THE JOURNAL OF CHEMICAL PHYSICS 136, 214901 (2012)

Adsorption-driven translocation of polymer chain into nanoporesShuang Yang1,2 and Alexander V. Neimark1,a)

1Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey,98 Brett Road, Piscataway, New Jersey 08854, USA2Beijing National Laboratory for Molecular Sciences, Department of Polymer Science and Engineeringand Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistryand Molecular Engineering, Peking University, Beijing 100871, China

(Received 5 February 2012; accepted 7 May 2012; published online 4 June 2012)

The polymer translocation into nanopores is generally facilitated by external driving forces, suchas electric or hydrodynamic fields, to compensate for entropic restrictions imposed by the confine-ment. We investigate the dynamics of translocation driven by polymer adsorption to the confiningwalls that is relevant to chromatographic separation of macromolecules. By using the self-consistentfield theory, we study the passage of a chain trough a small opening from cis to trans compartmentsof spherical shape with adsorption potential applied in the trans compartment. The chain transferis modeled as the Fokker-Plank diffusion along the free energy landscape of the translocation passrepresented as a sum of the free energies of cis and trans parts of the chain tethered to the pore open-ing. We investigate how the chain length, the size of trans compartment, the magnitude of adsorptionpotential, and the extent of excluded volume interactions affect the translocation time and its distribu-tion. Interplay of these factors brings about a variety of different translocation regimes. We show thatexcluded volume interactions within a certain range of adsorption potentials can cause a local mini-mum on the free energy landscape, which is absent for ideal chains. The adsorption potential alwaysleads to the decrease of the free energy barrier, increasing the probability of successful translocation.However, the translocation time depends non-monotonically of the magnitude of adsorption poten-tial. Our calculations predict the existence of the critical magnitude of adsorption potential, whichseparates favorable and unfavorable regimes of translocation. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4720505]

I. INTRODUCTION

How polymer chains pass through nanopores is an impor-tant issue in various biological processes,1 such as transloca-tion of RNA through nuclear pore complexes,2 injection ofDNA from a virus into a target cell, and transport of pro-teins across biological membranes.3, 4 Translocation throughsolid state nanopores is the founding mechanism of the var-ious attempts towards the sequence determination of DNAand proteins.5 Within last 15 years, the polymer transloca-tion phenomenon has been attracting constant attention in theliterature,6 including experiments,7–9 analytical theories,10–20

and computer simulations.21–31 Translocation of polymersthrough nanopores is controlled mainly by entropic barriersassociated with the restrictions on chain configurations im-posed by confinement.10–12, 32 In many practically relevantcases, translocation is driven by external driving forces, suchas electric or hydrodynamic fields, and it is not surprising thatsuch field-driven transport received most of attention.8, 9, 20, 33

Adsorption of chain segments to pore walls is another im-portant factor, which may counter-balance the loss of en-tropy and facilitate transport into nanopores.6, 21–23 Adsorp-tion plays a key role in polymer and biopolymer separationsand chromatography on nanoporous substrates.34 However,the literature on adsorption-driven translocation is limited and

a)Author to whom correspondence should be addressed. Electronic mail:aneimark@rci.rutgers.edu.

this translocation mechanism is poorly understood, especiallywith respect to the competition between repulsive exclusionvolume interactions in real chains and attractive adsorptioninteractions. The effect of adsorption on the translocation dy-namics of ideal Gaussian chains was studied in details by Parkand Sung,35 who used the self-consistent field theory (SCFT)and the Fokker-Plank (FP) formalism of random walk acrossthe barrier on the free energy landscape. This theoretical ap-proach was proved efficient in studies of translocation of idealand real chains;10, 15, 16 it is adopted with certain modifica-tions in this work. The existence of the free energy barriersmakes the translocation problem reminiscent to the classicalproblem of nucleation that is solved employing the Fokker-Plank formalism.15, 36 Milchev et al.21 employed Monte Carlosimulation of a bead-spring model chain threading througha membrane, which cis side is repulsive to the monomersand trans side is attractive. By varying the magnitude ofthe adsorption potential, they found two distinct dynamicsregimes with different scaling relationships between aver-age translocation time and chain length. Matsuyama et al.23

used a simple Flory theory to construct the free energy bar-rier of polymer translocation accounting for the polymer-poreinteractions. Furthermore, they employed a Langevin equa-tion to study the dynamics of polymer under additional ex-ternal electric field,22 and predicted an exponent relation-ship between translocation time and the number of polymersegments.

0021-9606/2012/136(21)/214901/15/$30.00 © 2012 American Institute of Physics136, 214901-1

214901-2 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

FIG. 1. Schematic representation of the process of polymer translocation from a semi-infinite space (cis) to a spherical pore (trans): (a) chain in cis compartmentmimics free chain in infinitely dilute solution, (b) chain in cis compartment tethered to the pore opening (beginning of translocation), (c) threading chaincomposed of cis and trans subchains, the trans subchain length determines the degree of translocation, (d) chain in trans compartment tethered to the poreopening (end of translocation), and (e) chain detached from the pore opening and completely confined to trans compartment.

In this paper, we investigate the adsorption effects onpolymer translocation drawing on the example of the simplestyet instructive model of the chain transfer from cis to transcompartments of spherical shape connected by a narrow open-ing, which accommodates just one chain segment in its crosssection. The radius Rcis of the cis compartment, which mim-ics the outer space, is taken significantly larger than the radiusRg of gyration of the chain, while the radius R of the transcompartment, which mimics the pore, is varied from severalto fractions of Rg. The pore surfaces exert an attractive po-tential so that chain segments tend to be adsorbed within thepore. Our approach is mainly based on earlier works of Parkand Sung35 and Kong and Muthukumar.16 While the formerconsidered adsorption driven translocation of ideal chains, thelatter considered purely diffusive translocation of real chainsand studied the effects of excluded volume interaction in theabsence of adsorption field. Here, we use the same approachof SCFT coupled with FP formalism to study the combinedeffects of adsorption and excluded volume interactions. Thegoal is to determine (i) conditions of adsorption equilibriumincluding stable and metastable chain conformations, (ii) theinfluence of excluded volume effect and adsorption potentialon the free energy barrier of polymer chain during its translo-cation, and (iii) the dynamics of translocation in terms of theprobability of translocation within given time and the dis-tribution of translocation times. Special attention is paid tothe so-called critical conditions of adsorption, which separateregimes of unfavorable and favorable adsorption in pores.

The solution of translocation problem requires two con-secutive tasks. First, we use the SCFT model to solve thestatistic-mechanical problem of determining the equilibriumdistribution of all possible chain conformations in the sys-tem of two connected compartments. Five characteristic typesof conformations represent five consecutive stages of thetranslocation process from cis to trans compartment that aresketched in Fig. 1. The key problem is to calculate the freeenergy of the chain in the process of translocation, stage (c)in Fig. 1. The chain conformations are quantified in terms ofmean field free energy using the degree of translocation, orthe length of the trans sub-chain as an order parameter. Thedependence of the free energy on the degree of translocationrepresents the free energy landscape with the minima corre-sponding to stable and metastable states and the maxima cor-responding to the energy barriers that should be overcome.Secondly, we model the process of translocation as a randomwalk along the free energy landscape by using the FP formal-ism, and determine how the distribution of translocation time

depends on the chain length, adsorption potential, strength ofexcluded volume interactions, and the pore radius.

