Int. J. Mol. Si. 2014, 15, 12714-12730; doi:10.3390/ijms150712714 Artile OPEN ACCESS Interntionl Journl of Moleulr Sienes ISSN 1422-0067 www.mdpi.om/journl/ijms Study on the Chrteristis of Gs Moleulr Men Free Pth in Nnopores y Moleulr Dynmis Simultions Qixin Liu 1, * nd Zhiyong Ci 2 1 2 Shool of Civil Engineering nd Arhiteture, Chongqing University of Siene nd Tehnology, Chongqing 401331, Chin Shool of Sfety Engineering, Chongqing University of Siene nd Tehnology, Chongqing 401331, Chin; E-Mil: zy677656@gmil.om * Author to whom orrespondene should e ddressed; E-Mil: lqx@qust.edu.n; Tel.: +86-23-6502-2325. Reeived: 12 My 2014; in revised form: 2 July 2014 / Aepted: 7 July 2014 / Pulished: 18 July 2014 Astrt: This pper presents studies on the hrteristis of gs moleulr men free pth in nnopores y moleulr dynmis simultion. Our study results indite tht the men free pth of ll moleules in nnopores depend on oth the rdius of the nnopore nd the gs-solid intertion strength. Besides men free pth of ll moleules in the nnopore, this pper highlights the gs moleulr men free pth t different positions of the nnopore nd the nisotropy of the gs moleulr men free pth t nnopores. The moleulr men free pth vries with the moleule s distne from the enter of the nnopore. The lest vlue of the men free pth ours t the wll surfe of the nnopore. The present pper found tht the gs moleulr men free pth is nisotropi when gs is onfined in nnopores. The rdil gs moleulr men free pth is muh smller thn the men free pth inluding ll moleulr ollisions ouring in three diretions. Our study results lso indite tht when gs is onfined in nnopores the gs moleule numer density does not ffet the gs moleulr men free pth in the sme wy s it does for the gs in unounded spe. These study results my ring new insights into understnding the gs flow s hrteristi t nnosle. Keywords: nnopore; gs moleulr men free pth; Knudsen numer; moleulr dynmis simultion
Int. J. Mol. Si. 2014, 15 12715 1. Introdution For nnosle gs flows, the flow flux is lrger thn the vlue sed on lssil fluid mehnis theory. Previous investigtions [1,2] found tht wter or gs flows muh fster inside ron nnotue thn in lssil mrosle tue. Due to inresed interest in nnofluidis nd nnoeletromehnil devies [3], these unusul flow phenomen hve ttrted muh interest from reserhers. When the system size eomes omprle to the men free pth of the moleules, the ontinuum ssumption should rek down nd the lssil Nvier-Stokes (N-S) equtions with no-slip oundry onditions nnot e diretly pplied [4]. The Knudsen numer is defined s Kn = λ / H, where λ is the moleulr men free pth nd H is the representtive length of the domin. The Knudsen numer is widely used to divide the gs flow into four regimes: Kn < 0.001 is the ontinuum regime, 0.001 < Kn < 0.1 is the slip regime in whih the Nvier-Stokes equtions with the slip oundry ondition re pproprite, 0.1 < Kn < 10 is the trnsition regime nd 10 < Kn is the free moleulr regime. Aording to the definition of Knudsen numer, the gs s moleulr free pth is very importnt in judging the flow regime. However, the gs moleulr men free pth t mrosle nd not its effetive men free pth is usully used to lulte the Kn numer. Thus, it is importnt to study the effetive moleulr free pth of gs t the nnosle. The men free pth onept is lso entrl to most models of the trnsport phenomenon in gses, suh s the momentum, energy nd mss exhnges whih re essentil to the gs flow nd het trnsfer hrter t nnosle spe. The most importnt differene etween gs flow t nnopore nd mrosle tues is tht the oundry of the nnopore limits the moleulr motions in the rdil diretion. Another importnt hrteristi of gs flow t nnosle is the uneven gs moleule numer density distriution in the whole flow spe [5,6]. Simple kineti theory indites tht the gs moleule s numer distriution would ffet the gs moleulr men free pth, hene its effetive men free pth should e modified due to the limittion of motions in the rdil diretion nd the uneven distriution of gs moleule s numer density. The effetive gs moleulr men free pth t the nnopore is therefore essentil to lrify. Stops [7] theoretilly studied the men free pth of gs moleules ounded y two prllel plnes. His study is sed on tht the possiility of moleule hving free pth r is desried s: ψ ( r) 1 = λ exp( r / λ ) (1) The study shows tht the men free pth of gs is redued. Dongri nd his o-workers [8] studied the gs men free pth in rrefied gs nd their results indite tht moleules perform Levy-type flights under rrefied onditions nd the free pth of gs moleules follows power-lw distriution. Tking into ount the solid oundry effets their further work [9] derived n effetive men free pth model for flows onfined y plnr surfes, whih otined good greement with the moleulr dynmis (MD) dt up to the erly trnsition regime. The men free pth of gs etween the ylinders ws studied y tking into ount the oundry-limit effets on the moleulr men free pth for surfes with oth onvex nd onve urvtures, though some ollisions were negleted [10]. The studies ove demonstrte tht limited studies hve een devoted to the gs men free pth t nnosle. At nnosle, the gs-solid intertion influenes the moleule s motion, whih nnot e
Int. J. Mol. Si. 2014, 15 12716 negleted. But to our knowledge, there re rre reports on the influenes of different gs-solid intertion strengths on the gs moleulr men free pth. On the other hnd due to the limittions on the gs moleulr motion y the nnopore s surfe, the gs moleulr free pth might e nisotropi. However, gs moleulr momentum nd energy exhnge hrteristis re highly dependent on the rdil gs moleulr men free pth in nnopores, so it is importnt to study the rdil gs moleulr men free pth. But to our knowledge, there is lk of the studies on the rdil gs moleulr men free pth in nnopores. Experimentl mesurements re optiml for these studies; it is however, very diffiult to rry out diret mesurements on the gs moleulr men free pth when it is onfined in nnopores. Thus, theoretil nlysis sed on the kineti theory is usully rried out to otin the men free pth. Some ssumptions should e mde to lulte the gs free pth in nnopores y kineti theory euse the intertions etween gs moleules nd the solid wll must e refully onsidered. Besides theoretil lultion, simultion methods suh s MD simultion llow us to ssess some fundmentl moleulr properties suh s the inter-moleulr ollisions nd the distnes etween two ollisions. MD is one very useful method to get the moleulr free pth t nnosle spe. This pper presents our numeril experiments on simple gs onfined t nnopores, whih re the most ommon nd most esily proessed. We highlight the profile of gs men free pth nd rdil men free pth nd their reltionships with the rdius of the nnopores nd the gs-solid intertion strength. 2. Results nd Disussion In this pper, we present our study results on nnopores of three different rdii, whih re 16.06, 21.4 nd 32.0 nm. For eh nnopore, three different gs-solid intertion strengths, tht is equls 0.2, 0.4 nd 0.6, were pplied to study its effets on the gs moleulr men free pth. 2.1. Men Free Pth of All Moleules in Nnopore Aording to the gs kineti theory, the men free pth n e expressed s Eqution (2) for gs t mrosle. where k B is the Boltzmnn onstnt, d is the dimeter of gs moleule nd P is the gs pressure. Sustituting the orresponding vlue in our studies we get tht the men free pth of rgon t the sme temperture nd density s our simultion ses ut t mrosle spe is 35.85 nm. Through lulting the integrtion desried s Eqution (3), in whih ψ (r) is the possiility moleule hs free pth etween r nd r + dr, we get the men free pth of ll moleules in the nnopore, whih is represented y λ in the following setion. ll 2 λ = k B T / 2πd P λ = rψ ( r) dr 0 Tle 1 lists the rtio of λ ll to the kineti theoretil vlue of the gs men free pth whih is represented y λ 0. From Tle 1, it n e lerly seen tht when gs is onfined t the nnopore is less thn its ounterprt vlue t mrosle spe. Tle 2 gives the omprison etween the Knudsen numer sed on the kineti theoreti vlue of men free pth (Kn 0 ) nd the tul Knudsen (2) (3) λ ll
