Confined diffusion in a random Lorentz gas environment
CConfined diffusion in a random Lorentz gas environment
Narender Khatri ∗ and P.S. Burada
1, 2, † Department of Physics, Indian Institute ofTechnology Kharagpur, Kharagpur - 721302, India Center for theoretical studies, Indian Institute ofTechnology Kharagpur, Kharagpur - 721302, India (Dated: July 15, 2020)
Abstract
We study the diffusive behavior of biased Brownian particles in a two dimensional confined geom-etry filled with the freezing obstacles. The transport properties of these particles are investigatedfor various values of the obstacles density η and the scaling parameter f , which is the ratio of workdone to the particles to available thermal energy. We show that, when the thermal fluctuationsdominate over the external force, i.e., small f regime, particles get trapped in the given environmentwhen the system percolates at the critical obstacles density η c ≈ .
2. However, as f increases, weobserve that particles trapping occurs prior to η c . In particular, we find a relation between η and f which provides an estimate of the minimum η up to a critical scaling parameter f c beyond whichthe Fick-Jacobs description is invalid. Prominent transport features like nonmonotonic behavior ofthe nonlinear mobility, anomalous diffusion, and greatly enhanced effective diffusion coefficient areexplained for various strengths of f and η . Also, it is interesting to observe that particles exhibitdifferent kinds of diffusive behaviors, i.e., subdiffusion, normal diffusion, and superdiffusion. Thesefindings, which are genuine to the confined and random Lorentz gas environment, can be useful tounderstand the transport of small particles or molecules in systems such as molecular sieves andporous media which have a complex heterogeneous environment of the freezing obstacles. ∗ [email protected] † Corresponding author: [email protected] a r X i v : . [ phy s i c s . b i o - ph ] J u l . INTRODUCTION Diffusion of small particles or molecules in a crowded environment is ubiquitous in sev-eral physical, chemical, and biological processes [1, 2]. Often, these particles encounter aheterogeneous environment [3–7] formed by obstacles distributed in an irregular fashion.This may control the diffusive behavior of the particles. Examples are, particle diffusion inbiological environments containing lipids and proteins [2, 8], diffusion in living cells crowdedwith cytoplasmic and nuclear environments [9], diffusion in porous soil columns [10], etc.From the theoretical perspective, the complex heterogeneous environment is modeled as arandom Lorentz gas [11–13], where the freezing obstacles are randomly distributed with agiven area fraction. The properties of this random Lorentz gas depend entirely on the ob-stacles density. For example, at the higher obstacles density, percolating clusters [14] willform, which effectively control the diffusive behavior of the particles. Note that when thesystem percolates at the critical obstacles density, the clusters can even arrest or trap theparticles [13, 15].On the other hand, when the particles diffuse in a confined geometry, their motion ishighly controlled by the structure of the geometry [16–22], e.g., ion channels [23], zeolites [24],microfluidic devices [25], ratchets [26–30] and artificial channels [31]. The irregular shapeof the structure gives rise to entropic barriers which play a prominent role in the diffusivebehavior of the particles [32–41]. In these confined geometries, the transport characteristicsare controlled by the effective free energy, which is a function of applied bias and the entropicpotential [19, 21, 34–36]. When the shape of the geometry is periodic and regular, toanalytically calculate the transport characteristics of the non-interacting particles, one canuse the Fick-Jacobs theory [32, 33], which assumes a faster equilibration of the diffusingparticles in the transversal direction of the channel compared to its longitudinal direction.The prominent transport features reported in these structures include a decrease in averageparticle velocity upon increasing the noise strength and exhibiting an enhanced effectivediffusion coefficient in highly confined geometries [34–36, 41]. However, quite often, particlesencounter a crowded environment while passing through the confined structures [9, 42, 43]such as biological cells, microfluidic channels, blood vessels, and porous media. Due to thecombined effect of density of obstacles and structure of the confined geometry, the transportproperties of the particles may exhibit interesting behaviors.2n this article, we study the diffusive behavior of point size particles, moving in a twodimensional confined channel filled with the freezing obstacles. These particles are subjectedto a constant external bias along the channel direction. Here, we consider the steric interac-tion between the diffusing particles and the freezing obstacles. We aim to find the transportcharacteristics of the Brownian particles, i.e., the nonlinear mobility and the effective diffu-sion coefficient, in the aforementioned conditions.Rest of this article is organized as follows. In section -II, we introduce our model for thebiased Brownian particles in a two dimensional confined channel with a random Lorentz gasenvironment. The impacts of such a confined and crowded heterogeneous environment onthe nonlinear mobility and the effective diffusion of the particles are presented in section-III and section -IV, respectively. Finally, we present our main conclusions in section -V.