The paper is structured as follows. In Sec. II, we rig-orously derive the SCFT equations for calculating the freeenergy of tethered chains confined to a spherical pore withattractive walls. In doing so, we obtained the integral rela-tionship between the free energy and the propagator func-tion, and the mean segment density, which differs from theone employed earlier in Ref. 16. While we consider the meansegment density of tethered chain as cylindrically symmetricand, thus, two dimensional, the authors16 adopted a sphericalsymmetry approximation and reduced the averaging problemto one dimensional. As shown in our calculations, this dif-ference, which vanishes for ideal chains, is significant andshould not be neglected in modeling translocation of realchains. The details of solution of the SCFT equations and thefree energy computations are given in Sec. III. In Sec. IV, wepresent the results of calculations of the free energy of teth-ered chains and analyze the free energy landscapes and energybarriers of translocation at different conditions. The calcu-lated free energy landscapes are incorporated in the FP equa-tion of translocation dynamics, which is considered in Sec. V.We calculate the probabilities and time distribution functionsof successful and failed translocation events for ideal and realchains at different chain length, and find a non-monotonic de-pendence of the translocation time on the magnitude of theadsorption potential. Our calculations predict the existence ofthe critical magnitude of adsorption potential, which separatesfavorable and unfavorable regimes of translocation and causesthe extremum of average translocation time. Conclusions aresummarized in Sec. VI.

II. SCFT MODEL OF TETHERED POLYMER CHAINS

The theoretical key problem consists in establishing thesystem of self-consistent equations determining the free en-ergy and the mean segment density of a tethered chain con-fined to a pore with adsorbing walls. Since the resulting re-lationship for the free energy differs from the one employedin earlier literature,16 we present below a detailed derivationstarting from the SCFT foundations.

A. The partition function, propagator, and free energyfor tethered Gaussian chains in external field

Within the SCFT framework, a polymer chain com-posed of N Kuhn segments, is modeled as a random walk

214901-3 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

trajectory of N steps, R(s) = {R0, R1, · · · , RN}, which origi-nates from position R0 and connects the endpoints of consec-utive segments Ri. The number of segments, s (s = 1, . . . , N),recounted from the chain origin represents a natural discreteco-ordinate along the trajectory. When N is large enough, thechain is approximated by a continuous trajectory with the co-ordinate s varying continuously from 0 to N. The true segmentdensity ρ̂(r) of the given chain R(s) is defined by

ρ̂(r) =∫ N

0δ[r − R(s)]ds. (1)

The starting point in SCFT is the Gaussian model of idealchains. For unconfined ideal chains, the probability distribu-tion function P[R(s)], which determines the statistics of chainconfigurations R(s) with endpoints R(0) = r0 and R(N) = r′′,is determined by the Wiener measure,36

P [R(s)]

=exp

[− 3

2b2

∫ N

0

(∂R(s)

∂s

)2

ds

]∫

�dr′′

∫ R(N)=r′′

R(0)=r0

D[R(s)] exp

[− 3

2b2

∫ N

0

(∂R∂s

)2

ds

] .

(2)

Here, b is the Kuhn length of each segment, D[R(s)] de-notes that the integration is carried out over the functionalspace of all possible configurations R(s), which starts at R(0)= r0 and ends at R(N) = r′′. R denotes that the integra-tion over the spatial coordinate r′′ is performed in the infinite

three-dimensional space. The Wiener measure reflects thechain connectivity and provides for the Gaussian distributionbetween any two segments of the chain. It is worth noting thatthe free Gaussian chain is considered as the reference state inthe SCFT thermodynamics.37 The denominator in the Wienerdistribution does not depend on the position r0 of the chainorigin and represents the configurational partition function ofa single Gaussian chain, ZG(N), and determines the excessfree energy, FG(N), of unconfined Gaussian chains of givenlength.

ZG (N )

=∫

�dr′′

∫ R(N)=r′′

R(0)=r0

D[R(s)] exp

[− 3

2b2

∫ N

0

(∂R∂s

)2

ds

](3)

and

FG (N ) = − ln ZG (N ) . (4)

Note that here and below kBT is used as the unit of energy.For an ideal Gaussian chain tethered at r0, and subjected

to the external field V (r), the calculation procedure is prettystandard.37, 38 The partition function,

ZtG[r0; N,V ] =

∫dr

∫ R(N)=r

R(0)=r0

D[R(s)]P [R(s)]

× exp

[−

∫ N

0V (R(s))ds

], (5)

is expressed through the chain propagator G(r, r0; N,V ),

G(r, r0; N,V ) =

∫ R(N)=r

R(0)=r0

D[R(s)] exp

[− 3

2b2

∫ N

0ds

(∂R∂s

)2

−∫ N

0V (R(s))ds

]∫

�dr

∫ R(N)=r

R(0)=r0

D[R(s)] exp

[− 3

2b2

∫ N

0

(∂R∂s

)2

ds

] , (6)

which is proportional to the probability of finding at position rthe Nth segment of the confined Gaussian chain tethered at r0

and being subjected to the external field V (r). This propaga-tor is symmetric, G(r, r0; N,V ) = G(r0, r; N,V ). The func-tional integral in the numerator is executed in confined space.With the propagator the partition function of Eq. (5) can beexpressed as

ZtG[r0; N,V ] =

∫�c

drG(r, r0; N,V ) (7)

and the free energy, F tG[r0; N,V ], as

F tG[r0; N,V ] = − ln

[∫drG(r, r0; N,V )

]. (8)

The mean segment density ρ(r; r0) can be also com-puted from the propagator G(r, r′; N,V ) according to the

relationship,

ρ(r; r0, V ) = 〈ρ̂(r)〉|V ;R(0)=r0

=

∫dr′

∫ N

0dsG(r, r0; s, V )G(r, r′; N − s, V )∫

dr′G(r′, r0; N,V ).

(9)

The derivation of Eq. (9) can be found in Ref. 37.The propagator formalism is practical, since the propa-

gator G(r, r0; s, V ) fulfills a diffusion type equation,39 wherethe chain co-ordinate s plays the role of time, and the meanfield potential V (r) acts as a rate of reaction,[

∂

∂s− b2

6∇2

r + V (r)

]G(r, r0; s, V ) = 0. (10)

214901-4 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

Here the operator ∇r acts on the spatial co-ordinate r.The external potential V (r) represents the adsorption field.Equation (10) can be solved numerically with natural initialand Dirichlet boundary conditions: G(r, r0; 0) = δ(r − r0),and G(r, r0; s) = 0 at the confinement boundaries. The Dirich-let boundary conditions correspond to a hard surface, at whichthe segment density must be zero. Other boundary conditionscan also be used, such as de Gennes boundary condition ∂(logG)/∂r|boundary = c adopted by Park and Sung.35 In case ofweak adsorption and long flexible chain, the two boundaryconditions are equivalent.

The above relationships present the properties ofGaussian chains and can be easily computed. Note that theserelationships depend on the external field and there are no re-strictions on the type of the function V (r). This later propertyis used in the SCFT model for real chains, where the two-body excluded volume interactions are approximated with aneffective mean field determined from the conditions of self-consistency.

B. The partition function for a tethered real chainin external field

The real chain is subjected to the excluded volume in-teractions. According to the Edwards’ prescription,40 thestrength of excluded volume interactions is accounted forby the two-body interaction parameter w which representsthe effective excluded volume of the segment. As such, thepartition function Zt

R[r0; N,V ] of a real chain tethered atR(0) = r0, which is confined to a finite volume Rc and sub-jected to the external field V (r), is given by38

ZtR[r0; N,V ] =

∫�c

dr′′∫ R(N)=r′′

R(0)=r0

D[R(s)]P [R(s)]

× exp

{− w

2

∫drρ̂(r)2 −

∫drρ̂(r)V (r)

}.

(11)

Respectively, the free energy of the tethered chain is

F tR[r0; N,V ] = − ln Zt

R[r0; N,V ]. (12)

Equations (11) and (12) for a particular case of w = 0correspond to the partition function (5) and free energy ofideal Gaussian chains tethered at r0, and subjected to the ex-ternal field V (r).

The main computational difficulty with real chains is re-lated to the density squared term in Equation (11), which ac-counts for two-body interactions. In the spirit of SCFT,37, 41

this term can be decoupled using the Hubbard-Stratonovichtransformation, which converts spatial integration into func-tional integration over the space of complex functions φ(r)defined within the confining volume Rc (Ref. 42)

exp

[−w

2

∫ρ̂(r)2dr

]

= 1

A

∫D[φ(r)] exp

{−i

∫drφ(r)�

ρ(r) − 1

2w

∫drφ(r)2

}.