Int. J. Mol. Si. 2014, 15 12717 numer whih is sed on the effetive men free pth. From Tle 2 we n see tht if the gs men free pth t mrosle spe is dopted to lulte the Knudsen numer we will get the overestimted Knudsen numer whih would ring dverse impt on the understnding of flow hrteristis t nnosle. Tle 1. Rtio of our simultions results λ ll to the kineti theoreti vlue of the gs moleulr men free pth. λ ll /λ 0 R = 16.06 nm R = 21.4 nm R = 32.0 nm = 0.2 0.321 0.388 0.519 = 0.4 0.209 0.248 0.371 = 0.6 0.128 0.179 0.237 Tle 2. Comprison etween the Kn 0 whih is sed on the kineti theoreti vlue of the gs men free pth nd the tul Knudsen numer. Atul Kn R = 16.06 nm R = 21.4 nm R = 32.0 nm Kn 0 = 1.116 Kn 0 = 0.837 Kn 0 = 0.560 = 0.2 0.358 0.325 0.291 =0.4 0.233 0.208 0.208 = 0.6 0.143 0.150 0.133 Another onlusion drwn from Tle 1 is tht oth the nnopore s rdius nd the gs-solid intertion strength will ffet λ. Aording to Tle 1, smller nnopore nd stronger gs-solid ll intertions will use shorter. A smller nnopore hs lrger rtio of surfe re to its volume λ ll nd the wll surfe will tke more onstrints on the gs moleule s movements so the men free pth nturlly eomes smller. When the gs-solid intertion eomes stronger more gs moleules will dhere to the wll of the nnopore; those moleules will experiene shorter trvel distne etween two suessive ollisions nd hene derese the men free pth of ll moleules in the nnopore. 2.2. Profile of Gs Moleulr Men Free Pth In this setion, we present our study results on the profile of the gs moleulr men free pth long the rdius of the nnopore. To get the profile, we divide the nnopore into mny onentri nnuluses nd the width of eh nnulus depends on its distne from the wll surfe of nnopore. For nnuluses whih re only 2δ prt from the wll surfe, their width is 0.125δ nd others hve the width of 4δ. The purpose of tking different widths is to otin detiled informtion out the zone only 2δ prt from the wll surfe. To get the gs moleulr men free pth in eh nnulus the lultion of Eqution (3) is done over eh nnulus. In this setion we present the profile of men free pth inluding gs moleule s ollisions in ll diretions, whih is expressed s λ. Our simultion results re shown in Figure 1 in whih the longitudinl oordinte represents the rtio of λ to the unonfined men free pth λ whose vlue is 35.85 nm. Figure 1 desries 0 severl properties onerning the vrition of λ with its distne from the enter of the nnopore. First, the lrgest vlue of the men free pth ours t the enter of the nnopore nd it is out the rdius of the nnopore. This phenomenon is physilly resonle euse the moleules t
Int. J. Mol. Si. 2014, 15 12718 the enter of nnopore hve the lrgest spe to move; Seond, the men free pth dereses long the nnopore s rdius diretion ut with different deresing rtes in different regions. For moleules in the region from the enter of the nnopore to 2 nm prt from the wll surfe, λ only delines out 5 nm. However, for moleules whih re ner the wll surfe, their men free pth shrply dereses; Thirdly, like the men free pth of ll gs moleules in the nnopore, the gs-solid intertion strength lso ffets the profile of λ. When the gs-solid intertion strength eomes stronger λ in the region tht suffers no diret effets from the wll surfe, eomes lrger. This so-lled region whih suffers no diret effets from the wll surfe in this pper mens the zone in whih gs moleules only experiene gs-gs ollisions nd no ollisions etween moleules nd the wll surfe ours. Stronger gs-solid intertions would use more gs-wll ollisions nd hene derese gs-gs ollisions, whih mke λ in the region whih suffers no diret effets from the wll surfe eome lrger. This phenomenon n e further illustrted y the moleulr ollision times dt reorded in our simultions. Tle 3 gives the omprison of moleule ollision times in the region tht suffers no diret effets from the wll surfe when different gs-solid intertion strengths re pplied. It should e noted tht the moleule ollision times in Tle 3 mens the verge numer of gs-gs ollisions per unit time. Our MD dt shows tht when is 0.2 the moleules in the region tht suffers no diret effets from the wll surfe experiene the most gs-gs ollisions, so we set it s the omprison stndrd. For the nnopores with the sme rdius the vlue in Tle 3 is the rtio of its tul moleule ollision times to the vlue of simultion results when is 0.2. Tle 3 lerly illustrtes tht when the gs-wll intertion strength inreses the gs moleules in the region whih suffers no diret effets from the wll experiene less ollisions nd hene uses the gs men free pth in this region to inrese. Figure 1. Vrition of gs moleulr men free pth with the distne from the enter of the nnopore. () R = 16.06 nm; () R = 21.4 nm; nd () R =32.0 nm.
