Accuracy of the blob model for single flexible polymers inside nano-slits that are a few monomer sizes wide Narges Nikoofard,∗ S. Mohammad Hoseinpoor, and Mostafa Zahedifar Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 51167-87317, Iran (Dated: January 14, 2015) The de Gennes’ blob model is extensively used in different problems of polymer physics. This model is theoretically applicable when the number of monomers inside each blob is large enough. For confined flexible polymers, this requires the confining geometry to be much larger than the monomer size. In this manuscript, the opposite limit of polymer in nano-slits with one to several 5 monomerswidthisstudied,usingmoleculardynamicssimulations. Extensionofthepolymerinside 1 nano-slits,confinementforceontheplates,andtheeffectivespringconstantoftheconfinedpolymer 0 areinvestigated. Despiteofthetheoretical limitations oftheblob model,thesimulation resultsare 2 explained with the blob model very well. The agreement is observed for the static properties and n thedynamicspringconstantofthepolymer. Atheoreticaldescriptionoftheconditionsunderwhich a thedynamicspringconstantofthepolymerisindependentofthesmallnumberofmonomersinside J blobsisgiven. Ourresultsonthelimitofapplicabilityoftheblobmodelcanbeusefulinthedesign 3 of nano-technology devices. 1 ] I. INTRODUCTION confining geometry, D. Inside blobs, polymer does not t f feel that it is confined. The number of monomers inside o s Advances in nanotechnology have enabled investiga- each blob is found from the relations of the self-avoiding 1 . tions of single polymers confined in the nano-scale[1, 2]. walk (SAW) in free space, g ∼ D ν. Here, ν is the t b a This problem has attracted interests due to its applica- Flory exponent in 3D and b is the monomer size. In m (cid:0) (cid:1) tions in the design of nanotechnology devices [3–5]. It lengthscaleslargerthantheblobsize,thepolymer(with - also helps in obtaining more knowledgeof polymers con- N realmonomers)behaveslikeaneffectivepolymerwith d fined in biological environments [6, 7]. Besides, it has N monomers of size D. This effective polymer has a n g o become possible to check accuracy of the old theories SAW in one or two dimensions for a nano-channel or a c and improve them [8, 9]. Single polymers confined in- nano-slit, respectively. The number of monomers inside [ side nano-spheres [10, 11], nano-channels [2, 8, 12–14] each blob, g, and total number of the blobs, N, should g and nano-slits (between two parallel plates) [9, 15–18] 1 be large enough, to be able to use the relations of sta- are studied. v tistical mechanics for SAW inside or outside the blobs, 5 The most known theory for confined polymers inside respectively [25]. 5 nano-channelsornano-slitsisthedeGennes’blobmodel. 9 Forsemi-flexiblepolymers,Odijktheoryisrelevantwhen A number of studies on confined polymers are per- 2 the sizeofthe confininggeometry,D, issmallerthanthe formedinconditionsthattheabovetworequirementsare 0 not satisfied. It is shown that the blob model is applica- persistencelengthofthepolymer,L [8,15]. Recently,it . p ble in micro-devices, where the number of blobs is very 1 isshownthatextendeddeGennes’theoryshouldbeused 0 L2 small [26]. Also in some practical situations, the poly- in the case of semi-flexible polymers when L <D < p 5 p w mer is confined in severe conditions that the number of 1 [19]. Here, w is the monomer width. monomersinsideeachblobisverysmall[27]. Speciallyin v: Most of the studies on confined polymers focus on nano-devicesfor DNA barcoding, full stretching of DNA double-stranded DNA, which is a semi-flexible polymer. i isrequired[5]. Computersimulationsshowthattheblob X Recently, single stranded DNA is introduced as a model modelcanexplainthe staticsofaflexible polymerinside r system for study on flexible polymers [20]. Also, many narrow channels [12, 14]. However, the polymer dynam- a proteins and synthetic polymers are flexible. For flexible icsinnano-channelsisonlydescribedwiththeblobmodel polymers,theonlyrelevanttheoryisthedeGennes’blob when the number of monomers inside each blob is large model and it would be useful to better understand its enough [13, 14]. limitations and regime of applicability. The blob model has broad applications in different problems of polymer In this manuscript, a flexible