(13)

Here, A is the normalization constant, A = ∫D[φ(r)]

exp{− 1

2w

∫drφ(r)2

}, and D[φ(r)] = limn → ∞dφ(r1)dφ(r2)

· · · dφ(rn) denotes the functional integration.Using the Hubbard-Stratonovich transformation,

Eq. (13), the partition function (11) can be converted into thefollowing functional integral:

ZtR[r0; N,V ]

= 1

A

∫dr′′

∫ R(N)=r′′

R(0)=r0

D[R(s)]P [R(s)]∫

D[φ(r)]

× exp

{−

∫dr[iφ(r) + V (r)]ρ̂(r) − 1

2w

∫drφ(r)2

}

= 1

A

∫D[φ′(r)] exp

(−H [φ′; r0]). (14)

The second equality in Eq. (8) introduces the functionalH[φ′; (r0)] that can be treated as the Hamiltonian of the chainsubjected to the complex external field φ′ = iφ + V . The ex-ternal field φ′(r) serves as a variable in the functional integral(14). The Hamiltonian H[φ′; (r0)] can be presented in a com-pact form as

H [φ′; r0] = − 1

2w

∫dr[φ′(r) − V (r)]2 − ln Zt

G[r0; N,φ′],

(15)

where ZtG[r0; N,φ′] represents the partition function (5) of a

Gaussian chain subjected to the external field φ′. Equation (7)relates Zt

G[r0; N,φ′] to the respective propagator G[r, r0; N,φ′] defined through Eq. (6) with the external field φ′.

The above relationships (11)–(15) formally present theproperties of real chains through the properties of idealchains, which are easily computed. Note that these relation-ships are exact, and as such not practical. The chain propaga-tor G[r, r0; N, φ′] depends of the field of complex functionsφ′(r) and cannot be explicitly determined without a certainapproximation, which implies the introduction of an effectivemean field.

C. Saddle-point approximation and SCFT equations

To evaluate the functional integrals like (14), one adoptsthe saddle-point approximation,38 which results in a set ofself-consistent field equations. Within this approximation, theHamiltonian H[φ′; r0] is replaced with its minimum achievedat a certain mean field φ′ = ω(r; r0), which is determinedfrom the condition of extremum,

δH [φ′; r0]

δφ′

∣∣∣∣φ′=ω

= − 1

w[ω(r; r0) − V (r)] − δ ln Zt

G[r0; N,ω]

δω(r; r0)= 0. (16)

Here, δδω

denotes the functional derivative. The mean fieldω(r; r0) depends on the tethering point r0.

214901-5 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

With the saddle-point approximation, the mean field freeenergy of the chain tethered at r = r0 is presented as

F tR[r0; N,V ] = − ln Zt

R[r0; N,V ] ∼= H [ω, r0]

= − 1

2w

∫V

dr[ω(r; r0) − V (r)]2

− ln ZtG[r0; N,ω]. (17)

Equation (13) gives the explicit relationship between themean field, ω(r; r0), and the mean segment density, ρ(r; r0,ω), of the chain subjected to this field. Indeed, Zt

G[r0; N,ω]defined by Eq. (15) can be presented using the true segmentdensity ρ̂(r; r0) as

ZtG[r0; N,ω] =

∫�

dr′′∫ R(N)=r′′

R(0)=r0

D[R(s)]P [R(s)]

× exp

[−

∫V

drω(r; r0)ρ̂(r)

]. (18)

Thus, the negative functional derivative − δ ln ZtG[r0;N,ω]

δω(r;r0) equalsthe mean segment density, ρ(r; r0, ω),

−δ ln ZtG[r0; N,ω]

δω(r; r0)

= 1

ZtG[r0; N,ω]

∫�

dr′′∫ R(N)=r′′

R(0)=r0

D[R(s)]ρ̂(r)P [R(s)]

× exp

[−

∫�

drω(r; r0)ρ̂(r)

]= 〈ρ̂(r; r0)〉� = ρ(r; r0, ω). (19)

Finally, the mean field ω(r; r0) can be expressed through bythe mean segment density, the excluded volume parameter,and the external potential,

ω(r; r0) = wρ(r; r0) + V (r). (20)

On the other hand, the mean segment density ρ(r; r0, ω) canbe computed from the propagator G(r, r′; N, ω) for the chainsubjected to the external field ω(r; r0), according to Eq. (9), inwhich the external field V(r) is substituted by ω(r; r0) definedthrough Eq. (20),

ρ(r; r0, ω) =

∫dr′

∫ N

0dsG(r, r0; s, ω)G(r, r′; N − s, ω)∫

dr′G(r′, r0; N,ω).

(21)

The propagator G(r, r0; N, ω) fulfills Eq. (10) with thesubstitution of V(r) by ω(r; r0),[

∂

∂s− b2

6∇2

r + ω(r; r0)

]G(r, r0; s, ω) = 0. (22)

Equation (22) is solved numerically with the respectiveinitial G(r, r0; 0, ω) = δ(r − r0) and boundary G(R, r0; s, ω)= 0 conditions. As such, Eqs. (20)–(22) represent the closeself-consistent system of three equations for the chain seg-ment density ρ(r; r0, ω), mean field ω(r; r0), and propagatorG(r, r0; N, ω), which is solved for given confinement geom-etry and external potential. Note that all these functions are

cylindrically symmetric since the tethering point is fixed, anddifferential equation (22) is essentially two-dimensional, seedetails in Sec. III.

Note that since, both ZtG[r0; N,ω] and G(r, r0; N, ω)

are the properties of the Gaussian chain tethered at r0, themean density defined by (21) should be treated as the den-sity of the ideal chain subjected to the mean field ω(r; r0).By this way, the self-consistent field approximation representsthe real chain as the ideal chain subjected to the mean field,which accounts for the excluded volume effect.

D. The mean field free energy of real chains

Substituting Eqs. (7) and (20) into Eq. (17), the free en-ergy (17) is expressed through the propagator G(r, r0; N, ω)and the mean segment density ρ(r; r0, ω) in the followingform:

F tR[r0; N,V ] = − ln

[∫drG(r, r0; N,ω)

]

− w

2

∫drρ(r; r0, ω)2. (23)

The derived formula is similar to that obtained by Netzand Shick for polymer brushes.43

In Ref. 16 authors ignored the second term in RHS ofEquation (23) when they modeled the process of unforcedtranslocation of real chains between spherical pores. How-ever, as shown in numerical examples given below, the sec-ond term in Eq. (20) is significant and has to be taken intoaccount in calculations of free energy of tethered real chains.In this paper, the cylindrical symmetry of functions G(r, r0;N, ω) and ρ(r; r0, ω) is rather important during the integra-tion over the spherical volume in Eq. (23). Neglect of thetwo-dimensional nature of the properties of tethered chainsmay significantly underestimate the contribution of the sec-ond term, especially when a large pore is considered. Whenthe radius of gyration of tethered chain is much larger thanthat of pore, the anisotropic character of density profiles canbe ignored due to strong spatial confinement, and the two-dimensional model derived here can be reduced to the onedimensional model.

III. CALCULATION DETAILS

In order to determine the energy landscape of the translo-cation chain, we have to solve the SCF equations (20)–(22) forthe cis and trans subchains. A numerical iteration process44 isadopted to solve the set of equations until G, ω, and ρ satisfythe self-consistent condition. Then the respective free ener-gies can be calculated according to Eq. (20). For simplicity offurther equations, we introduce the end-integrated propagator,

q(r; s, ω) =∫

dr′G(r, r′; s, ω), (24)

which satisfies the same Eq. (22) as G(r, r0; s, ω) but withdifferent initial condition q(r; 0, ω) = 1 in terms of its defini-tion. Note that the propagator G(r, r′; s, ω) satisfies Eq. (22)for any part of chain of length s. With the definition (24), themean segment density of a tethered chain of length N can be

214901-6 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

expressed as

ρ(r; r0, ω) =

∫ N

0dsG(r, r0; s, ω)q(r; N − s, ω)

q(r0; N,ω). (25)

The external adsorption potential is modeled as thesquare potential of depth V at the pore surface with the widthequal to Kuhn’s length b. In calculations, we use the dimen-sionless units of length reduced to Kuhn’s length b.

