aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Knudsen Diffusion in Silicon Nanochannels
Simon Gruener and Patrick Huber ∗ Faculty of Physics and Mechatronics Engineering,Saarland University, D-66041 Saarbr¨ucken, Germany
Measurements on helium and argon gas flow through an array of parallel, linear channels of 12 nmdiameter and 200 µ m length in a single crystalline silicon membrane reveal a Knudsen diffusion typetransport from 10 to 10 in Knudsen number Kn. The classic scaling prediction for the transportdiffusion coefficient on temperature and mass of diffusing species, D He ∝ √ T is confirmed over a T -range from 40 K to 300 K for He and for the ratio of D He /D Ar ∝ p m Ar /m He . Deviations of thechannels from a cylindrical form, resolved with electron microscopy down to subnanometer scales,quantitatively account for a reduced diffusivity as compared to Knudsen diffusion in ideal tubularchannels. The membrane permeation experiments are described over 10 orders of magnitude in Kn,encompassing the transition flow regime, by the unified flow model of Beskok and Karniadakis. PACS numbers: 47.45.n, 47.61.-k, 47.56.+r
Knudsen diffusion (KD) refers to a gas transportregime where the mean free path between particle-particle collisions λ is significantly larger than at leastone characteristic spatial dimension d of the system con-sidered [1]. By virtue of the negligible mutual particlecollisions transport in such systems takes place via a se-ries of free flights and statistical flight direction changesafter collisions with the confining walls. Because of thedependency of λ on pressure p and temperature T givenby the kinetic theory of gases, λ ∝ T /p , this transportregime is only observable in macroscopic systems at verylow p or elevated T . By contrast, for transport in spa-tially nanoconfined systems with d = O (10 nm), theKnudsen number Kn = λ/d , which quantifies the gasrarefaction, is larger than 1 even for ambient pressuresand temperatures, e.g. λ (He) = 118 nm at p = 1 barand T = 297 K. Thus, for many processes involvinggas transport in restricted geometries, e.g. gas catalysisand storage[2] or equilibration phenomena via gas flow inmeso- and nanopores[3, 4], KD plays a crucial role.Albeit the phenomenon of KD has been known for al-most a hundred years, details on its microscopic mecha-nisms are still at debate, in particular how wall rough-ness down to atomic scales and particle-wall interactionparameters affect the statistics of the diffusion processand thus the value of the KD transport coefficient D [5].In the past, a comparison between theory and experi-ment has often been hampered by complex pore networktopologies, tortuous transport paths and poorly charac-terized pore wall structures in available nanoporous ma-trices. Nowadays, membranes with better defined geome-tries, such as carbon nanotube bundles[6, 7], porous alu-mina [8], and tailored pores in mesoporous silicon [3, 9],allow us to address such fundamental questions in moredetail. A better understanding of, and more in-depth in-formation on, transport in such systems is not only ofacademic interest, but also of importance for the archi-tecture and functionality in the emerging field of nano-electromechanical and nanofluidic systems [10, 11]. FIG. 1: (color online) He pressure relaxations in R1 and R2at T = 297 K for starting pressures p s = 0 . τ as determined frommeasurements without (squares) and with SiNC membrane(circles) versus Knudsen number in the capillaries Kn c (bot-tom axis) and in the SiNCs Kn (top axis). The lines in (a),(b) and (c) represent calculated p ( t ) and τ ( Kn ) values. Insetsin (c): TEM cross section views of two SiNCs in comparisonwith circular perimeters of 12 nm diameter (dashed lines). In this Letter, we report the first rigorous experimen-tal study on transport of simple rare gases, i.e. He andAr through an array of parallel aligned, linear channelswith ∼