Int. J. Mol. Si. 2014, 15 12719 Tle 3. Comprison of moleule ollision times in the zone whih suffers no diret effets from the wll etween different gs-solid intertion strength. Comprison of Moleule Collision Times R = 16.06 nm R = 21.4 nm R = 32.0 nm = 0.2 1 1 1 = 0.4 0.937 0.968 0.981 = 0.6 0.893 0.911 0.947 To further study the vrition of men free pth ner the wll surfe of the nnopore, Figure 2 gives the profile of λ within the zone of 0.7 nm prt from the wll surfe. Figure 2 shows tht in the ner-wll zone the men free pth does not vry monotonously nd flututions n e lerly seen. The lest men free pth ours t the wll surfe whose vlue rnges from out 0.9 to 4.0 nm. When the gs-solid intertion strength is identil, the lest vlue of λ for different nnopores is nerly the sme, whih shows the gs-solid intertion strength plys key role in determining λ t the wll surfe. Figure 2. Profile of gs moleulr men free pth in the zone 0.7 nm prt from the wll surfe of the nnopore; the dotted line mens the dividing line of slip regime. () R = 16.06 nm; () R = 21.4 nm; nd () R = 32.0 nm.
Int. J. Mol. Si. 2014, 15 12720 Aording to its definition, the vlue of the Knudsen numer should hve the sme vrition tendeny with the men free pth. Sustituting the men free pth to the definition of Kn, the line of Kn = 0.1 is shown in Figure 2. Aording to Figure 2 it is interesting tht the Knudsen numer in the zone ner the wll surfe of the nnopore is less thn 0.1 nd in other prts of the nnopore the Knudsen numer is greter thn 0.1. 2.3. Profile of the Rdil Gs Moleulr Men Free Pth In mrosle tues, the gs moleulr men free pth is isotropi due to lk of surfe effets on the moleule s motion. But for nnopore, the gs moleule s motion is limited in the rdil diretion, so is the gs moleulr men free pth still isotropi? In this setion we will give our simultion results on the rdil gs moleulr men free pth in nnopores. The rdil gs moleulr men free pth expressed s λ r in this pper is defined s the men distne in the rdil diretion etween gs moleule s two suessive ollisions, whih inludes oth gs moleule-gs moleule nd gs moleule-wll surfe ollisions. Figure 3 gives the vrition of rdil gs moleulr men free pth with the distne from the enter of the nnopore. In Figure 3 the longitudinl oordinte stnds for the rtio of λ to the unonfined r men free pth λ whose vlue is 35.85 nm. In the region fr from the wll surfe 0 λ, like r λ, lso inreses with the distne from the wll surfe, ut it hnges linerly with its distne from the enter of the nnopore. This hnge tendeny long the rdius is different from tht of λ. The liner fit of reltionship etween λ r / λ nd r / R in the region fr wy from the wll surfe is lso 0 presented in Figure 3 nd it is found tht it n e desried y one liner funtion with the form of λ r / λ0 = A + B ( r / R). The vlues of A nd B only vry with the nnopore s rdius. For the sme nnopore, even the gs-solid intertion strength is different from the reltionship etween λ r / λ nd 0 r / R n e desried y the sme funtion. Figure 3. Vrition of rdil gs moleulr men free pth with the distne from the enter of the nnopore; the line represents the fitting results. () R = 16.06 nm; () R = 21.4 nm; nd () R = 32.0 nm.