First let’s consider the trans subchain, which is confinedin spherical pore of radius R, with its end is tethered at thesurface of spherical pore (the opening). Then the density dis-tribution of chain is anisotropic. In principle we have to solveall SCF equations in spherical polar coordinates(r, θ , ϕ) withthe origin is located at the center of pore. However, the sit-uation will reduce to two-dimensional problem with the az-imuthal symmetry if we set the axis (r, θ = 0, ϕ) passingthrough the tethering point r0. All the quantities need only tobe determined in (r, θ ) space, where r ∈ [0, R], θ ∈ [0, π ]. Thetethering point coordinates is (r0, θ = 0). With above choice,in (r, θ ) space equation (22), which is the central quantityneed to be determined, transforms to45 (for tethered point r0

we ignore θ index)

∂

∂sG(r, θ, r0; s, ω)

= b2

6

[∂2

∂r2G(r, θ, r0; s, ω) + 2

r

∂

∂rG(r, θ, r0; s, ω)

+ 1

r2 sin θ

∂2

∂θ2G(r, θ,r0; s,ω)+ 1

r2 sin θ

∂

∂θG(r,θ,r0; s,ω)

]−ω(r, θ ; r0)G(r, θ, r0; s, ω). (26)

At the surface of pore, the boundary condition is G(r, θ ,r0; s, ω)|r = R = 0. Also, we set the boundary conditions of∂G/∂r = 0 at θ = 0 and θ = π to reflect the azimuthal sym-metry. The initial condition of G(r, r0; s = 0, ω) = δ(r − r0) isnontrivial. The tethering point has to be set at some distancea near the surface to avoid the conflict with zero boundaryconditions. According to Kong and Muthukumar,16 the coor-dinates of tethered point are chosen as (r0 = R − a, θ = 0)with a = b/2. Similarly, the same equation and boundary con-ditions as G apply to q(r, θ ; s). G(r, r′; s, ω) and q(r0; s, ω)can be solved using ADI technique.44

Once G and q are obtained, the mean density is deter-mined by

ρ(r, θ ; r0, ω)

=

∫ N

0dsq(r, θ ; s, ω)G(r, θ, r0; N − s, ω)

q(r = r0, θ = 0; N,ω). (27)

At last, free energy for a tethered chain in the spherical poreis given by

F t (r0; N,V ) = − ln[q(r = r0, θ = 0; N,ω)]

−πw

∫ R

04πr2dr

∫ π

0sin θdθρ(r, θ ; r0, ω)2.

(28)

Next we consider cis subchain and will use two settings.One kind of cis compartment corresponds to a spherical pore.In this case, the situation is completely the same as that oftrans compartment. Then we only need to do the same calcu-lation as trans subchain.

Another kind of cis compartment corresponds to a semi-infinite space, in which case we can choose different coordi-nates to solve those SCF equations. Spherical coordinates arestill used, but this time the position of opening of the pore ischosen as the coordinate origin and the radius is infinite. Sim-ilarly, all the quantities are only determined in (r, θ ) space.The integral space becomes r ∈ [0, ∞), θ ∈ [0, π /2], whereasthe coordinate of tethered point is (r0 = a, θ = 0). When solv-ing the equations for G(r, r0; s, ω) and q(r; s, ω), the Dirich-let boundary conditions are used at the boundaries, G = 0and q = 0 if r = ∞ or θ = π /2. For the special boundaries,θ = 0, the first derivative of propagator G or q with respect tor disappears. Namely, ∂G/∂r = 0 or ∂q/∂r = 0 if θ = 0.

The free energy of the chain tethered at the repulsive sur-face in semi-infinite space is given by

F t (r0; ∞, N) = − ln[q(r = a, θ = 0; N )]

−πw

∫ ∞

04πr2dr

∫ π/2

0sin θdθρ(r, θ ; r0, ω)2.

(29)

IV. FREE ENERGY LANDSCAPE

The above method provides for accurate computation ofthe free energy of tethered chains within the mean field frame-work and, as such, it allows one to investigate how variouscompetitive factors affect the free energy barrier during poly-mer translocation.

A. Chain length dependence of the free energyof confined tethered chain

Before determining the free energy barrier, it is importantto investigate the free energy of one chain tethered at a hard(non-adsorbing) surface, which corresponds to a semi-infinitecis compartment. Fig. 2 displays the chain length dependenceof the free energy for Gaussian and real chains. The solutionfor Gaussian chains is exact, and the free energy is equal tothe conformational entropy loss due to the presence of thehard wall. The excluded volume effect in real chains leadsto higher free energies compared to Gaussian chains, and thedifference progresses with the increase of the chain length.

The free energy of a tethered chain confined to a spher-ical pore shows an interesting behavior. Fig. 3 gives the freeenergy of Gaussian and real chains confined to a small spher-ical pore of radius 5 times larger than the Kuhn segmentlength (R = 5) as a function of chain length N at differentvalues of adsorption potential V. The free energy of Gaussianchain is proportional to the chain length except for very shortchains, which results from the linear dependence of confinedconformational entropy loss and adsorption energy gain onthe chain length. With the increase of the adsorption poten-tial magnitude, |V|, the free energy decreases. At a particular,so called, critical adsorption potential Vc, the enthalpy gain

214901-7 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

FIG. 2. Chain length dependence of the free energy of Gaussian and realchains tethered at the plain surface without adsorption.

compensates for the entropy loss. At such critical adsorptionconditions, the free energy of adsorbed chain is independentof its length. The critical adsorption potential separates unfa-vorable and favorable conditions of chain adsorption. Uponfurther increase of the adsorption potential, the chain free en-ergy decreases with its length, and, as such the chain translo-cation becomes more and more energetically auspicious. Onemay expect that the critical adsorption condition would bring

FIG. 3. Chain length dependence of the free energy of tethered chains con-fined to spherical pore of radius R = 5 at different magnitudes of the ad-sorption potential (V = 0, −0.2, −0.5, −0.7). (a) Gaussians chains. Criticalconditions are achieved at Vc = −0.442, at which the free energy is indepen-dent of the chain length. (b) Real chains (w = 0.5). Critical conditions do notexist. Note a non-monotonic behavior at V = −0.7 with a shallow minimum.

about two characteristic translocation regimes: slow translo-cation at weak adsorption, |V| < |Vc|, and fast translocationat |V| > |Vc|. The phenomenon of critical adsorption plays animportant role in polymer separation on porous substrates; itconstitutes the foundation of a special branch of modern chro-matography named liquid chromatography at critical condi-tions (LCCC).34 A detailed study of this phenomenon as re-lated to polymer chromatography is the subject of anotherpaper.46

The free energy of real chains is qualitatively differentdue to the contribution from excluded volume interactions,which depend on the chain length, pore size, and adsorptionpotential in a complex manner. The linear relationship be-tween the free energy and the chain length does not hold forreal chains. Although there is a prominent transition from un-favorable to favorable adsorption conditions with the increaseof the adsorption potential, the free energy within a certainrange of adsorption potentials is not monotonic. The free en-ergy of the trans subchain achieves a minimum at a certainsubchain length. One may expect that such a minimum maybring about a local equilibrium between cis and trans sub-chains that may significantly affect the translocation dynam-ics, as discussed in Sec. V C.