II. MODEL
Consider an overdamped Brownian particle suspended in a two dimensional symmetricchannel consisting of a heat bath of friction coefficient γ at the temperature T . Also, aheterogeneous environment of freezing obstacles (spatial Poisson point process) [44], modeledby a random Lorentz gas, is present inside the channel as illustrated in Figure 1. The shapeof the two dimensional symmetric and spatially periodic channel is described by its half-width (see Fig. 1) ω ( x ) = a sin (2 πx/L ) + b, where L corresponds to the periodicity of thechannel, and the parameters a and b control the slope and channel width at the bottleneck,respectively. Here, we choose a = 1 / π , b = 1 . / π , and L = 1.The particle is driven by a constant external force (cid:126)F along the direction of the channeland the interaction force (cid:126)F int due to the obstacles. In the overdamped regime [45], theequation of motion of the particle is given by the 2D Langevin equation, γ d(cid:126)rdt = (cid:126)F + (cid:126)F int + (cid:112) γk B T (cid:126)ξ ( t ) , (1)where (cid:126)r is the position of the particle in two dimensions and k B is the Boltzmann constant.The thermal fluctuations due to the coupling of the particle with the surrounding heat bathare modeled by a zero-mean Gaussian white noise (cid:126)ξ ( t ), with the autocorrelation function (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = 2 δ ij δ ( t − t (cid:48) ) for i, j = x, y .As the channel geometry is periodic along the longitudinal direction, we use the periodic3 IG. 1: (Color online) Schematic illustration of a two dimensional symmetric channel in whichthere is a heterogeneous environment of freezing obstacles modeled by a random Lorentz gas, withthe periodicity L confining the motion of a Brownian particle which is subjected to a constant force (cid:126)F along the x − direction. The reflecting channel boundaries assure the confinement of a particleinside the channel. boundary conditions. If each obstacle has the same radius R , then the obstacles occupy anarea fraction η = nπR /A , where A = 2 bL is the area of a single cell of the channel and n denotes the total number of obstacles in the cell. However, the obstacles can fully overlapeach other, thus the actual area fraction is given by [46] φ = 1 − e − η . (2)The interaction force on a particle i due to the freezing obstacles is assumed to be of thelinear spring form [47], which reads (cid:126)F int = k s n (cid:88) j =1 ( R − r ij )ˆ r ij , for r ij < R , (3)where the sum is taken over all the obstacles within the cell in which the particle is presentat the given instant of time, k s denotes the spring constant, and r ij denotes the center tocenter distance between the particle i and the obstacle j . For r ij < R , the obstacle stronglyrepels the particle, however, for r ij ≥ R , there is no interaction between the obstacle andthe particle. In order to mimic hard-core pure volume exclusion, we use a large value of k s ,ensuring that the particle does not overlap with the obstacles.For the sake of a dimensionless description, we henceforth scale all lengths by the peri-odicity of the channel L and time by τ = γL / ( k B T ). In dimensionless variables, the 2D4angevin equation (Eq. 1) reads d(cid:126)rdt = (cid:126)f + (cid:126)f int + (cid:126)ξ ( t ) , (4)where (cid:126)f = f ˆ x ( f = F L/k B T ) denotes the dimensionless external force, which is the ratioof work done on the particle due to the external force and the available thermal energy,and the dimensionless interaction force becomes (cid:126)f int = (cid:126)F int L/k B T . For