Int. J. Mol. Si. 2014, 15 12721 Figure 3. Cont. Compring Figure 3 with Figure 1, there is gret differene seen etween the vlue of λ r nd λ. This phenomenon lerly illustrtes tht the gs moleules motions re nisotropi when they re onfined in nnopores. The most importnt point is tht the men free pth will ffet the momentum nd energy exhnge, so it ould e dedued tht the momentum nd energy exhnge hrteristis re lso nisotropi in nnopores. The rtio of λ r to λ inreses when the nnopore eomes lrger ut is not een ffeted y the gs-wll intertion strength. So it seems tht the nisotropy of moleulr trnsporttion is only ffeted y the size of the nnopore. Figure 4 gives the detils of rdil gs moleulr men free pth in the zone 0.7 nm prt from the nnopore s wll surfe. It shows tht the vlue of λ r is muh smller thn tht of λ in the sme zone. Compring Figure 4 with Figure 2 we n find tht in this zone oth λ r nd λ show nerly the sme hnge tendeny with the distne from the wll surfe. This phenomenon illustrtes tht in the zone diretly ffeted y the wll surfe the wll surfe s effets dominte oth λ nd r λ. Figure 4. Profile of rdil gs moleulr men free pth in the zone 0.7 nm prt from the nnopore s wll surfe. () R = 16.06 nm; () R = 21.4 nm; nd () R = 32.0 nm.
Int. J. Mol. Si. 2014, 15 12722 Figure 4. Cont. 2.4. Reltionship etween Gs Numer Density nd Moleulr Men Free Pth in Nnopores It is well known tht the gs numer density shows n uneven distriution ner the wll surfe when it is onfined in nnopores. The ove results show tht the gs moleulr men free pth lso vries with its distne from the enter of the nnopore, so it is neessry to study the reltionship etween the gs numer density nd the gs moleulr men free pth. In this pper we tke the nnopore whose rdius is 16.06 nm s exmple to study the reltionship etween the gs numer density nd the moleulr men free pth. Figure 5 shows the profiles of gs numer density nd moleulr men free pth inluding ll ollision diretions. Figure 5 shows tht there is pek numer density vlue ner the wll surfe. Unlike the gs moleulr men free pth, the gs numer density in most spes of the nnopore does not vry oviously, whih illustrtes tht in the onstnt density zone the vrition of the gs moleulr men free pth is not used y the gs numer density. Figure 5. Profiles of gs moleulr men free pth nd gs numer density. () = 0.2; () = 0.4; nd () = 0.6.
Int. J. Mol. Si. 2014, 15 12723 Figure 5. Cont. To refully study oth the vritions of gs numer density nd moleulr men free pth in the zone ner the wll surfe we give the detils in the zone 0.7 nm prt from the nnopore s wll surfe in Figure 6. Aording to the kineti theory, for gs moleules in the unounded spe the lrger gs numer density will redue the gs moleulr men free pth. However, ording to Figure 6 t the position where the highest vlue of gs numer density ours, the gs moleulr men free pth dose not show the lowest vlue. When the numer density delines, the gs moleulr men free pth lso delines. These phenomen illustrte tht for gs onfined in nnopores the wll surfe hs stronger effets on the gs moleulr men free pth thn gs numer density does. Figure 6. Profiles of gs moleulr men free pth nd gs numer density in the zone 0.7 nm prt from the nnopore s wll surfe. () = 0.2; () = 0.4; nd () = 0.6.
Int. J. Mol. Si. 2014, 15 12724 As fr s the rdil gs moleulr men free pth is onerned, Figures 7 nd 8 give the profiles of rdil gs men free pth nd numer density. Through these two figures we n esily find tht like the gs moleulr men free pth inluding ll diretion s ollisions, the gs numer density does not ffet the rdil gs moleulr men free pth in the sme wy s it ffets the gs moleulr men free pth in the unounded spe. Figure 7. Profiles of rdil gs moleulr men free pth nd gs numer density. () = 0.2; () = 0.4; nd () = 0.6. Figure 8. Profiles of rdil gs moleulr men free pth nd gs numer density in the zone 0.7 nm prt from the nnopore s wll surfe. () = 0.2; () = 0.4; nd () = 0.6.
Int. J. Mol. Si. 2014, 15 12725 Figure 8. Cont. 2.5. The Body Fore s Effets on the Gs Moleulr Men Free Pth In our simultion runs one onstnt fore ws dded on every moleule in the system in order to generte mrosopi gs flow. In this setion we will study the effets y the ody fore on our simultion results. We tke the nnopore whose rdius is 16.06 nm nd the vlue of is 0.2 s n exmple to ompre the differene etween simultion results when the ody fore is dded on or sent from every gs moleule. Figure 9 give our omprison results, from Figure 9 we n lerly see tht the ody fore ted on every moleule in the simultion system hs no ovious effets on the simultion results of oth λ nd r λ. Similr results re lso otined for other nnopore rdii tht we studied. This result is physilly sound euse the gs moleulr men free pth is only dependent on the gs moleules reltive motions nd hs nothing to do with moleulr mss motion, whih is generted y the ody fore. Figure 9. Comprison of simultion results whether the ody fore is dded on every gs moleule. () gs moleulr men free pth inluding ll diretions ollisions; () rdil gs moleulr men free pth.