B. Free energy dependence on the adsorptionpotential

The conditions of critical adsorption and favorable andunfavorable regimes of translocation are demonstrated inFig. 4, where we present the free energy of tethered chainsof different length (N = 50, 100, and 200) in the pore of ra-dius R = 5, as a function of adsorption potential magnitude|V|. As a comparison, we also calculate the radius of gyra-tion of free polymer chains. For Gaussian chains it is givenbyRg = √

N/6 (in unit of b). For real chains we adopt an ap-proximate relationship47 of Rg ≈ 0.297N3/5. With these for-mulas, the radii of gyration of Gaussian chains of N = 50,100, and 200 are Rg = 2.89, 4.08, and 5.77, respectively. Thegyration radii of correspond real chains are Rg = 3.1, 4.7, and7.13, respectively. In the case of ideal chains, these functionsintersect in one point at the critical adsorption potential, Vc

= −0.442, at which the free energy of Gaussian chains is in-dependent of chain length N, as shown in Fig. 4(a). For realchains, in contrast, a single intersection point is not observedthat points toward a non-existence of true critical conditions.Also for sufficiently long chains, the free energy of real chainis much higher than that of Gaussian chain of the same length,which clearly indicates the significance of the excluded vol-ume interactions in confined real chains.

C. Free energy landscape of translocating chain

The free energy landscape represents the dependence ofthe free energy of the translocating chain on the degree oftranslocation, or the translocation coordinate, characterizedby the length of trans subchain n. The free energy of thetranslocating chain is presented as a sum of the free energiesof cis and trans subchains,

Ftotal(N, n) = F ttrans[n] + F t

cis[N − n]. (30)

214901-8 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

FIG. 4. Free energy of tethered chain confined to spherical pore of radius R= 5 as a function of the magnitude of adsorption potential for Gaussian (a)and real (b) chains of length N = 50, 100, and 200. The gyration radii of freeGaussian chains are Rg = 2.89, 4.08, and 5.77, respectively, for N = 50, 100,and 200. The gyration radii of free real chains are Rg = 3.1, 4.7, and 7.13,respectively.

Equation (30) is based on an assumption that both sic andtrans subchains are equilibrated; discussion on the validity ofthis assumption at experimental conditions of translocation isbeyond the scope of this paper.

In Fig. 5, we present the free energy landscape ofGaussian chain of length N = 200 at three different adsorptionpotentials (V = 0, −0.5, −0.65) from a semi-infinite space to aspherical pore of R = 5. In the absence of adsorption, the free

FIG. 5. Free energy landscape for Gaussian chain of length N = 200 (withRg = 5.77). Translocation from a semi-infinite space into spherical pore ofradius R = 5 at different adsorption potentials (V = 0, −0.5, −0.65).

energy increases to a maximum achieved near N = 200. Thismaximum corresponds to the free energy barrier of translo-cation. Since F t

trans[n]∣∣n→N

F tcis[N − n]

∣∣n→0 the translo-

cation of non-adsorbing Gaussian chain implies an increaseof the chain free energy and, as such, is an unfavorable pro-cess due to the large conformational entropy loss that can becompared to “up-hill” diffusion against the “entropic” force.When the adsorption energy is strong enough (V = −0.65),the free energy decreases with the degree of translocation thatcan be compared to “down-hill” diffusion accelerated by thedriving force. At the adsorption potential (V = −0.5) slightlysmaller that the critical adsorption potential, the free energyis almost flat and translocation can be compared with purediffusion.

In Fig. 6, we present characteristic examples of the freeenergy landscape for real chain (N = 200, w = 0.5) fortwo sizes of cis compartment: a semi-infinite space (a) anda spherical pore of Rcis = 10 (b). Trans compartment is thesame as in the above examples, a spherical pore of Rtrans

= 5. In contrast to ideal chains, the energy landscapes forreal chains are generally convex. The convexity is due tothe increase of the excluded volume interactions in transcompartment as the translocation progresses. One can distin-guish two characteristic regimes. At weak adsorption (−0.5< V < 0 in case of semi-infinite cis compartment), the freeenergy of cis subchain is significantly smaller than that oftrans subchain, F t

trans[n]|n→N F tcis[N − n]|n→0. The free

FIG. 6. Free energy landscape of translocation of real chain (w = 0.5) oflength N = 200 (with Rg = 7.13) into spherical pore of Rtrans = 5 at differentmagnitudes of the adsorption potential, V = 0, −0.5, −0.7, and −1.0. (a)Semi-infinite cis compartment and (b) cis compartment represented by a largespherical pore of Rcis = 10.

214901-9 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

energy increases monotonically with the degree of transloca-tion setting a high-energy barrier; as such, translocation im-plies an increase of the chain free energy and, as such, isunfavorable. At strong adsorption (V < −0.7), F t

trans[n]|n→N

� F tcis[N − n]|n→0, the free energy gradually decreases and

translocation is effectively facilitated. In the intermediateregime, (−0.7 < V < −0.5), the free energy landscape hasa minimum at a certain degree of translocation, which corre-sponds to a long-living conformation of the chain composedof cis and trans subchains. Such a minimum may cause verylong translocation time since the chain can be trapped in thisconformation. However, in contrast to ideal chains, the criti-cal adsorption potential, at which the free energy of adsorbedpolymer is independent of the polymer length, does not existfor real chains.

Another worth noting feature is that the free energy bar-riers are different between Rcis = 10 and Rcis = ∞ at the sameadsorption strength. The case of Rcis = 10 corresponds to ap-parent lower barrier, which result from the smaller conforma-tional entropy loss due to the change of spatial confinementfrom cis to trans.

D. The free energy barrier between two poresof equal size

Unforced translocation between two pores of equal sizewithout any external force involved is an instructive exampleto demonstrate interplay between entropic and excluded vol-ume effects. Fig. 7(a) displays the free energy landscape forGaussian chains, which is concave disregarding of the size ofconfinement. It is symmetric with a flat maximum at n = N/2that corresponds to the energy barrier that should be crossedin the course of successful translocation. The minimum isachieved at the initial and final conformation, F t

trans[n]|n→0 =F t

trans[n]|n→N .The energy landscapes for real chains are quite different

(Fig. 7(b)). The size of small compartments of R = 5 and 8is now comparable with the free chain size, Rg = 7.13, andthe exclusion volume interactions are significant. At theseconditions, the most energetically favorable conformation isachieved at n = N/2, when the chain is equally distributed be-tween cis and trans compartments reducing the exclusion vol-ume penalty. This mostly stretched conformation correspondsto the free energy minimum, which gets deeper as the poresize decreases. At R = 5, there is only one minimum of thesymmetric energy landscape, and one may expect equal prob-ability of successful and fail translocation events and long res-idence times due to long-living equilibrium symmetric con-formation at n = N/2. At R = 8, in addition to the minimumat n = N/2, there are two symmetric maxima to small(n N/2) and large (N – n N/2) degrees of translocationswhich posed energy barrier for entering into and exiting fromtrans compartment. In this case, one may expect smaller prob-ability of translocation and longer translocation time. A spe-cial case should be observed at a certain size of compartmentwhen the free energy of symmetric conformation at n = N/2equals the free energy of initial and final conformations. Forsufficiently large confinements of R = 12 and 20, which canaccommodate real chains of Rg = 7.13 without severe exclu-

FIG. 7. Free energy landscape for (a) Gaussian (w = 0, N = 200, Rg =5.78) and (b) real chains (w = 0.5, N = 200, Rg = 7.13) as a function oftranslocation coordinate n. Translocation between two equal spherical poreswithout adsorption (V = 0). Different curves correspond to different poresizes R given in the figures.

sion volume penalty, the energy landscapes are qualitativelysimilar to those for ideal chains and have one flat maximum.The described dependence of the energy landscape on the sizeof confinement is consistent with Monte Carlo simulation re-sults of Cifra.47

V. TRANSLOCATION DYNAMICS

A. The Fokker-Plank formalism

The FP formalisms model the diffusion in real space as arandom walk process over the free energy landscape associ-ated with the variation of an order parameter, which character-ized the progression of the transport process. The FP equationrelates the rate of transfer to the free energy gradient in thedirection transfer. As related to the problem of translocation,this order parameter is the degree of translocation, or the num-ber of chain segments transferred from cis to trans compart-ment. The free energy landscape is the free energy Ftotal(N,n) of the chain composed of cis and trans fragments tetheredto the pore opening, Eq. (30). The translocation dynamics wasinvestigated by the FP formalism in earlier papers by Park andSung35, 48 and Muthukumar,15, 16 which we follow.

