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Geometry and Di usion. Comment on \Tracer Di usion in a Dislocated LamellarSystem"1, 2, yDoru Constantin and Robert Ho lyst1Laboratoire de Physique de l’ENS de Lyon, 46 Allee d’Italie, 69364 Lyon Cedex 07, France2Institute of Physical Chemistry PAS, Dept. III, Kasprzaka 44/52, 01-224 Warsaw, PolandPACS numbers: 61.30.Jf, 66.30.JtIn a recent Letter, Gurarie and Lobkovsky [1] a rm shown that :that the movement of tracer particles across the layers2pin a lamellar system with screw dislocations is super- D = D (2)? 2 2(2r) + pdi usiv e (ballistic when dislocations of one sign are in ex-2cess and with normal displacement hz (t)i / t log t whenClearly, the particle always exhibits normal di usionthe average dislocation charge is zero). We argue that,along z, with a di usion constant D D, even when?when surface geometry is properly taken into account,r ! 0 (incidentally, in this limit one obtains the \pipethe normal displacement is di usiv e and the associateddi usion" along the dislocation core). On the contrary, ifdi usion constant D is always smaller than the in-plane? 2one neglects the p term in the denominator of equationt D.(2) (as in reference [1]), the ratio D =D arti cially di-?Particle di usion on a curved surface extending indef-verges with vanishing r. This example is very relevant toinitely in all three directions in space is characterized by2 2 2 the discussion in [1] because, as the authors acknowledge,a displacement hx +y +z i = D t, where ...

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Geometry and Diffusion.Comment on “Tracer Diffusion in a Dislocated Lamellar System”

1,∗2,† DoruConstantinandRobertHolyst 1 LaboratoiredePhysiquedel’ENSdeLyon,46All´eed’Italie,69364LyonCedex07,France 2 Institute of Physical Chemistry PAS, Dept.III, Kasprzaka 44/52, 01-224 Warsaw, Poland

PACS numbers: 61.30.Jf, 66.30.Jt

In a recent Letter, Gurarie and Lobkovsky [1] affirmshown that : that the movement of tracer particles across the layers 2 p in a lamellar system with screw dislocations is super- D ⊥ =D (2) 2 2 (2πr) +p diffusive (ballistic when dislocations of one sign are in ex- 2 cess and with normal displacementhz(t)i ∝tlogtwhen Clearly, the particle always exhibits normal diffusion the average dislocation charge is zero).We argue that, alongz, with a diffusion constantD ⊥ ≤D, even when when surface geometry is properly taken into account, r→0 (incidentally, in this limit one obtains the “pipe the normal displacement is diffusive and the associated diffusion” along the dislocation core).On the contrary, if diffusion constantD ⊥ is always smaller than the in-plane 2 one neglects thepterm in the denominator of equation diffusion constantD. (2) (as in reference [1]), the ratioD ⊥ /Dartificially di- Particle diffusion on a curved surface extending indef- verges with vanishingr. Thisexample is very relevant to initely in all three directions in space is characterized by 2 2 2 the discussion in [1] because, as the authors acknowledge, a displacementhx+y+zi=D eff t, where the effective trajectories tightly wound around the core dominate the diffusion coefficientD eff ≤D, withDthe local diffusion statistics. Inour opinion, this is the cause of the diver- coefficient in the plane of the surface [2].Equality is gence in their equation (11), and not the fact that the achieved when the surface has zero mean curvature ev- trajectories enclose several dislocations, as they imply. erywhere (minimal surface) [3].Thus, the displacement along any particular direction (sayz) has an upper bound z 2 hz(t)i ≤Dt(1) This can be shown by taking a cutoff at a finite distance from the core (as done in [1]) and introducing reflecting s(t) boundary conditions on a helix at distanceafrom the p core. Thesurface is smooth and one can perfectly define the diffusion equation locally.As discussed in [3], the r diffusion tensor is diagonal in a local reference frame, of valuesDin the plane of the surface and 0 along the nor- FIG. 1:Helical trajectory of radiusrand pitchparound a mal to the surface [4].Thus, diffusion cannot be faster screw dislocation (the core is along thezpositionaxis). The thanDin any particular space directionzˆ, as one can see of the particle is given by the curvilinear coordinates(t), with by averagingzˆDzˆ over an arbitrary surface.The limit is s= 0 in thez= 0 plane. given by thelocalproperties of the surface and does not depend on its detailed overall configuration.In refer- R. H. was supported by KBN grant 2P03B00923. ence [1], the authors artificially decouple the coordinates so that the particle diffuses in the (x, y) plane, but the “cost” associated with displacement alongzis not taken into account, leading to the violation of (1). ∗ To illustrate these observations on a very simple case, Electronic address:[e-mail] † let us consider the case of a tracer particle that is con- Electronic address:[e-mail] strained to remain at a distancerfrom the dislocation [1] Victor Gurarie and Alexander E. Lobkovsky, Phys. Rev. Lett. , 178301 (2002). core (FIG. 1) and consequently diffuses along a helix with [2]R.Holyst,D.Plewczyn´ski,A.AksimentievandK.Bur- pitchpequal to the lamellar spacing. dzy, Phys. Rev. E60ewPly´cz29(93109D.,)..iksnRdna Suppose the particle starts atz= 0 at timet= 0; the Holyst,J.Chem.Phys.113, 9920 (2000). statistical distribution of its curvilinear coordinate along [3]D.M.AndersonandH.Wennerstro¨m,J.Phys.Chem.94, the helixs(t) is then described by :hΔs(t)i= 0 and 8683 (1990) (see end of section III). 2 hΔs(t)i= 2Dt. When itmoves one layer up (toz=p), the factor 3[4] Note/2 between our definition ofDand their 2 2 its coordinate iss= (2πr) +pis then readily. It definition ofD 0 .