Int. J. Mol. Si. 2014, 15 12726 3. Numeril Experimentl Setion 3.1. Simultion Model In our present pper, moleulr dynmis simultions re used to study the gs moleulr free pth in nnopores. The physil model for MD simultions is illustrted in Figure 10. The gs moleules re represented y the dots in Figure 10 nd the nnopore is represented y the drk zone. In our simultion studies, rgon ws hosen s the representtive gs euse the Lennrd-Jones (L-J) potentil is well known for rgon. As fr s the solid nnopore is onerned, there re usully two type models, one is the sttionry solid toms nd nother lterntive system is omposed y moile solid toms. If the moile solid system is utilized the omputtion lod would e very lrge euse the motions of wll toms must e lulted. The min funtion of the wll surfe is to surround the gs moleules through the intertions etween gs moleules nd wll toms. There is little differene etween these sttionry solid toms nd the rel situtions, ut this pper highlights the men free pth of gs moleules nd the wll solid toms minly serve s one potentil wll to surround the gs moleules. In order to redue the omputtionl lod, the present study keeps the solid toms sttionry nd for the reson of simpliity the nnopore mteril is lso set s rgon, ut in the solid stte. The solid nnopore is omposed of rgon toms tht re rrnged in terms of FCC (fe enter ue) struture. The length of the nnopore is fixed t 53 nm. The rdius of the nnopores is hngele in our simultions to study the size effets on the gs moleulr men free pth. Figure 10. Cross-setion snpshot of the prtile lotions in the nnopores of the MD (moleulr dynmis) simultions. The MD simultion is sed on the integrtion of Newton s seond lw y pproprite integrl methods suh s the Verlet method. In MD method, the fore etween two moleules is desried y the potentil funtion, for rgon Eqution (4) gives the funtion. U () r gs gs δ = 4ε r 12 6 δ r (4) where r is the distne etween two moleules, the energy prmeter for rgon is ε = 1.67 10 21 J nd δ is the moleulr length sle with vlue of 3.405 10 10 m [11]. The purpose of our studies on the gs moleulr men free pth is to further understnd the hrteristis of gs flow t nnosle, so the flow should exist. In this pper, the flow is generted y
Int. J. Mol. Si. 2014, 15 12727 dding one onstnt ody fore on every moleule in the simultion domin. The vlue of ody fore is sed on the following eqution: ΔP πr = N F ody 2 (5) where Δ P is the pressure differene etween two ends of the nnopore, r is the rdius of the nnopore nd N is the totl numer of gs moleules in the nnopore. The potentil funtion etween the gs nd solid toms lso uses eqution like Eqution (4) tht is: U () r gs wll δ = 4ε r 12 6 δ r (6) ut with ε = 2.82 10 21 J nd δ = 3.405 10 10 m where is one vrile prmeter whih represents the strength of intertions etween gs moleules nd wll toms, when the vlue of is lrger the gs moleule-wll tom intertion strength is stronger. So for gs moleules without intertion with the wll, the fore on every gs moleule is desried s Eqution (7) nd for gs moleules interting with the wll, is desried s Eqution (8). U ( r) r gs gs In our present studies, the initil positions of gs moleules re rrnged ording to FCC struture with the density of rgon gs t the men pressure of two ends of the nnopore t 300 K. The veloities of gs moleules were rndomly initilized ording to the Mxwell veloity distriution t 300 K nd the totl kineti temperture is equl to the kineti energy in the lultion domin: + F ody N 3 1 2 2 2 Nk BTgs = m ( vi, x + vi, y + vi, z) 2 2 i= 1 In Eqution (9) N is the totl numer of gs moleules in the nnopore, whih rnges from 2124 to 8592 when the dimeter of the nnopore vries from 32.12 to 64 nm in this pper. T is lwys equl gs to 300 K nd k is the Boltzmnn onstnt, m is the mss of gs moleule nd B v i, x, vi, y, v re gs i, z moleule veloities in the x, y nd z diretions. Vlues for the initil fore ted on eh gs moleule were determined in terms of their initil positions r (0). 