Consider the function W (n, t ; n0, 0), which representsthe probability of a polymer chain of length N, which hadn0 segments in trans compartment (and N – n0 segmentsin cis compartment) at time 0, to have n segments in trans

214901-10 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

compartment at time t. As such, n is the variable order pa-rameter, which defines the state of the chain. In other words,W (n, t ; n0, 0) is the probability of translocation of n – n0 seg-ments within time t. The equation for W (n, t ; n0, 0) is derivedin a standard way, which we present below for the sake ofmethodological clarity. The balance equation in finite differ-ences for a small time interval �t reads

W (n, t + �t ; n0, 0) = W (n, t ; n0, 0)[1 − (k+n + k−

n

)�t]

+ k+n−1W (n − 1, t ; n0, 0)�t

+ k−n+1W (n + 1, t ; n0, 0)�t. (31)

Here, k+n is the rate constant of chain translocation by one

segment into trans compartment and transition from n to n+1state, and k−

n is the rate constant of chain translocation back-wards into cis compartment and transition from n to n–1 state.

Defining a probability flux from n to n – 1 state as

Jn,n+1 = knW (n, t ; n0, 0) − k′n+1W (n + 1, t ; n0, 0). (32)

Equation (31) is transformed into

∂

∂tW (n, t ; n0, 0) = Jn−1,n − Jn,n+1 = −∂J

∂n. (33)

The main assumption is that of equilibrium conforma-tions of cis and trans fragments of the translocating chain.We suppose that the polymer translocation process is slowenough compared to the thermodynamic relaxation time ofchain conformations in cis and trans compartments at anytime. As such, the rate constants for direct and reverse transi-tions between states n and n + 1 satisfy the detailed balancecondition,

k′n+1

kn

= exp(Fn+1 − Fn), (34)

where Fn = Ftotal(N, n)is the free energy of the chain with ntranslocated segments defined by Eq. (30). Using this equa-tion, we arrive at the FP equation in the following form:

∂

∂tW (n, t ; n0, 0)

= ∂

∂n

[kn

∂Fn

∂nW (n, t ; n0, 0) + kn

∂

∂nWn(n, t ; n0, 0)

]. (35)

The boundary and initial conditions for the partial differ-ential equation (35) are

W (n, t ; n0, 0) = 0 at n = 0 and n = N;

and W (n, t ; n0, 0) = δ(n − n0) at t = 0. (36)

The rate constant kn is referred to as a local frictionparameter36 and is taken in the following calculations as anindependent of m constant (kn = k0). Introducing the dimen-

sionless time τ = k0t, Eq. (35) can be expressed as

∂

∂τW (n, τ ; n0, 0)

= ∂

∂n

[∂Fn

∂nW (n, τ ; n0, 0) + ∂

∂nWn(n, τ ; n0, 0)

]. (37)

The time of successful translocation is determined as theprobability flux J|n = N at n = N, which represents the prob-ability per unit time that the chain will successfully passthrough the opening in time t starting from n0 segments lo-cated in trans compartment. Respectively, the time of unsuc-cessful translocation attempt, or the time of return to cis com-partment, is determined as the probability flux −J|n = 0 at n= 0. In terms of the probability flux, the translocation timeprobability distribution is given as

PT (τ )=J |n=N

=−[k0

∂Fn

∂nW (n, τ ; n0, 0)+k0

∂

∂nW (n, τ ; n0, 0)

]n=N

,

(38)

and the return time probability distribution is given as

PR(τ )=−J |n=0

=[k0

∂Fn

∂nW (n, τ ; n0, 0) + k0

∂

∂nW (n, τ ; n0, 0)

]n=0

.

(39)

The average times of successful 〈τ T〉 and unsuccessful〈τR〉 translocation attempts are given by

〈τT/R〉 =

∫ ∞

0τPT/R(τ )dτ∫ ∞

0PT/R(τ )dτ

. (40)

By the definition 〈τ T/R〉 depends on the initial state n0.In this paper, we consider the translocation process, whichstarts from the initial state with one translocated segment, andalways choose n0 = 1.

B. Translocation time distribution

As an instructive example of detailed calculations of thecharacteristics of the translocation process, we present inFig. 8, the normalized distribution function PT (τ )/P total

T fortranslocation time of a real polymer of length N = 200 at dif-ferent adsorption potentials transferred from a semi-infinitespace to the adsorbing spherical pore of radius R = 5. Thefree energy landscapes are shown in Fig. 6(a). The normal-ization constant equals to the total probability of successfultranslocation attempt,

P totalT =

∫ ∞

0PT (τ )dτ. (41)

Note the qualitative difference between the differenttranslocation regimes depending on the magnitude of theadsorption potential: week adsorption (−0.5 < V < 0)with monotonically increasing free energy; strong adsorption(V < −0.8) with monotonically increasing free energy, and

214901-11 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

FIG. 8. (a) Distribution functions of the translocation time for real chain oflength N = 200 at different adsorption potentials, V = 0, −0.2, −0.5, −0.7,−0.8, and −1.0. Note the qualitative difference between different transloca-tion regimes: week adsorption (−0.5 < V < 0) with monotonically increas-ing free energy; strong adsorption (V < −0.8) with monotonically increasingfree energy, and intermediate regime, exemplified by V = −0.7, with a non-monotonic free energy landscape having a minimum that corresponds to along-living metastable state. (b) Dependence of the mean translocation timeon the adsorption potential. (c) Dependence of the translocation probabilityon the adsorption potential.

intermediate regime, exemplified by V = −0.7, with a non-monotonic free energy landscape having a minimum that cor-responds to a long-living metastable state. In the latter case,the distribution function is the widest and the mean transloca-tion time is the largest. This is explained by the fact that thechain fluctuates around the metastable state for long time. Inthe absence of adsorption (V = 0), the probability of translo-cation is lowest, yet the translocation time is the shortest. Thisapparent contradiction is explained by the analogy of climb-ing up a slippery hill: the sharper the hill the faster should

be an attempt to be successful; a slow goer is doomed to fail.With the increase of adsorption driving force both the proba-bility of translocation and the translocation time increase, andthe time distribution becomes wider. While the former obser-vation is logical, the fact that the translocation time increaseswith the increase of the driving force is peculiar. The explana-tion is similar to that discussed above: as the hill becomes lesssteep, less rapid walkers may get a chance to succeed. How-ever, upon achievement the intermediate regime with transientmetastable states, the qualitative behavior changes. With thefurther increase of the driving force, the behavior becomesmore “logical”: the probability of translocation increases andthe translocation time decreases. This observation is impor-tant for theoretical models of translocation, which are basedon the scaling analysis and relate the translocation time to thedriving force using the monotonically increasing power func-tions. As shown in this example, such scaling approach maybe justified only when the driving force is large enough sothat the process of translocation is reminiscent to down-hilldiffusion.

C. Chain length dependence of the meantranslocation time

The chain length dependence of the mean translocationtime provides an instructive information for a better under-standing of the specifics of the translocation process. Themean translocation time as a function of the chain length forboth ideal and real chains at different adsorption potentials aregiven in Figs. 9 and 10, respectively. The corresponding freeenergy landscapes are given by Figs. 5 and 6(a). For idealchains, the translocation time is a concave function of thechain length N. If the adsorption potential is strong enough(|V| = 1.0) we find a linear dependence τ ∼ N . However,real chains do not follow a simple linear relationship betweentranslocation time and chain length: they might be concave,convex, or even of sigmoidal shape depending on the magni-tude of the adsorption potential.

