r (cid:48) ) we obtain (cid:104) u ( r ) (cid:105) v = χ ( r , r (cid:48) ) ◦ χ − ( r (cid:48) , r (cid:48)(cid:48) ) ◦ v . (15)For r ∈ δV , we can use Eq. (12) to show that (cid:104) u ( r ) (cid:105) v = v as required. Fig. 1(a) shows schematically how theimposed surface velocity generates the flow at r in thesolvent through a concatenation of the tensors χ − and χ .The presence of the solute not only causes a flow butalso alters the solvent velocity fluctuations. The correla- r r r v u ( r ) r r ! ✓ ✓ R p cos ✓ ) (a) (b) FIG. 1: (a) Schematic representation of Eq. (15): The flow u around the spherical particle at r is connected to the flowon the surface (given by the particle velocity v ) through aconcatenation of the tensors χ − and χ . (b) Coordinatesystem and symbols used in the appendix: r and r (cid:48) are twopoints on the particle surface and ω is the angular velocity. tions are calculated as[ χ ( r , r (cid:48) )] ij = δ ln Qδφ i ( r , t ) δφ j ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12)(cid:12) φ =0 (16)and read χ ( r , r (cid:48) ) = (cid:104) δ u ( r ) δ u T ( r (cid:48) ) (cid:105) v = χ ( r , r (cid:48) ) − χ ( r , r (cid:48)(cid:48) ) ◦ χ − ( r (cid:48)(cid:48) , r (cid:48)(cid:48)(cid:48) ) ◦ χ ( r (cid:48)(cid:48)(cid:48) , r (cid:48) ) (17)for δ u ( r ) ≡ u ( r ) − (cid:104) u ( r ) (cid:105) v . For either r ∈ δV or r (cid:48) ∈ δV ,i.e., directly on the particle surface, the correlations van-ish. Far away from the particle in the bulk they becomethose of the pure solvent, χ = χ . IV. BROWNIAN MOTION
Comparing the result Eq. (15) with the linear responserelation Eq. (4), we see that it corresponds to a forcedensity that is confined to the solute-solvent interfaceand given by f ( r ) = 2 T χ − ( r , r (cid:48) ) ◦ v , r ∈ δV. (18)The total force exerted by the particle follows from inte-grating this force density over the particle surface, F = ◦ f = 2 T ( ◦ χ − ◦ ) · v ≡ Γ · v . (19)This is a linear relation between force and velocity, allow-ing us to identify Γ as the friction tensor. In appendix A, Γ is calculated explicitly from Eq. (19) for a sphericalparticle.So far, we have treated the particle velocity v as given.Inspecting Eq. (6), we see that for φ = 0 the generatingfunction reduces to the path probability to observe spon-taneous solvent velocity fluctuations that are compatiblewith the boundary condition of a uniform velocity on the solute surface. Hence, P [ v ( t )] ∼ Q [ φ = 0] is the pathweight of a history of particle velocities with P [ v ( t )] ≡ exp (cid:26) − (cid:90) d t v ( t ) · D − · v ( t ) (cid:27) , (20)which corresponds to the path weight of a free, over-damped Brownian particle. The diffusion tensor D − ≡ ◦ χ − ◦ ) = Γ /T is given by the well-known Einstein relation connectingdiffusion to friction through the fluid temperature. V. CONCLUSIONS
The formalism introduced here is not restricted tospherical particles but can be used to obtain the fric-tion tensors for arbitrarily shaped objects, e.g., colloidalclusters with complex shapes [13]. Its practical use issomewhat hampered by the difficulties to determine χ − explicitly, which amounts to the inversion of χ on thetwo dimensional sub-manifold of the solute surface. Ex-cept for simple shapes (cf. appendix A) this has to bedone numerically, possibly through an expansion of χ s into a suitable matrix basis. Rotational diffusion, andits coupling to translational diffusion, can be treatedstraightforwardly as demonstrated in appendix B. An-other advantage of the presented formalism is that weare not restricted to use the correlations Eq. (5) holdingfor an unbounded solvent. For example, one might calcu-late the pure solvent correlations χ ( r , r (cid:48) ) in some geome-try numerically (e.g., using fluctuating lattice Boltzmannsimulations [14]) and use these correlations to obtain thediffusion coefficients of differently shaped objects movingin this geometry. Acknowledgments
I thank David Chandler for introducing me to Gaussianfield theory. I acknowledge financial support from theAlexander-von-Humboldt foundation.