3.2. Boundry Conditions U ( r) gs wll U ( r) gs gs Due to the limittion of omputtionl ility, it is impossile to simulte ll of the moleules in the rel flow system, so periodi oundry ondition is pplied to the flow diretion, tht is the x diretion. In the rdil diretion of nnopore, no speil oundry onditions re pplied to gs moleules, euse the movements of gs moleules re restrited y the wll of the nnopore. 3.3. Method of Gs Moleulr Men Free Pth Clultion r + Aording to the definition of men free pth, eh moleule s trvel distne etween two suessive ollisions is required to e reorded. But when the periodi oundry ondition is pplied, i r + F ody (7) (8) (9)
Int. J. Mol. Si. 2014, 15 12728 it is not pproprite to diretly mesure the trvel distne of moleule s the differene etween its urrent position nd the position of its lst ollision. Insted, we lulte the moleule s trvel distne etween two suessive ollisions s where Δ t is time differene etween two suessive ollisions nd v urrent is the urrent speed of the moleule. The moleule s trvel distne etween two suessive ollisions otined through the ove mentioned wy is reorded to lulte the men free pth distriution ψ (r), whih is the possiility moleule hs free pth etween r nd r + dr. One ψ (r) is otined, the effetive gs moleulr men free pth equls the vlue desried s Eqution (3). In lulting the gs moleulr men free pth, how to determine the ollision is tking ple is very importnt. The L-J potentil is used in our studies, we judge the ollision to tke ple if two moleules re loser together thn distne r ol. In the present pper, the vlue of r ol is set s δ. It should e noted tht the moleule s ollisions with wll surfe re lso onsidered, we lso judge the gs moleule-wll ollision hppens when the distne etween the moleule nd wll surfe is less thn r ol. The vlidtion of the r ol is tested through the lultion of the men free pth of unonfined rgon gs presented in Setion 3.5. 3.4. Other Simultion Detils In our MD study, n equilirium moleulr dynmis simultion for gs with initilly ssigned numer density t 300 K ws onduted for the first 1,000,000 run steps nd then the ody fore ws dded to every gs moleules in the nnopore. The temperture of gs needs to e ontrolled t 300 K, so the veloities of eh gs moleules re ontrolled to mintin the desired temperture 300 K. In the present study, the Verlet method ws used to solve the motion eqution of ll moleules in the simultion system nd the tom list method ws used to redue simultion run time. The simultion run time step ws set s 1.0 10 15 s. 3.5. Certifition of the Methodology r = Δt v urrent To onfirm the vlidity of our methodology for lulting the gs moleulr men free pth using Eqution (3), the unonfined rgon gs moleulr men free pth is lulted first. To simulte the unonfined gs, the ui sptil geometry with periodi oundry onditions pplied in three diretions is dopted. To ensure the orretness of the simultion the side of the ui is lrger thn the men free pth nd two different ui lengths re used to test the vlidity of our simultion results. These two different ui lengths re 42.4 nd 53.0 nm. The moleulr density in the ui is equl to the rgon s density t 0.2 MP nd 300 K. The MD simultions in the present pper re sed on our own developed solver. To firstly test the vlidity of our MD solver, the totl energy inluding potentil energy nd kineti energy per moleule nd temperture profile in the se of unonfined rgon gs t 0.2 MP is presented in Figure 11. From Figure 11 we n see tht oth the totl energy per moleule nd temperture flutute round one fixed vlue, ut their devitions re very smll ompred with the desire vlue. Aording to the (10)