As discussed above in Sec. V B with example ofdata presented in Fig. 8, the mean translocation time is anon-monotonic function or the magnitude of the adsorption

FIG. 9. The average translocation time of Gaussian chain (w = 0) as a func-tion of N is plotted for different adsorption potentials V. Translocation fromsemi-finite space (RA = infinite) to a small adsorbing pore (RB = 5). Thevalues of V are 0, −0.2, −0.4, −0.6, −0.8, and −1.0.

214901-12 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

FIG. 10. The average translocation time for real chain (w = 0.5) as a functionof N for different adsorption potentials. Translocation from semi-finite space(RA = infinite) to a small adsorbing pore (RB = 5). The values of V are (a) 0,−0.2, −0.4, −1.0; (b) −0.6, −0.7, −0.8, and −1.0.

potential. This effect is clearly seen in Figs. 9 and 10. Forideal chains, Fig. 9, the mean translocation time for unforcedtranslocation (V = 0) is comparable with that for adsorptiondriven translocation at V = −0.6, and the maximum time isobserved at V = −0.4. In Fig. 11, we present the transloca-tion time as a function of the magnitude of adsorption poten-tial for chains of different length. The non-monotonic behav-ior is more pronounced for longer chains. In the case of realchains, the maximum of the translocation time is related tothe existence of a shallow minimum on the respective free en-ergy landscapes shown in Fig. 6, which reflects the presenceof transient metastable states. In the case of Gaussian chains,the maximum corresponds to the values of the adsorption po-tential, at which the energy landscape is almost flat as seen inFig. 5, so that the driving force proportional the free energygradient is negligibly small.

D. Non-monotonic dependence of the translocationtime

The non-monotonic dependence of the translocation timeon the magnitude of the driving force deserves an additionaldiscussion beyond a qualitative allegory with up-hill climbingsuggested above. In order to get a better understanding on aqualitative level, let us consider a simple example.

Assume a linearly decreasing (down-hill) free energylandscape, which favors the translocation, Fn = −υ0n and∂Fn/∂n = −υ0. If at the beginning τ = 0 there are n0 segments

FIG. 11. The average translocation time for different chain length as a func-tion of the magnitude of adsorption potential. Translocation of when Gaus-sian chain (a) or real chain (b) from semi-finite space (RA = infinite) to asmall adsorbing pore (RB = 5).

in trans compartment, then the solution of Fokker-Plank equa-tion is given by20

W (n, τ ; n0, 0) = 2

N

∞∑m=1

exp(−λmτ ) exp(−υ0n0/2) sin(hmn0)

× exp(−υ0n/2) sin(hmn). (42)

Here hm = mπ /N and λm = υ20/4 + h2

m is the eigenvalues ofthe operator,

[∂2/∂n2 − υ0∂/∂n]fm(n) = −λmfm(n). (43)

As such, we can express the translocation time distributionfunction as

PT (τ ) = J |n=N = −2k0

Nexp

[υ0

2(N − n0)

]

×∞∑

m=1

exp(−λmτ )hm sin(hmn0) cos(hmN ). (44)

The integral of PT(τ ) represents the probability of trans-location,

P totalT =

∫ ∞

0PT (τ )dτ = −2k0

Nexp

[υ0

2(N − n0)

]

×∞∑

m=1

hm

λm

sin(hmn0) cos(hmN ). (45)

214901-13 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

Let us now consider the opposite situation with a linearlyincreasing free energy landscape, which favors the transloca-tion, Fn = υ0n and ∂Fn/∂n = υ0. The time distribution func-tion in this case differs from that given by Eq. (42) only by theprefactor, which is time independent, as the eigenvaluesλm arethe same. The probability of up-hill translocation is equal to

P totalT =

∫ ∞

0PT (τ )dτ = −2k0

Nexp

[−υ0

2(N − n0)

]

×∞∑

m=1

hm

λm

sin(hmn0) cos(hmN ). (46)

This result means that the normalized distributions ofthe translocation time for these two opposite situation are thesame, while the probability of up-hill translocation is signifi-cantly smaller than that of down-hill translocation. It is worthnoting that Park and Sung35 used the reflection boundary con-ditions, which prevent the chain return into the cis compart-ment, and at these conditions the translocation time monoton-ically decreases with the increase of driving force. However,the definition of the translocation time that we adopted re-flects the experimental measurements, in which the success-ful and unsuccessful translocation effects are separated duringthe recording.

E. Probability of successful translocation

In Fig. 12, we present the probability of successfultranslocation P total

T (41), as a function of the magnitude ofadsorption potential for ideal (a) or real (b) chains of dif-ferent length with example of translocation from semi-finitespace into spherical pore of radius R = 5, for which the freeenergy landscapes are given in Figs. 5 and 6(a). These de-pendencies are as expected. The probability of translocationincreases with the increase of the driving force. The proba-bility of translocation for ideal chains is larger than that forreal chains at the same conditions, since the excluded volumeinteraction hinders translocation into confining compartment.For reasonably long chains (N > 50) and large driving forcesin the regime of strong adsorption ((|V| > 0.8), the proba-bility of translocation does not any significantly depend onthe chain length. This is understandable, since in the regimeof strong adsorption, we deal with down-hill free energylandscapes, and the probability of return sharply decreaseswith the increase of the degree of translocation, as shown inSec. V F.

Our method can be generalized easily to the case withpore openings, which can accommodate more than one seg-ment. This situation for Gaussian chains has been studied byMuthukumar et al.17 and the application of our approach isstraightforward. Since the opening is not a planar hole but achannel, one only needs to add the adsorption energy relatedto chain block in the channel in Eq. (30). The free energylandscape apparently will be affect by the channel length es-pecially at the beginning and the end of translocation. How-ever, we will not pursue the detailed investigation about thiseffect.

FIG. 12. Probability of successful translocation as a function of the mag-nitude of adsorption potential for ideal (a) or real (b) chains of differentlength. Translocation from semi-finite space into spherical pore of radiusR = 5.

F. Distribution of translocation and return times

The distribution functions of successful translocationand return (failed translocation attempt) times are shown inFig. 13 drawing on the example of Gaussian and real chainsof length N = 30 transferred from the semi-infinite spaceto the spherical pore of radius R = 5 with adsorbing po-tential V = −1.0. Note that despite the significantly higher

FIG. 13. Distribution functions of successful translocation and returntimes for Gaussian (w = 0) and real (w = 0.5) chains of length N= 30. Translocation from semi-finite space into spherical pore of radiusR = 5 with adsorbing potential V = −1.0. Probabilities of successful translo-cation are 0.245 and 0.187 for Gaussian and real chains, respectively.

214901-14 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

probability of translocation for Gaussian chains, the time dis-tribution functions are pretty similar and reflect qualitativelygeneral features of the translocation process. The probabilityof return sharply decreases with the time of observation. Theprobability of successful translocation is negligibly small atshort times, which are not sufficient for the passage of thechain through the opening; it has a prominent maximum andlong tail. These features of the time distribution functions aresimilar to the experimental results on the forced translocationof single stranded DNA chains through a lipid bilayer withα-hemolysin pores.8 The translocation event in these experi-ments was detected as a prominent drop of the ionic currentdue to a partial pore blockage by moving chains. The dura-tion of the current drop was associated with the transloca-tion time. The authors distinguished “short-time” and “long-time” blockages, which may be related to failed and success-ful translocation attempts.