our simulations, weconsider the Brownian particles of point size with random initial conditions and overlappingwith the obstacles is taken care by the interaction force (cid:126)f int . The Langevin equation (Eq.4) is solved by using the standard stochastic Euler algorithm over 5 × trajectories witha time step 10 − . The reflecting channel boundaries assure the confinement of particlesinside the channel. For the simulations, we have chosen the dimensionless spring constant k ( k = k s L/k B T ) = 5 × and the dimensionless radius of the obstacle R = 0 . III. NONLINEAR MOBILITY
In the absence of a random Lorentz gas environment, i.e., η = 0, the nonlinear mobilityof biased Brownian particles in a symmetric confined environment has been studied earlierboth analytically and numerically by H¨anggi and co-workers [34–36, 41]. It has been shownthat under the assumption of a fast equilibration in the transverse direction of the channel,the 2D Smoluchowski equation can be reduced to an effective 1D equation (the Fick-Jacobsequation), reading in the dimensionless form [34, 41] ∂P ( x, t ) ∂t = ∂∂x D ( x ) (cid:26) ∂P ( x, t ) ∂x + A (cid:48) ( x ) P ( x, t ) (cid:27) , (5)where P ( x, t ) denotes the reduced probability density, D ( x ) = 1 / (1 + ω (cid:48) ( x ) ) / is the po-sition dependent diffusion coefficient for a 2D system [48], the dimensionless free energy isgiven by A ( x ) = − f x − ln(2 ω ( x )), and 2 ω ( x ) is the local width of the two dimensionalchannel. Note that, here, the free energy assumes the form of a periodic tilted potentialwhose barrier height is a function of the temperature [34]. Using the mean first passage time(MFPT) approach [34, 35], the nonlinear mobility can be obtained as µ := (cid:104) ˙ x (cid:105) f = 1 − e − f f (cid:90) I ( z ) dz , (6)5 . . .
751 1 10 100 1000 µ f . . f = 1 η η = 0 η = 0 . η = 0 . η = 0 . η = 1 η = η c = 1 . FIG. 2: (Color online) The nonlinear mobility µ versus the scaling parameter f for various values ofthe obstacles density η . The dashed line corresponds to the analytical findings for η = 0. The insetdepicts the dependence of µ on η for f = 1. The other set parameters are a = 1 / π , b = 1 . / π ,and R = 0 . where the integral I ( z ) = e A ( z ) /D ( z ) (cid:82) zz − e − A ( y ) dy . The same quantity is calculated usingthe numerical simulations for the 2D channel as µ := lim t →∞ (cid:104) x ( t ) (cid:105) t f . (7)Figure 2 depicts the nonlinear mobility µ as a function of the scaling parameter f forvarious values of the obstacles density η . The nonlinear mobility is greatly influenced bythe obstacles density. For lower values of η ( < . µ monotonically increases with f andattains the bulk value in the limit f → ∞ . As expected, the numerical simulations resultsare in good agreement with the analytical findings when η → f . It is due to the failure of the thermal equilibrationassumption in this limit. Interestingly, on further increasing η , µ exhibits a nonmonotonicbehavior. It is because, for higher f values, the freezing obstacles slow down the motion ofparticles and the mobility becomes zero. Surprisingly, in the lower f limit, the nonlinearmobility is unaltered and agrees well with the analytical findings up to the obstacles density η ≈ .
92 (see inset of Fig. 2). This indicates that, in this limit, the obstacles do not havemuch impact on the diffusion of particles and the thermal equilibration assumption is stillvalid (see Fig. 3(b)). Here, the thermal fluctuations dominate, and the particles can movefreely in the 2D channel. However, at the critical obstacles density η c ≈ .