Int. J. Mol. Si. 2014, 15 12729 theory of sttistil physis, ny therml property would show flututions whih re relted with the numer of moleules in the system, so our simultion results re physilly sound. Figure 11. Therml property s vrition with the simultion time for the se of unonfined gs t 0.2 MP. () totl energy per moleule; () temperture. For the unonfined rgon, our MD dt relxes to men free pth vlue of 36.0 nm, whih differs from the kineti theoretil vlue only +0.55%. Our MD simultion lso ounts the numer of multiple ollisions nd two ody ollisions, the results show tht for unonfined rgon gs t 0.2 MP nd 300 K the multiple ollisions ount for only 0.07% of totl ollisions, so the rgon gs t 0.2 MP n e regrded s idel gs nd the effets on the men free pth y the multiple ollisions n e ignored. This illustrtes tht our method of lulting the men free pth through the integrtion of ψ (r) nd the hoie of the vlue r ol re vlid. 4. Conlusions The gs moleulr free pth in nonpores with different rdius nd gs-wll intertion strengths ws studied using moleulr dynmis simultions nd severl onlusions hve een otined. The men free pth of ll moleules in the nnopore is smller thn its ounterprt vlue t mrosle spe nd is relted with the nnopore s rdius nd the gs-wll intertion strength. Smller nnopore size nd stronger gs-solid intertions would use shorter men free pth of ll moleules. The rdil gs moleulr men free pth is smller thn the moleulr men free pth inluding ll diretion s ollisions, whih mens the gs moleulr motions in the nnopore re nisotropi. Both λ r nd λ ll diretion vry with the distne from the nnopore s enter. The lowest vlue of men free pth ours t the wll surfe of the nnopore. For the sme nnopore, the reltionship etween λ r / λ nd 0 r / R n e desried y the sme liner funtion. Ner the wll surfe the gs-solid intertion plys key role in determining oth λ nd r λ, the rdius of the nnopore hs only minor effets on ll diretion them. For gs onfined in nnopores, the gs numer density does not ffet the gs moleulr men free pth in the sme wy s it does in the unounded spe.
Int. J. Mol. Si. 2014, 15 12730 Aknowledgments The study of this pper is funded y the Ntionl Nturl Siene Foundtion of Chin (51206201) nd Reserh Foundtion of Chongqing University of Siene & Tehnology, the projet No. is CK2011B26. Author Contriutions The o-uthor did some nlytil work on the MD simultion results. Conflits of Interest The uthors delre no onflit of interest. Referenes 1. Hummer, G.; Rsih, J.C.; Noworyt, J.P. Wter ondution through the hydrophoi hnnel of ron nnotue. Nture 2001, 414, 188 190. 2. Holt, J.K.; Prk, H.G.; Wng, Y.M. Fst mss trnsport through su-2-nnometer ron nnotues. Siene 2006, 312, 1034 1037. 3. Co, B.Y.; Sun, J.; Chen, M.; Guo, Z.Y. Moleulr momentum trnsport t fluid-solid interfes in MEMS/NEMS: A review. Int. J. Mol. Si. 2009, 10, 4638 4706. 4. Trvis, K.P.; Todd, B.D.; Evns, D.J. Deprture from Nvier-Stokes hydrodynmis in onfined liquids. Phys. Rev. E 1997, 55, 4288 4295. 5. Song, X.; Chen, J.K. A omprtive study on poiseuille flow of simple fluids through ylindril nd slit-like nnohnnels. Int. J. Het Mss Trnsf. 2008, 51, 1770 1779. 6. Murt, B.; Beskok, A. Moleulr dynmis simultions of sher-driven gs flows in nno-hnnels. Mirofluid. Nnofluid. 2011, 11, 611 622. 7. Stops, D.W. The men free pth of gs moleules in the trnsition regime. J. Phys. D Appl. Phys. 1970, 3, 685 696. 8. Dongri, N.; Zhng, Y.H.; Reese, J.M. Moleulr free pth distriution in rrefied gses. J. Phys. D Appl. Phys. 2011, 44, doi:10.1088/0022-3727/44/12/125502. 9. Dongri, N.; Zhng, Y.H.; Reese, J.M. Modeling of Knudsen lyer effets in miro/nnosle gs flows. J. Fluids Eng. 2011, 7, doi:10.1115/1.4004364. 10. Dongri, N.; Brert, R.W.; Emerson, D.R.; Stefnov, S.K.; Zhng, Y.; Reese, J.M. The effet of Knudsen lyers on rrefied ylindril Couette gs flows. Mirofluid. Nnofluid. 2013, 14, 31 43. 11. Rpport, D.C. The Art of Moleulr Dynmis Simultion; Cmridge University Press: Cmridge, UK, 2004; p. 15. 2014 y the uthors; liensee MDPI, Bsel, Switzerlnd. This rtile is n open ess rtile distriuted under the terms nd onditions of the Cretive Commons Attriution liense (http://retiveommons.org/lienses/y/3.0/).