polymer confined inside physics, such as polyelectrolytes [21], polymers under a nano-slit is studied using molecular dynamics (MD) tension [22], and polymers adsorbed to surfaces [23] or simulations. The extreme condition that the number of confined near surfaces unfer external fields [24]. monomers inside each blob is small is considered. It is Intheblobmodel,theconfinedpolymerisdividedinto shownthatourresultsfortheradiusofgyrationandcon- smaller sections, called blobs, that have the size of the finementfreeenergyofthepolymercanbeexplainedvery well,usingtheblobmodel. Despitepreviousstudies(Ref. [16]and referencestherein), no correctionis made to the slit width (distance between the two plates). Agreement ∗ [email protected] between simulation results and the blob model is ob- 2 served even for very narrow slits with 1-2 monomer(s) width. These widths are completely below the regime of applicability of the blob model. Extensionof the polymer inside the nano-slithas fluc- tuationsarounditsmeanvalue. Fromthesefluctuations, an effective spring constant can be obtained. It is a dy- namic variable for the polymer and is a measure of elas- ticity of the polymer. Our results for the effective spring constantareingoodagreementwiththeblobmodel,un- like the caseofthe polymer inside the nano-channel[12– 14]. This discrepancy is due to the difference between the origins of the effective spring constant of a polymer con- fined in a nano-channel and a nano-slit. Two factors FIG. 1. (Color online) (a) Blob model for a flexible polymer governthe fluctuations of a polymer confined in a nano- confined inside a nano-channel. The blobs of the polymer in slit: fluctuations in the size of blobs and fluctuations in the nano-channel are arranged one after the other along the nano-channel. (b) Top view of a polymer confined inside a the self-avoidingwalkofthe blobs. Itisobviousthatthe nano-slit. Theblobsof thepolymerin thenano-slit span the latter factor is dominant. However, for a polymer con- 2Dspacebetweentheplates. Thenumberofmonomersinside fined ina nano-channel,the blobs arearrangedone after one blob and total number of the blobs are clear in the two the other along the channel. So, fluctuations in the blob cases (a) and (b). size are the only source of fluctuations in the extension of the polymer (see Fig. 1). This makes the internal dynamics of the blobs important, in the dynamics of the whole polymer. As a result, it is determining to have the sufficient number of monomers inside each blob, for 3 D −41 a polymer confined inside a nano-channel. R|| ∼bN4 b . (1) (cid:18) (cid:19) Inthe nextsection, the statics anddynamics ofa flex- Free energy of confining the polymer is k T per blob, ible polymer confined in a nano-slit is reviewed briefly. B F ≈k TN. Substituting g in this relation gives In Sec III, our simulation method is described. Results conf B g of the simulation and their compatibility with the the- ory are discussed in Sec. IV. Finally, a summary of the 5 b 3 results is presented in the last section. F ≈k TN . (2) conf B D (cid:18) (cid:19) The force exerted on the plates is found by differenti- ating the confinement free energy with respect to D, II. THEORY 8 k T b 3 B f ≈ N . (3) Considera flexible polymerconfined betweentwo par- b D (cid:18) (cid:19) allel plates (Fig. 1). Confinement extends the polymer The polymer has fluctuations around its average ex- on the walls, to a radius of gyration larger than its free tension. The mean square deviation from the average radius of gyration R ∼ bNν. Here, N is the number g extension is described by an effective spring constant of monomers of the polymer, b is the monomer size and ν = 3 is the Flory exponent in 3D. Sections of the chain h∆R|2|i = h R||(t)−R||av 2i ≈ kBT/keff. The simplest 5 smaller than the distance between the plates, D, do not way to calc(cid:0)ulate the spri(cid:1)ng constant, keff, is to elimi- feel the constraints. These sections, called blobs, have a nateD betweentherelations1and2. Thisgivesthefree statistics similarto a chaininfree space D ∼bg53, where energy as a function of extension. Differentiating the g is the number of monomers inside each blob. polymer free energy twice with respect to its extension andsubstituting theequilibriumextensiongivesaspring The polymeris composedof N