2, obstacles form6 = 0 η = 0 . η = η c = 1 . η = 0 η = 0 . η = 1 . η = 0 η = 0 . η = 0 . f = 1 f = 100 f = 1000(a) (b) (c)(d) (e) (f)(g) (h) (i) FIG. 3: (Color online) Steady state evolution of a Brownian particle, within a finite time window,for various values of the obstacles density η . Here, the position of the particle is mapped intoa single cell of the 2D channel filled with the heterogeneous distribution of obstacles. Top panel(a)-(c) for f = 1, middle panel (d)-(f) for f = 100, and bottom one (g)-(i) for f = 1000. The otherset parameters are a = 1 / π , b = 1 . / π , and R = 0 . percolating clusters that span the cell of the 2D channel. As expected, the particles gettrapped or localized in these clusters, and the nonlinear mobility becomes zero (see insetof Fig. 2). Note that in a two dimensional square box the critical obstacles density for thepercolation of disks is η c ≈ .
128 [14], while for our system it is obtained as η c ≈ .
2. Thisindicates that η c depends on the shape of the channel structure.In general, according to the ergodic hypothesis [49], the steady state evolution of a Brow-nian particle within a time window is equivalent to the steady state distribution of theparticles. Figure 3 depicts the steady state behavior of a single particle in a finite timewindow. Note that since the channel is periodic and the distribution of obstacles in a givencell is heterogeneous, we have mapped the position of the particle into a single cell. When f value is small, the particle can explore uniformly in the transversal direction of the cell,and this indeed satisfies the thermal equilibration assumption [35, 41] up to η ≈ .
92 (see(a)-(b) in Fig. 3). This reflects the fact that, in the lower f limit where the thermal fluctu-7 . . .
751 0 . . . η f Numerical dataFitting
FIG. 4: (Color online) The η versus f relation which provides an estimate of the minimum η up toa critical scaling parameter f c ∼
51 beyond which the Fick-Jacobs description is not valid. Symbolsrepresent numerical data, and the solid line is an empirical relation η ( f ) = c − c erf(( f − λ ) / Σ),where the parameters are c = 0 . , c = 0 . , λ = 9, and Σ = 5. The other set parameters are a = 1 / π , b = 1 . / π , and R = 0 . ations dominate, the obstacles do not influence the particle diffusion. Whereas, by furtherincreasing η , the thermal equilibration assumption is no longer valid because the percolatingclusters hinder the diffusion of the particle in both the longitudinal and transversal direc-tions. Note that when the system percolates at the critical obstacles density η c ≈ .
2, theparticle gets trapped in the given environment (see Fig. 3(c)). However, for the moderateand higher f values, i.e., when the thermal fluctuations are less dominant, the particle getstrapped in the given environment well before the critical obstacles density η c (see (f) and (i)in Fig. 3). For the moderate f value the particle gets trapped at η ≈ .
1, and for the higher f value this occurs at η ≈ .
4. As reported earlier [41], in η → f values, the Brownian particle evolution tends to focus at the middle and exit of thechannel evidencing the failure of the thermal equilibration assumption (see (d) and (g) inFig. 3). This is because the influence of the external force, which tries to drag the particlein its direction, is much more effective than the thermal noise present in the system.In the absence of a random Lorentz gas environment, i.e., η = 0, Burada et al. [41] analyti-cally obtained a critical scaling parameter, which is given by f c ∼ L [1 −(cid:104) ω (cid:48) ( x ) (cid:105) ] / (2 (cid:104) ω ( x ) (cid:105) ),where (cid:104)· · · (cid:105) denotes the average over the period L of the channel. This critical scaling pa-8ameter f c provides an estimate of the minimum forcing beyond which the Fick-Jacobsdescription is expected to fail in providing an accurate description of the dynamics of thesystem. For the considered channel structure, we find that f c ∼