12 nm diameter and 200 µ m length in single crys-talline Si over a wide temperature (40 K < T <
300 K)and Kn -range (10 < Kn < ). We explore the clas-sical scaling predictions given by Knudsen for D on T ,mass of diffusing species, m , and explore its dependencyn Kn . Attention shall also be paid to the morphologyof the channel walls, resolved with transmission electronmicroscopy (TEM), and how it affects the diffusion.Linear, non-interconnected channels oriented along the h i Si crystallographic direction (SiNCs) are electro-chemically etched into the surface of a Si (100) wafer[12]. After the channels reach the desired length L of200 ± µ m, the anodization current is increased by afactor of 10 with the result that the SiNC array is de-tached from the bulk wafer [13]. The crystalline Si wallscall for irregular channel perimeters as one can see inthe TEM pictures of Fig. 1(c), recorded from a mem-brane part Ar ion milled to ∼ ∼
12 nm, in ac-cordance with the analysis of an Ar sorption experimentat T = 86 K, which yields additionally a SiNCs densityof Φ = (1 . ± . · cm − .Our experimental setup consists of a copper cell withan inlet and outlet opening [14]. Inlet and outlet areconnected via stainless steel capillaries of radius r c =0 . l c = 70 cm with two gas reservoirs,R1 and R2, of an all-metal gas handling. Additionallytwo pneumatic valves V1 and V2 are used to open andclose the connections between the sample cell and R1 andthe sample cell and R2. Four thermostatted capacitivepressure gauges allow us to measure the gas pressuresin R1 and R2, p and p , resp., over a wide pressurerange (5 · − mbar < p < − mbar. The cell is mounted in a closed-cycle Hecryostat to control the temperature between 40 K and300 K with an accuracy of 1 mK.Our goal of studying gas transport dynamics over alarge Kn range necessitates a thorough understanding ofthe intrinsic flow characteristics of our apparatus. There-fore, we start with measurements on He flow throughthe macroscopic capillaries and the empty sample cell atroom temperature T = 297 K. We record the equilibra-tion of p ( t ) and p ( t ) towards a pressure p eq as a functionof time t after initial conditions, p = p s > p = 0 mbarat t = 0 s. In Fig. 1 (a) p ( t ) /p s and p ( t ) /p s are plottedfor starting pressures of p s = 0 . p eq = 0 . · p s . Thevalue 0.4 is dictated by the volume ratio of R1 to R2.From the p ( t )-curves we derive characteristic relaxationtimes τ according to the recipe p ( t = τ ) − p ( t = τ ) ≡ / · p s . It is understood that p is monotonically decreas-ing downstream from R1 to R2. In order to quantify thegas rarefaction we resort, therefore, to a calculation of amean Knudsen number Kn c = λ ( p ) /r c in the capillariesassuming a mean pressure, p = p eq . This simplificationis justified by the analysis provided below and by theo-retical studies which indicate differences of less than 1%between flow rates calculated with an exact and an av-eraged treatment for Kn [10, 15]. In Fig. 1 (c) we plotmeasured τ values versus Kn c corresponding to a p s vari- ation from 0 .
01 to 100 mbar. For Kn c < . τ increaseswith increasing Kn c . In an intermediate Kn c range,0 . < Kn c < .