blobsthatcannotpen- g constant [8]. This calculation does not give a correct re- etrateintoeachother. So,itcanbe regardedasaneffec- sult,becauseequation2isthefreeenergyofthepolymer tive two-dimensional polymer with N monomers of size g at its equilibrium extension. Indeed, the free energy of D. We use the exponent of a self-avoiding walk in two the polymer for any desired extension should be used in dimensions, ν = 3, to find the extension of the polymer 4 the calculation. parallel to the plates R|| ∼ D(Ng)43. This gives the rela- Suppose the two ends of the confined polymer are tion between the average extension of the polymer with under tension and it is elongated to the extension R . f the total number of monomers and the distance between Free energy of a two- or three-dimensional polymer with the plates, N monomers of size b which is extended to size R is f 3 1 F ≈ k T Rf 1−ν [25]. ν is equal to 3 for 3D and tens B bNν 5 equalto 3 for(cid:16)2D.B(cid:17)ydifferentiatingthisfreeenergytwice 4 with respect to R and substituting the equilibrium ex- f tension R ∼ bNν, the spring constant is obtained as eq k = ∂2Ftens| ≈ kBT ≈ kBT. This is in agree- eff ∂R2f Req (bNν)2 R2eq ment with the relation used in Ref. [28]. Here, the confined polymer is described as an effec- tive polymer with N monomers of size D in two di- g mensions (ν = 3). So, the spring constant becomes 4 k ≈ kBT . Substituting for g, we get the spring eff 3 2 D(N)4 (cid:18) g (cid:19) constant for the confined polymer FIG. 2. (Color online) Side view of a polymer confined in a 1 nano-slit. The surface of each plate is defined such that the k T D 2 k T k ≈ B N−3/2 = B . (4) closest distance of the center of a monomer to the plate is eff b2 b R2 equal to b/2. Distance between the plates in our paper, D, (cid:18) (cid:19) || is shown in the figure. An effective width for the nano-slit is In the final relation, R is substituted from eq. 1. To also shown, to examine the effect of adding correction to the || derivethis relation,the confinedpolymeris simply mod- slit width. eledasaneffectivetwo-dimensionalpolymercomposedof blobsandtheeffectoftheexternaltensionistomovethe in a nano-slit. A schematic of the system is shown in confinementblobstowardalinearalignment. Thissimple Fig. 2. The polymer is modeled as a bead-spring chain. description is valid for small tensions with R <D(N). f g Monomers of the polymer are connected by the FENE The blob model is accurate when the number of potential, monomers inside each blob, g, and total number of the blobs, N, are considerable. This is necessary to be able 1 r 2 g U (r)=− Kr2ln 1− . (5) to use statistical mechanics relations for a self-avoiding FENE 2 0 r " (cid:18) 0(cid:19) # walk inside and outside the blobs, respectively. The size Shifted-truncatedLennard-Jonesisusedfortheexcluded spannedbyaself-avoidingwalkerinspaceisproportional volume interactions between the monomersand between tothenumberofstepstothepoweroftheFloryexponent the monomers and the plates, in two or three dimensions, when the number of steps is largeenough. Forg tobe largeenough,theconfiningge- 12 6 b b 1 1 ometry(nano-slitornano-channel)shouldbemuchlarger ULJ(r)=4ǫ − + , r <26b. (6) r r 4 than one monomer size. "(cid:18) (cid:19) (cid:18) (cid:19) # It is instructive to compare the effect of an external ǫandbaretheenergyandlengthscalesofthesimulation. tensiononapolymerconfinedinsideanano-channelora K =100ǫandr =1.5bareusedfortheFENEpotential. 0 nano-slit. Prior to tension, the polymer inside the nano- The plates are infinite and the closest distance of the slitarrangesitsblobsintwodimensions. Theeffectofthe center of a monomer to the plate surface is b/2. This externaltensionistomovetheblobstowardaonedimen- means that the spherical surface of the monomer can sional arrangement. However, the blobs of the polymer touch the plate, in agreement with our intuition from inside a nano-channel have already lost two degrees of a wall (Fig. 2). In previous studies, the center of a freedom and lie on a one dimensional line. So, the poly- monomers was allowed to touch the impenetrable plate mer has to stretch its blobs to respond to the external surface[16]. So,itwasneededtoaddthemonomersizeb tension. One can conclude that the elasticity of a poly- to the distance between the walls to compare the results mer