51, up to which the numer-ical simulations results are in good agreement with the analytical findings when η → η up to f c below which the thermal equilibration assumption is naturally satisfied. In otherwords, there exists a η versus f boundary below which the numerical results agree well withthe analytical finding for η = 0 (see Fig. 4). Unfortunately, in the presence of a heteroge-neous environment inside the channel formed by freezing obstacles in an irregular fashion,an explicit analytical expression of η up to f c beyond which the Fick-Jacobs description isnot valid cannot be obtained. However, by looking at the behavior of η versus f in Figure 4,an empirical relation between η and f can be obtained as η ( f ) = c − c erf(( f − λ ) / Σ).After fitting this function to the numerical data, the constant parameters values can be readas c = 0 . , c = 0 . , λ = 9, and Σ = 5. Thus, the rate at which η decays exponentiallywith f is 1 / Σ. IV. EFFECTIVE DIFFUSION
The effective diffusion is characterized by the mean-squared deviation (variance) of theparticle position x ( t ), i.e., (cid:104) ∆ x ( t ) (cid:105) = (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) . This variance obeys a power law ∼ t α , where the power α decides the nature of the particle diffusion. If the long timebehavior assumes a linear function of time, i.e., α = 1, it is a normal diffusion [50]. On theother hand, any deviation from the strict linear behavior at asymptotic times is termed asanomalous diffusion [50, 51]. For example, if 0 < α < < α < α = 0 corresponds to zero diffusion, i.e.,the trapped state.In the η = 0 limit, the effective diffusion coefficient can be obtained using the Fick-Jacobsequation [34, 35]. It is given by D eff D = (cid:90) (cid:90) xx − D ( z ) D ( x ) e A ( x ) e A ( z ) [ I ( z )] dx dz (cid:20)(cid:90) I ( z ) dz (cid:21) , (8)where I ( z ) is same as mentioned before in equation (6). Numerically, for any arbitrary9
12 0 . . . α ( t ) t .
12 0 . . . . . α ( t ) . . . . . . α ( t ) . . . . ηηη (a)(b)(c) f = 1 f = 100 f = 1000 FIG. 5: (Color online) Local exponent α ( t ) for various values of the obstacles density η and thescaling parameter f . The other set parameters are a = 1 / π , b = 1 . / π , and R = 0 . obstacles density, the local exponent and the corresponding local effective diffusion coefficientare, respectively, calculated as [13] α ( t ) := d log( (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) )d log t , (9) D eff ( t ) := 12 d( (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) )d t . (10)Note that the steady state values of the exponent and the corresponding effective diffusioncoefficient can be obtained as α = lim t →∞ α ( t ) and D eff = lim t →∞ D eff ( t ), respectively.Results for the local exponent α ( t ) are depicted in Figure 5 for various values of theobstacles density η and the scaling parameter f . Note that for the small f value, when10 < η c ≈ .
2, the local exponent fluctuates around one in the long time limit. In thislimit, as discussed earlier, the thermal fluctuations dominate over the external force, andthe particles exhibit normal diffusive behavior. Whereas, when η ≥ η c , particles get trappedin the percolating clusters, long chains with length in the order of periodicity of the channel.As a result, the local exponent becomes zero in the steady state limit.On the other hand, for the moderate f value, the obstacles do not influence the normaldiffusion of the particles up to η < .
94, thus the local exponent fluctuates around unityin the long time limit. Interestingly, by further increasing η , the diffusing particles aretrapped partially in cavities formed by the obstacles and escape from them due to thermalfluctuations. As a result, particles exhibit long jumps, i.e., so called L´evy flights [52].Correspondingly, the particles show superdiffusive behavior. For η ≥ .
1, as mentionedearlier, the particles get trapped completely in the percolating clusters. Therefore, the localexponent becomes zero in the long time limit.At the higher value of f , for η < .