6, we observe a cross-over regime towardsa saturation plateau with τ ∼
140 s that extends to thelargest Kn c studied. These changes in the p relaxationand hence flow dynamics are reminiscent of the threedistinct transport regimes known to occur for gases as afunction of their rarefaction [10, 16]: For Kn c < . τ due to the 1 /Kn -scaling of the particle num-ber density in gas flows. For Kn c > D Hec dependent, however a Kn c independentparticle flow rate,˙ n Kn = π r l c p o − p i k B T D
Hec , (1)which is responsible for the plateau in τ for the larger Kn c investigated. In Eq. 1 p i , p o refers to the inletand outlet pressure of the capillary considered, resp.,and k B to the Boltzmann factor. In the intermediate Kn -range, the interparticle collisions occur as often asparticle-wall collisions which gives rise to the cross-overbehavior found for τ .We elucidate this behavior in more detail by divid-ing the flow path within our apparatus into two flowsegments (up- and downstream capillary) and calculatethe particle number changes along the flow path andthe resulting p ( t ), p ( t ) with a 1 ms resolution usingthe unified flow model of Beskok and Karniadakis (BK-model)[10, 17, 18] and a local Kn -number for each flowsegment, Kn l = Kn (( p i + p o ) / n BK = πr ( p − p )16 l c k B T α Kn l µ ( T ) (cid:18) Kn l − b Kn l (cid:19) . (2)Eq. 2 comprises a Hagen-Poiseuille term, a term whichtreats the transition of the transport coefficient fromcontinuum-like, i.e. the bare dynamic viscosity µ ( T ),to the KD transport coefficient, D Hec with α =2 . /π tan − ( α Kn β ), and a generalized velocity slipterm, which is second-order accurate in Kn in the slipand early transition flow regimes ( Kn < . Kn → Kn ≫
1. As veri-fied by comparison of the BK-model with Direct Simula-tion Monte Carlo and solutions of the Boltzmann equa-tion, a choice of b = − Kn -range, including the transition flow regime. Afteroptimizing the free parameters in Eq. 2, α and β , we ar-rive at the p ( t )- and τ ( Kn c )-curves depicted in Fig. 1 (a)and (c), resp. The good agreement of the BK-model pre-ictions with our measurements is evident and the ex-tracted parameters α = 10 ± . β = 0 . ± . D Hec = 0 . ± .
06 m /s) agree with BK-modelling ofHe gas flow as a function of rarefaction [10, 19]. An anal-ogous analysis for Ar gas flow yields α = 1 . ± .
05 and β = 0 . ± .
02 ( D Arc = 0 . ± .
02 m /s). FIG. 2: (color online) Pressure relaxation time τ for He (cir-cles) and Ar (triangles) measured with built-in SiNC mem-brane at T = 297 K versus Knudsen number in the SiNCs Kn (bottom axis) and in the capillaries Kn c (top axis). The linesrepresent calculated τ ( Kn )-curves. Inset: TEM cross sectionof a SiNC in comparison with a circular channel perimeterof 12 nm diameter (dashed line). The SiNC’s perimeter ishighlighted by a line. Having achieved a detailed understanding of the intrin-sic flow characteristics of our apparatus, we can now turnto measurements with the SiNC membrane. The mem-brane is epoxy-sealed in a copper ring [14] and special at-tention is paid to a careful determination of the accessiblemembrane area, A = 0 . ± .
016 cm in order to allowfor a reliable determination of the number of SiNCs in-serted into the flow path, N = Φ · A = (11 . ± · . Asexpected the pressure equilibration is significantly sloweddown after installing the membrane - compare panel (a)and (b) in Fig. 1. Choosing selected p s within the range0.005 mbar to 100 mbar, which corresponds to a variationof Kn in the SiNCs from 10 to 10 , we find an increasein τ of ∼
170 s, when compared to the empty cell mea-surements - see Fig. 1 (c). From the bare offset in τ for Kn c >