Appendix A: Friction tensor of a spherical particle
To demonstrate the validity of the present formalism,we explicitly calculate the friction tensor for a sphericalparticle with radius R via Eq. (19). We start by calcu-lating the matrix X ≡ ◦ χ s = (cid:90) δV d r χ ( r − r (cid:48) ) , which is the integral of the free correlations Eq. (5) overthe surface of the sphere with r (cid:48) hold fixed. Without lossof generality, we place the particle center at the originand use spherical coordinates with r (cid:48) pointing along the z -axis, see Fig. 1(b). The off-diagonal components vanishafter integration over the azimuth angle. Substituting z = cos θ , the diagonal components read X xx = X yy = T R η (cid:90) − d z (cid:112) − z ) (cid:20) − z − z ) (cid:21) ,X zz = T R η (cid:90) − d z (1 − z ) (cid:112) − z ) , and therefore X = 4 T R η . As expected, this matrix does not depend on the vector r (cid:48) due to the spherical symmetry. Following Eq. (12), wefind χ − ◦ = X − and thus recover Stokes’ expression Γ = 2 T ( ◦ χ − ◦ ) = 2 T πR X − = 6 πηR for the friction tensor of a sphere. Appendix B: Rotational diffusion
In order to include rotational diffusion, we have tomodify the boundary condition. The fluid velocity onthe surface now reads u ( r , t ) = v ( t ) + ω ( t ) × r , r ∈ δV, where ω = ω e is the angular velocity with speed ω aboutan axis given by the normalized vector e . Repeatingthe steps leading to Eq. (20), we obtain the path weight P ∼ e − A with stochastic action A = 12 (cid:90) d t [ v + ω × r ] ◦ χ − ( r , r (cid:48) ) ◦ [ v + ω × r (cid:48) ] . We now calculate the rotational diffusion coefficientfor a spherical particle with radius R . We use a strategysimilar to the previous section by multiplying Eq. (12) bythe vector e × r from the left, and e × r (cid:48) from the rightside followed by integrations, (cid:90) δV d r L ( r ) · L − ( r ) = (cid:90) δV d r ( e × r ) · ( e × r ) = 8 π R . Here, we have defined the vector field L ( r (cid:48) ) ≡ (cid:90) δV d r χ ( r (cid:48) − r ) · ( e × r ) = 23 T Rη e × r (cid:48) . It is convenient to calculate the integral using sphericalcoordinates as sketched in Fig. 1(b), where the z -axispoints along r (cid:48) . For the inverse field we thus find L − ( r (cid:48) ) ≡ (cid:90) δV d r χ − ( r (cid:48) , r ) · ( e × r ) = ηT R e × r (cid:48) . Using this vector, the stochastic action can be written A = (cid:90) d t (cid:26) v · D − · v + v · ( ◦ L − ) ω + Γ r T ω (cid:27) . Clearly, ◦ L − = 0, i.e., translational and rotationaldiffusion decouple. The rotational diffusion coefficientreads Γ r = 2 T (cid:90) δV d r ( e × r ) · L − ( r ) = 8 πηR as expected. [1] A. Einstein, Ann. Phys. , 549 (1905).[2] L. Landau and E. Lifshitz, Fluid mechanics , vol. 6 (Perg-amon Press, Oxford, 1987), 2nd ed.[3] E. H. Hauge and A. Martin-L¨of, J. Stat. Phys. , 259(1973).[4] M. Schindler, Chem. Phys. , 327 (2010).[5] D. Chandler, Phys. Rev. E , 2898 (1993).[6] X. Song, D. Chandler, and R. A. Marcus, J. Phys. Chem. , 11954 (1996).[7] K. Lum, D. Chandler, and J. D. Weeks, J. Phys. Chem.B , 4570 (1999).[8] A. Widom, Phys. Rev. A , 1394 (1971).[9] T. Franosch, M. Grimm, M. Belushkin, F. M. Mor, G. Foffi, L. Forro, and S. Jeney, Nature , 85 (2011).[10] F. Detcheverry and L. Bocquet, Phys. Rev. Lett. ,024501 (2012).[11] R. Kubo, M. Toda, and N. Hashitsume, StatisticalPhysics II (Springer-Verlag, Berlin, 1991), 2nd ed.[12] J. K. G. Dhont,
An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996).[13] D. J. Kraft, R. Wittkowski, B. ten Hagen, K. V. Edmond,D. J. Pine, and H. L¨owen, arXiv:1305.1253 (2013).[14] R. Adhikari, K. Stratford, M. E. Cates, and A. J. Wagner,Europhys. Lett.71