inside a nano-slit has its origin in the arrangement with the blob model. It is important to note that this of the blobs. This is in contrast to the elasticity of a is not attributed to the finite size of the chains in these polymerinside a nano-channelwhichoriginatesfromthe simulations. Here, it will be shown that the simulation elasticityofeachblob. Asaresult,tousetheblobmodel results for similar chains are in agreement with the blob for the former case, the number of blobs should be suf- model, without any corrections in the wall distance. ficient; but for the latter case, the number of monomers The equations of motion are integrated using the Ve- inside each blob is also important. locityVerletalgorithm,withthestepsizeequalto0.01τ . 0 τ = mσ2 is the MD time scale and m is the monomer 0 ǫ III. SIMULATION METHOD mass.qThe system is kept at the constant temperature T =1.0 ǫ , using the Langevinthermostat with the fric- kB We use MD simulations to check the limits of accu- tion coefficient 1.0τ−1. The simulations are performed 0 racy of the blob model for a flexible polymer confined using ESPResSo [29]. 4 1.5 26 24 1.4 22 1.3 20 18 b) 1.2 R / b||16 0.2 g (R / ||1.1 14 0.15 lo 12 z)0.1 1 ( n 10 0.05 0.9 8 0 −1 0 1 6 z 0.8 20 30 40 50 60 70 2 2.1 2.2 2.3 2.4 2.5 0.75 −0.25 log (N) N (D / b) FIG. 4. (Color online) log10-log10 plot of radius of gyration FIG. 3. (Color online) Simulation results for R|| versus the of the polymer parallel to the plates R|| versus the polymer theoretical relation N0.75D−0.25. #, 2, ×, , +, ▽, △, ⊳ length, N, for different nano-slit widths. #, 2, ×, , +, and⊲showthesimulation resultsforD/b=1-9,in ascending ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9, in order. Inset: Distribution of the monomers in the distance ascending order. The corresponding α values are 0.74, 0.78, between the plates. The solid, dashed and dash-dotted lines 0.79,0.76, 0.72,0.74, 0.74,0.78and0.79. Theaverageofαis correspondtothethreenarrowestslitsD/b=1-3,inascending 0.76. order. IV. SIMULATION RESULTS A. Radius of gyration Inoursimulations,thenano-slitwidthischangedfrom Radiusofgyrationofthepolymerparalleltotheplates b to 9b. Polymers with 100, 200 and 300 monomers are versus the theoretical relation in equation 1 is shown in investigated. Forslitswithwidth3-9b,themonomersare Fig. 3. All data lie on a single line, in the whole range arranged in an initial structure close to a self-avoiding of parameters studied. This shows agreement between walk. Then, the system is warmed up to remove any the simulation results and the blob model. Distribution overlap between the monomers. For slits with width b of the monomers in the distance between the plates is and 2b, the monomers are initially arrangedon one line. shownintheinsetofFig. 3,forthethreenarrowestslits. This initial configurationreaches equilibrium in a longer Because of the soft nature of the potential, monomers timeinterval,butitisinevitablebecauseoftheverysmall can penetrate the walls rarely. width of the slits. To further investigate the correspondence of the sim- ulation results with the theory, exponents α and β in Allsimulationsareperformedforatotaltimeof5N2τ0. the relation R|| ∝ NαDβ are obtained from simulation. Data points to find time averages of the static and dy- These exponents are found from the slopes of the log10- namic properties of the polymer are collected after the log10 plots of the radius of gyration versus the polymer time N2τ , in time intervals equalto τ . In all plots, the length and the nano-slit width, respectively. As can be 0 0 size of errorbarat each point is smaller than the symbol seen in Fig. 4, α is close to the theoretical value even size. for very narrow slits (1-2b). α is not a decreasing or in- creasing function of the nano-slit width. So, the small numberofmonomersinside eachblobis notdetermining The radius of gyration of the polymer parallel to inthis exponent. The averageofα is0.76,whichhas%1 the plates is calculated using the relation R|| = deviationfromthetheoreticalvalue. Fig. 5(a)showsthe 1 (X −X )2+(Y −Y )2. X and X are the exponentβ fordifferentlengthsofthepolymer. Itisseen N i CM i CM i cm x components of the positions of the ith monomer and that finite size of the chains has no effect on the simula- p the center of mass, respectively. Y and Y are