1, we could calculate D He in the SiNCs. We are,however, interested in the D He behavior in a wide Kn -range, therefore we modify our flow model by insertinga segment with KD transport mechanism characteristicof N tubular channels with d = 12 nm and L = 200 µ min between the capillary flow segments. Adjusting thesingle free parameter in our simulation, the value of theHe diffusion coefficient D He in a single SiNC, we arriveat the p ( t )- and τ ( Kn )-curves presented as solid lines inFig. 1 (b) and (c), resp. The agreement with our mea-surement is excellent and the model yields a Kn inde-pendent D He = 3 . ± . /s. It is worthwhile to compare this value with Knudsen’sprediction for D . In his seminal paper he derives an ex-pression for D with two contributions, a factor G , charac-teristic of the KD effectivity of the channel’s shape, anda factor solely determined by the mean thermal velocityof the particles v given by the kinetic theory of gases, D = 13 G v = 13 G r k B Tπm . (3)Interestingly, by an analysis of the number of particlescrossing a given section of a channel in unit time aftercompletely diffuse reflection from an arbitrary element ofwall surface and while assuming an equal collision acces-sibility of all surface elements, Knudsen derived an ana-lytical expression for G for a channel of arbitrary shape.Knudsen’s second assumption is, however, only strictlyvalid for circular channel cross-sections, as pointed outby v. Smoluchowski [1] and elaborated for fractal porewall morphologies by Coppens and Froment [5]. Giventhe roughly circular SiNCs’ cross section shapes, we nev-ertheless resort to Knudsen’s formula, here quoted nor-malized to the KD form factor for a perfect cylinder:1 G = 14 L Z L o ( l ) A ( l ) dl. (4)The integral in Eq. 4 depends on the ratio of perimeterlength o ( l ) and cross sectional area A ( l ) along the chan-nel’s long axis direction l , only. This ratio is optimizedby a circle, accordingly the most effective KD channelshape is a cylinder, provided one calls for a fixed A alongthe channel. Eq. 4 yields due to our normalization just G = d and we would expect a D He of 5 mm /s for He KDin a SiNC, if it were a cylindrical channel of d = 12 nm.A value which is ∼
30% larger than our measured one.If one recalls the TEM pictures, which clearly indicatenon-circular cross-sections this finding is not surprising.In fact these pictures deliver precisely the informationneeded for an estimation of the influence of the SiNC’sirregularities on the KD dynamics. We determine the ra-tio o/A of 20 SiNC cross sections, and therefrom valuesof G . The values of the SiNC in Fig. 1 (top), (bottom)and Fig. 2 corresponds to G s of 9.1 nm, 10.3 nm, and10.6 nm, resp. Tacitly assuming that the irregularitiesexhibited on the perimeter are of similar type as alongthe channel’s long axis we take an ensemble average overthe 20 G values and arrive at a mean G of 9.9 nm, whichyields a D He of 4 . /s, a value which agrees withinthe error margins with our measured one.To further explore the KD transport dynamics in theSiNCs we now focus on the m and T dependency of D . In Fig. 2 the τ ( Kn ) curve recorded for Ar and Heat T = 297 K are presented. Both exhibit a similarform, the one of Ar is, however, shifted up markedlytowards larger τ . Our computer model can quantita-tively account for this slowed-down dynamics by a fac-tor 2 . ± .
25 decrease in D as compared to the Heeasurements, which confirms the prediction of Eq. 3, D He /D Ar = p m Ar /m He = 3 .
16. For the exploration ofthe T behavior we choose again He as working fluid due toits negligible physisorption tendency, even at low T . Weperform measurements at selected T s from 297 K down to40 K, shown in Fig. 3. We again perform computer calcu-lations assuming KD in the SiNC array superimposed tothe transport in the supply capillaries, presented as linesin Fig. 3, and optimize the single free parameter D ( T ) inorder to match the observed τ ( Kn, T ) behavior. Despiteunresolvable deviations at larger Kn c and decreasing T ,which we attribute to thermal creep effects, characteristicof T gradients along the flow path [10], we