VI. CONCLUSIONS

We investigated the effect of adsorption on the dynam-ics of polymer translocation by means of the self-consistentfield theory and Fokker-Plank formalism. In doing so, we de-rived the SCFT equations for the free energy of a real chaintethered at the surface of confining compartment in the pres-ence of external field, calculated the free energy landscapesof the chain translocation from cis to trans compartment, anddetermined the distribution of translocation times by solutionof the FP equation of the chain diffusion along the free en-ergy landscape. The goal was to investigate how the exclusionvolume interactions, the chain length, and the magnitude ofthe adsorption potential affect the mechanism of the translo-cation process. We considered transfer of ideal (Gaussian)and real (with exclusion volume interactions) chains from thesemi-infinite space into the spherical pore through an open-ing, which can accommodate only one Kuhn segment. It isworth noting that the suggested method can be generalized tothe case of wider pore openings, following the approach em-ployed by Muthukumar et al.17 in studies of Gaussian chains.

The main results can be summarized as follows. There isa qualitative difference between the behavior of ideal and realchains. For ideal chains, the energy landscapes are linear. De-pending on the magnitude of the adsorption potential, one canidentify two regimes: weak adsorption, when the driving forcedoes not compensate for the loss of entropy, and the translo-cation represents an up-hill diffusion; strong adsorption, whenthe gain of enthalpy due to adsorption overweighs the entropyloss, and the translocation process represents a down-hill dif-fusion. These two regimes are separated by the critical ad-sorption potential, at which the free energies of the chain incis and trans compartments equal each other. The excludedvolume interactions bring about nonlinear features of free en-ergy landscapes. One can identify the intermediate range ofadsorption potentials, at which the energy landscape is non-monotonic and possesses a minimum, which corresponds toa long-living metastable state of the translocating chain. Atthese conditions, the process of translocation may take longtime, since the chain tends to fluctuate around this metastablestate prior to escape from the opening either into trans or

back into cis compartment. With the increase of the magni-tude of the adsorption potential, the probability of success-ful translocation sharply increases, as expected, in all cases.However, the mean translocation time and the distribution oftranslocation times depend on the adsorption strength non-monotonically. The longest translocation times correspond tothe intermediate regime between weak and strong adsorption.We also found nonlinear dependences of the translocationtime on the chain length, which qualitatively differ depend-ing on the adsorption strength. These observations show thatthe standard interpretation of the parameters of the translo-cation process in terms of scaling relationships, which implypower dependencies between the translocation time on onehand and the chain length and the driving force on anotherhand, have a limited applicability. They may be justified onlyfor sufficiently strong driving forces. In addition, we derivethe distribution functions for the times of successful and failtranslocation attempts, which are qualitative similar to thoseobserved in the experiments.

ACKNOWLEDGMENTS

This work was supported in parts by CBET National Sci-ence Foundation (NSF) grant “Multiscale Modeling of Ad-sorption Equilibrium and Dynamics in Polymer Chromatog-raphy,” NSF ECR “Structured Organic Particulate Systems,”and PRF-ACS grant “Adsorption and Chromatographic Sepa-ration of Chain Molecules on Nanoporous Substrates.”

1B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson,Molecular Biology of the Cell (Garland, New York, 2002).

2J. O. Bustamante, J. A. Hanover, and A. Liepins, J. Membr. Biol. 146, 239(1995).

3S. Casjens, Virus Structure and Assembly (Jones and Bartlet, Boston, 1985).4G. Schatz and B. Dobberstein, Science 271, 1519 (1996).5D. W. Deamer and M. Akeson, Trends Biotechnol. 18, 147 (2000).6A. Milchev, J. Phys. Condes. Matter 23, 103101 (2011).7M. Akeson, D. Branton, J. J. Kasianowicz, E. Brandin, and D. W. Deamer,Biophys. J. 77, 3227 (1999).

8J. J. Kasianowicz, E. Brandin, D. Branton, and D. W. Deamer, Proc. Natl.Acad. Sci. U.S.A. 93, 13770 (1996).

9S. E. Henrickson, M. Misakian, B. Robertson, and J. J. Kasianowicz, Phys.Rev. Lett. 85, 3057 (2000).

10W. Sung and P. J. Park, Phys. Rev. Lett. 77, 783 (1996).11M. Muthukumar, Phys. Rev. Lett. 86, 3188 (2001).12E. A. DiMarzio and A. J. Mandell, J. Chem. Phys. 107, 5510 (1997).13J. L. Viovy, Rev. Mod. Phys. 72, 813 (2000).14J. Chuang, Y. Kantor, and M. Kardar, Phys. Rev. E 65, 011802 (2002).15M. Muthukumar, J. Chem. Phys. 111, 10371 (1999).16C. Y. Kong and M. Muthukumar, J. Chem. Phys. 120, 3460 (2004).17C. T. A. Wong and M. Muthukumar, J. Chem. Phys. 128, 154903

(2008).18C. T. A. Wong and M. Muthukumar, Biophys. J. 95, 3619 (2008).19J. K. Wolterink, G. T. Barkema, and D. Panja, Phys. Rev. Lett. 96, 208301

(2006).20D. K. Lubensky and D. R. Nelson, Biophys. J. 77, 1824 (1999).21A. Milchev, K. Binder, and A. Bhattacharya, J. Chem. Phys. 121, 6042

(2004).22A. Matsuyama, M. Yano, and A. Matsuda, J. Chem. Phys. 131, 105104

(2009).23A. Matsuyama, J. Phys. Condes. Matter 17, S2847 (2005).24M. G. Gauthier and G. W. Slater, J. Chem. Phys. 128, 065103 (2008).25M. Fyta, S. Melchionna, S. Succi, and E. Kaxiras, Phys. Rev. E 78, 036704

(2008).26S. S. Chern, A. E. Cardenas, and R. D. Coalson, J. Chem. Phys. 115, 7772

(2001).

214901-15 S. Yang and A. V. Neimark J. Chem. Phys. 136, 214901 (2012)

27M. Bernaschi, S. Melchionna, S. Succi, M. Fyta, and E. Kaxiras, Nano Lett.8, 1115 (2008).

28A. Bhattacharya, W. H. Morrison, K. Luo, T. Ala-Nissila, S. C. Ying,A. Milchev, and K. Binder, Eur. Phys. J. E 29, 423 (2009).

29K. F. Luo, R. Metzler, T. Ala-Nissila, and S. C. Ying, Phys. Rev. E 80,021907 (2009).

30W. C. Yu and K. F. Luo, J. Am. Chem. Soc. 133, 13565 (2011).31S. Alapati, D. V. Fernandes, and Y. K. Suh, J. Chem. Phys. 135, 055103

(2011).32M. Muthukumar, J. Chem. Phys. 118, 5174 (2003).33S. Matysiak, A. Montesi, M. Pasquali, A. B. Kolomeisky, and C. Clementi,

Phys. Rev. Lett. 96, 118103 (2006).34H. Pasch and B. Trathnigg, HPLC of Polymers (Springer-Verlag, Berlin,

1999).35P. J. Park and W. Sung, J. Chem. Phys. 108, 3013 (1998).36H. Riskin, The Fokker-Plank Equation (Spinger-Verlag, Berlin, 1984).

37K. F. Freed, Adv. Chem. Phys. 22, 1 (1972).38G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers

(Clarendon, Oxford, 2006).39E. Helfand, J. Chem. Phys. 62, 999 (1975).40M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford

University Press, New York, 1986).41S. F. Edwards, Proc. Phys. Soc. 85, 613 (1965).42S. F. Edwards, Proc. Phys. Soc. 88, 265 (1966).43R. R. Netz and M. Schick, Europhys. Lett. 38, 37 (1997).44S. A. Yang, D. D. Yan, H. G. Tan, and A. C. Shi, Phys. Rev. E 74, 041808

(2006).45R. Kumar and M. Muthukumar, J. Chem. Phys. 131, 194903 (2009).46S. Yang and A. V. Neimark, “Critical condition in chromatography of poly-

mers” (unpublished).47P. Cifra, Macromolecules 38, 3984 (2005).48P. J. Park and W. Sung, Phys. Rev. E 57, 730 (1998).