the tion results. The results are summarized in the relation i CM corresponding y components. The force on each plate in R|| ∝N0.76D−0.25. all steps of the simulation is sum of the Lennard-Jones For comparison with Ref. [16], an effective width is forces from all monomers interacting with the plate. definedforthenano-slitbyaddingamonomersizetothe 5 1.6 300 N = 100, β = − 0.25 1.5 N = 200, β = − 0.24 250 N = 300, β = − 0.25 1.4 200 1.3 b) 150 R / ||1.2 f og ( 1.1 100 l 1 50 0.9 (a) 0 0.8 0 5 10 15 0 0.2 0.4 0.6 0.8 1 − 2.7 log (D / b) N (D / b) 1.6 N = 100, β = − 0.33 FIG. 6. (Color online) Simulation results for f versus the 1.5 N = 200, β = − 0.34 theoretical relation ND−2.7. #, 2, ×, , +, ▽, △, ⊳ and ⊲ N = 300, β = − 0.33 show thesimulation results for D/b=1-9, in ascending order. 1.4 4.5 b) 1.3 R / ||1.2 4 g ( 3.5 o l 1.1 3 1 0.9 og ( f ) 2.25 l (b) 0.8 1.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 log [(D+δ) / b] 1 0.5 FIG.5. (Coloronline)log10-log10 plotofR|| versusthenano- 0 slit width. (a) and (b) show two different definitions for the 2 2.1 2.2 2.3 2.4 2.5 nano-slitwidth(seeFig. 2). Ascanbeseen,omittingδresults log ( N ) in an excellent improvement in thescaling exponent. FIG.7. (Coloronline)log10-log10plotoff versusthepolymer nano-slit width (see Fig. 2) in Fig. 5(b). The exponent length, N, for different nano-slit widths. #, 2, ×, , +, β deviates largely from the theoretical value, with this ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9, in change in the definition of the wall surface. ascendingorder. Thecorrespondingγvaluesare1,0.99,0.99, 0.98, 0.99, 0.96, 0.98,0.95 and0.99. Theaverageofγ is0.99. B. The confinement force theory and simulation. The exponents γ and ω in the relation f ∝NγDω are It is not easy to measure the free energy of confining obtainedfromsimulation. AccordingtoFig. 7,deviation the polymer in the nano-slit. Instead, the force that the in γ from the theoretical value (=1) is less than %1. confined polymer exerts on the plates is studied in the In Fig. 8(a), the value of ω is found from the log - 10 simulation. The results are compared with equation 3 log plot of force versus the nano-slit width. Deviation 10 from the theory. The simulation results for the force are of the exponent from the theoretical value (-2.7) is less plottedversusthe theoreticalrelationinFig. 6. Alinear than %8. The simulation results are summarized in f ∝ behavior is observed, which shows agreement between N0.99D−2.9. Again, to check our definition for the nano- 6 2.5 x 10−3 N = 100, ω = − 2.9 3.5 N = 200, ω = − 2.9 N = 300, ω = − 2.9 3 2 2.5 1.5 og ( f ) keff 2 l 1 1.5 1 0.5 0.5 (a) 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00 0.1 0.2 0.3 0.4 0.5 0.6 log(D / b) N − 1.5 (D / b)− 0.5 2.5 N = 100, ω = − 3.5 FIG. 9. (Color online) Simulation results for the effective N = 200, ω = − 3.6 springconstantversusthetheoreticalrelation N−1.5D0.5. #, N = 300, ω = − 3.6 2, ×, , +, ▽, △, ⊳ and ⊲ show the simulation results for 2 D/b=1-9, in ascending order. 1.5 g ( f ) verTshues stihmeutlhaetoiornetriceasullrteslafotriotnhe(eeqffueacttiiovnes4p)rainrge cdoenpsictatendt o in Fig. 9. The linear curve confirms the excellent agree- l 1 ment between theory and simulation. This agreement is observed even when the nano-slit width is very small. The exponents η and ζ in the relation k ∝ NηDζ 0.5 eff are found from the simulation results. Fig. 10(a) shows the log -log plot of k versus the polymer length. 10 10 eff (b) The average slope of the plots is -1.3. This exponent 0 has the largest deviation, %13, among the exponents in- 0.5 0.6 0.7 0.8 0.9 1 log [(D+δ) / b] vestigated in this manuscript. The exponent is not a monotonic function of the nano-slit width and it cannot be deduced that an improvedagreement between theory FIG. 8. (Color online) log10-log10 plot of f versus the nano- and simulation would be observed with increasing the slit width. (a) and (b) show two different definitions for the nano-slit width. nano-slitwidth(seeFig. 