find, in agree-ment with the experiment, increasingly faster dynamicswith decreasing T . Note, this counter-intuitive findingfor a diffusion process results from the 1 /T scaling inEq. 1, which reflects the T -dependency of the particlenumber density in gas flows. By contrast, the D He ( T ),determined by our simulations, scales in excellent agree-ment with √ T , see inset in Fig. 3, confirming Knudsen’sclassic result down to the lowest T investigated. FIG. 3: (color online) Pressure relaxation time τ for He gasflow through SiNC membrane (symbols) and simulated τ (Kn)(lines) versus Knudsen number in the SiNC membrane Kn (bottom axis) and in the supply capillaries Kn c (top axis) atselected temperatures T . Inset: He diffusion coefficient D He in SiNCs in comparison with the √ T prediction of Eq. 3 (line)plotted versus √ T . We find no hints of anomalous fast KD here, as was re-cently reported for a bundle of linear carbon nanotubes[7] and explained by an highly increased fraction of spec-ularly reflected particles upon wall collisions [1]. The al-tered collision statistics was attributed to the crystallinestructure and the atomical smoothness of the nanotubewalls. The SiNC walls are also crystalline, however, notatomical flat, as can be seen from our TEM analysis.Along with the formation of a native oxide layer, typicalof Si surfaces, which renders the near surface structureamorphous, silica like, this, presumably, accounts for thenormal KD observed here contrasting the one found forthe graphitic walls of carbon nanotubes.We presented here the first detailed study of gas trans- port in linear SiNCs. Our conclusions are drawn from acorrect treatment of gas flow over 10 orders of magnitudein gas rarefaction, which is, to the best of our knowledge,the largest Kn range ever explored experimentally. Thecharacteristic properties of KD, an independency of D on Kn , its scaling predictions on m , T , and on details ofthe channel’s structure, resolved with sub-nm resolution,are clearly exhibited by our measurements.We thank A. Beskok for helpful discussions and theGerman Research Foundation (DFG) for support withinthe priority program 1164, Nano- & Microfluidics (Hu850/2). ∗ E-mail: [email protected][1] M. Knudsen, Ann. Phys. (Leipzig) , 75 (1909), M. v.Smoluchowski, ibid. , 1559 (1910).[2] J. K¨arger and D. Ruthven,
Diffusion in Zeolites and Mi-croporous Solids (Wiley & Sons, New York, 1992), S. M.Auerbach, Int. Rev. Phys. Chem. , 155 (2000).[3] D. Wallacher et al. , Phys. Rev. Lett. , 195704 (2004).[4] P. Huber and K. Knorr, Phys. Rev. B , 12657 (1999);R. Paul and H. Rieger, J. Chem. Phys. , 024708(2005); R. Valiullin et al. , Nature , 965 (2006).[5] M. O. Coppens and G. F. Froment, Fractals , 807(1995); A.I. Skoulidas et al. , Phys. Rev. Lett. , 185901(2002); G. Arya, H.C. Chang, and E.J. Maginn, ibid. ,026102 (2003); S. Russ et al. , Phys. Rev. E , 030101(R)(2005).[6] B. J. Hinds et al. , Science , 62 (2004).[7] J. K. Holt et al. , Science , 1034 (2006).[8] S. Roy et al. , J. Appl. Phys. , 4870 (2003); K.J.Alvine et al. . Phys. Rev. Lett. , 175503 (2006).[9] C.C. Striemer et al. , Nature , 749 (2007).[10] G. Karniadakis, A. Beskok, and N. Aluru, Microflowsand Nanoflows (Springer, 2005).[11] H.A. Stone, A.D. Stroock, and A. Ajdari, Ann. Rev.Fluid Mech. , 381 (2004); P. Huber et al. , Eur. Phys.J. Spec. Top. , 101 (2007); M. Whitby and N. Quirke,Nature Nanotechnology , 87 (2007).[12] V. Lehmann and U. G¨osele, Appl. Phys. Lett. , 856(1991); V. Lehmann, R. Stengl, and A. Luigart, Mat. Sc.Eng. B , 11 (2000).[13] A. Henschel et al. , Phys. Rev. E , 021607 (2007); P.Kumar et al. , J. Appl. Phys. , 2933 (1994).[16] P. Tabeling Introduction to Microfluidics (Oxford Uni-versity Press, New York, 2005).[17] A. Beskok and G. Karniadakis, Nanosc. Microsc. Ther-mophys. Eng. , 43 (1999).[18] S. Gr¨uner, Diploma thesis, Saarland University (2006).[19] S. Tison, Vacuum44