12, as expected, the local exponent fluctuates aroundone in the long time limit, illustrating that the particles exhibit normal diffusion. An increasein η leads to partial trapping of particles at the obstacles boundaries for long times becausethe external force dominates over the thermal fluctuations present in the system which triesto drag the particle along its direction, i.e., a straight line motion. As a result, the particlesexhibit subdiffusive behavior. By further increasing η , particles are trapped partially incavities formed by the obstacles and while escaping, they exhibit long jumps which result intothe superdiffusive behavior. As mentioned earlier, the particles get trapped when η ≥ . α and the correspondingscaled effective diffusion coefficient D eff /D as a function of the obstacles density η forvarious values of the scaling parameter f . For small f , as discussed before, the particlesexhibit normal diffusion for η < η c , and by further increasing η , the particles get trappedinstantaneously (see Fig. 6(a)). Interestingly, the corresponding D eff /D remains unalteredand agrees well with the analytical findings (for η = 0, dashed line) up to η ≈ − . . . D e ff / D η . . α (a)(b) f SuperdiffusionSubdiffusionTrapped state → FIG. 6: (Color online) The steady state behaviors of the exponent α (a) and the correspondingscaled effective diffusion coefficient D eff /D (b) as a function of the obstacles density η for variousvalues of the scaling parameter f . The dashed line in (b) corresponds to analytical finding for η = 0 and f = 1. The other set parameters are a = 1 / π , b = 1 . / π , and R = 0 . Fig. 6(b)), reflecting the fact that for this parameter range when the thermal fluctuationsdominate over the external force, the obstacles do not have any influence on the effectivediffusion of the particles. Whereas, D eff /D drops to zero for η ≥ η c , indicating that theparticles are trapped in the given environment. Note that new features start to emergeas the scaling parameter is increased to moderate and higher values. For the moderate f ,as explained earlier, the normal diffusion is followed by the superdiffusion, and then theparticles get trapped in the given environment well before η c . Whereas, for the higher f values, i.e., when f ≥ η c . In particular, for some intermediate η values, particles exhibit a normaldiffusive behavior during the cross over between either subdiffusion to superdiffusion or12uperdiffusion to trapped state. Here, it is important to point out that the obstacles formpercolating clusters inside the channel at the higher obstacles density η , which slow downthe motion of particles for the moderate scaling parameter, i.e., f = 100, even though theprocess is still normal. Therefore, the average survival time of the particles in a cell increases.This behavior plays an important role in the diffusion process, which decreases the nonlinearmobility (see Fig. 2) and results in an enhancement of the effective diffusion coefficient (seeFig. 6(b)). This fact has already been reported in references [20, 21]. Interestingly, theeffective diffusion coefficient is greatly enhanced when the particles exhibit subdiffusive orsuperdiffusive behavior for various values of η at the higher scaling parameter values. Thisis because of the partial trapping of particles for long times at the obstacles boundaries orinside the cavities formed by the freezing obstacles. However, as expected, D eff /D dropsdown to zero when the particles are trapped. V. CONCLUSIONS
In this work, we have studied the diffusive behavior of the biased Brownian particles ina two dimensional confined and heterogeneous environment of freezing obstacles modeledby a random Lorentz gas. We have shown that the transport properties and the dynamicsof the particles exhibit peculiar features for various values of the obstacles density η andthe scaling parameter f . Using Brownian dynamics simulations, we have observed thatthe particles get trapped or localized in the given environment for any value of the scalingparameter when the system percolates at the critical obstacles density η c ≈ .
2, hence for η ≥ η c , the nonlinear mobility and the scaled effective diffusion coefficient become zero.Further, we have numerically demonstrated that the particles can be easily trapped priorto the critical obstacles density η c for the moderate and higher scaling parameter values.Interestingly, the nonlinear mobility exhibits a nonmonotonic behavior as a function of thescaling parameter for various values of the obstacles density. In addition, the effectivediffusion coefficient shows a greatly enhanced behavior for various values of the obstaclesdensity at the moderate and higher scaling parameter values. Moreover, we have obtaineda relation between η and f which provides an estimate of the minimum η up to a criticalscaling parameter f c beyond which the Fick-Jacobs description is invalid. Also, it has beenobserved that the particles exhibit normal diffusion, anomalous diffusion, and trapped state,13hich are genuine to the confined and random Lorentz gas environment, depending on thescaling parameter for various values of the obstacles density.The approach of effectively controlled diffusion of particles in a confined and randomLorentz gas environment by varying the obstacles density could be applied to a wide rangeof applications, including biochemical reactions in living cells which occur in a complexheterogeneous media [2], particles transport in crowded cellular environments [53], dispersivetransport in disordered semiconductors [54], controlled drug release [55], etc. In the future,the current study can be easily extended in different directions. Important examples wemention, (i) the generalization of freezing obstacles to mobile obstacles with kinetic rate κ [56–59] and (ii) the channel walls can be considered as fluctuating walls [60]. VI. ACKNOWLEDGEMENTS
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