2). Ascanbeseen,omittingδresults Dependence of the effective spring constanton the slit in an excellent improvement in thescaling exponent. width is investigated in Fig. 10(b). The average of the exponentis0.48. Althoughdeviationoftheaveragefrom the theory is around %4, the exponents themselves are slit width, the exponent ω is found from the log -log 10 10 not very close to it. Overall, a good agreement between plot of force versus D+δ (Fig. 8(b)). Deviation of the theory andsimulationis observedand the simulationre- exponent becomes more than %30. sults are summarized in the relation k ∝N−1.3D0.48. eff To check the effect ofcorrelationsin the calculationof h∆R2i, a different averaging scheme was also examined. C. The effective spring constant || Data points with a time interval of 100τ were collected 0 fromthewholedata. Startingfromdifferentinitialtimes, Fluctuations in the radius of gyration h∆R2i can be || 100 different series of data points can be defined. The usedto findaneffective springconstantfor the polymer, valuesofthespringconstantobtainedfromeachofthese keff ≈ h∆kBRT2i. Thiseffectivespringconstantisameasure seriesareveryclosetotheonesshownonFigs. 10(a)and || of the dynamics of the confined polymer. For example, 10(b). the relaxation time of the confined polymer is found by Furtherinvestigationisneededtoexplainthelargerde- dividing the friction coefficient from the solvent to this viations of the dynamic exponents from theory, relative effective spring constant [15]. to the static ones. One can attribute these deviations to 7 −0.2 V. SUMMARY AND DISCUSSION (a) −0.4 In this manuscript, we checked the accuracy of the de Gennes’ blob model for a flexible polymer inside a nano- −0.6 slit, using MD simulation. Theoretically,the blob model is accurate for wide slits and long polymers. However, )eff−0.8 wNeaneox-asmlitisnewditthhewmidotdhesl ienquthale tliom1it-9oftvimereysnaarmroownosmlitesr. k g ( −1 size were investigated. Simulation results both for the o l static properties and the dynamic spring constant of the polymer were in excellent agreement with the theory. −1.2 Our results showed that the blob model can describe theeffectivespringconstantoftheconfinedpolymereven −1.4 for very narrow slits. This is in contrast to a recent study on single flexible polymers inside nano-channels. −1.6 Indeed, the blob model is only applicable for the spring 2 2.1 2.2 2.3 2.4 2.5 constant of the polymer in nano-channels with at least log ( N ) 10monomerswidth. Here,itwasexplainedthatthisdis- crepancyisaresultofthedifferentoriginsofthepolymer 0.2 N = 100, ζ = 0.40 (b) dynamics in the two cases. The internal dynamics of a blob is determining in the overalldynamics of the chain, 0 N = 200, ζ = 0.46 N = 300, ζ = 0.58 for a polymer confined in a nano-channel. But, for a −0.2 polymer confined in a nano-slit, the dynamics is domi- nated by the two-dimensional arrangement of the blobs −0.4 between the plates. )eff−0.6 weIrne ouuserds.imIutlwataiosnosb,speorvlyemdetrhsatwitthhe1e0ff0e-c3t00ofmtohneofimneitres k g ( −0.8 size of the chains is not considerable. Our results on lo the accuracy of the blob model evenfor narrowslits and −1 shortchainscanbeveryusefulinthestudiesofpolymers in confinement. Polymer confinement occurs in nano- −1.2 technologydevices,suchasnano-poresequencing[3,30], DNA barcoding [5] and polymer separation devices [4]. −1.4 The main result of this manuscript was that the blob −1.6 modelissometimesaccuratebeyonditstheoreticallimits, 0 0.2 0.4 0.6 0.8 1 in practical situations. Considering that the blob model log (D / b) isusedinmanyproblemsofpolymerphysics[25],thisre- sult is applicable in different circumstances. A polymer FIG.10. (Coloronline)(a)log10-log10 plotofkeff versusthe adsorbed to a surface [23] or compressed on a surface polymer length, N, for different nanoslit width. #, 2, ×, , by an external field [24] is confined to a region near the +, ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9, surface. For these problems with 2D confinement of the inascendingorder. Thecorrespondingη valuesare-1.6,-1.6, polymer, the above result can be used directly. How- -1.7, -1.4, -1.1, -1.5, -1.3, -1.3, -1.6. The average η is -1.3. ever, for a polymer under external tension more caution (b)log10-log10 plot of keff versustheslit width, for different polymer lengths. The average of ζ is 0.48. isneeded,sincetheblobsarearrangedin1Dandthe dy- namicsmaybemoresensitivetothenumberofmonomers inside one blob. the finite length of the chains or insufficient simulation times for equilibration of the dynamic quantities. 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