"Paris car parking problem" for partially ordered discorectangles on a line
Nikolai I. Lebovka, Mykhailo O. Tatochenko, Nikolai V. Vygornitskii, Yuri Yu. Tarasevich
““Paris car parking problem” for partially ordered discorectangles on a line
Nikolai I. Lebovka,
1, 2, ∗ Mykhailo O. Tatochenko, Nikolai V. Vygornitskii, and Yuri Yu. Tarasevich † Department of Physical Chemistry of Disperse Minerals,F. D. Ovcharenko Institute of Biocolloidal Chemistry, NAS of Ukraine, Kyiv 03142, Ukraine Department of Physics, Taras Shevchenko Kiev National University, Kyiv 01033, Ukraine Mykolaiv Professional Shipbuilding Lyceum, Mykolaiv 54011, Ukraine Laboratory of Mathematical Modeling, Astrakhan State University, Astrakhan 414056, Russia (Dated: June 26, 2020)The random sequential adsorption (RSA) of identical elongated particles (discorectangles) on aline (“Paris car parking problem”) was studied numerically. An off-lattice model with continuouspositional and orientational degrees of freedom was considered. The possible orientations of thediscorectanles were restricted between θ ∈ [ − θ m ; θ m ] while the aspect ratio (length-to-width ratio)for the discorectangles was varied within the range ε ∈ [1; 100]. Additionally, the limiting case ε = ∞ (i.e., widthless sticks) was considered. We observed, that the RSA deposition for the problem underconsideration was governed by the formation of rarefied holes (containing particles oriented along aline) surrounded by comparatively dense stacks (filled with almost parallel particles oriented in thevertical direction). The kinetics of the changes of the order parameter, and the packing density arediscussed. Partial ordering of the discorectangles significantly affected the packing density at thejamming state, ϕ j , and shifted the cusps in the ϕ j ( ε ) dependencies. This can be explained by theeffects on the competition between the particles’ orientational degrees of freedom and the excludedvolume effects. I. INTRODUCTION
The random sequential adsorption model (RSA) iswidely used for modeling the packing of particles inspaces of different dimensionalities [1, 2]. In this model,particles are placed randomly and sequentially. Theiroverlapping with previously placed particles is strictlyforbidden, i.e., excluded volume interaction between par-ticles is assumed. The basic variant of the RSA modelalso assumes the absence of any relaxation, diffusion anddesorption. Sequential placing of particles leads to theformation of a jammed state where no additional parti-cle can be added due to the absence of appropriate holes.Different variants of one-dimensional (1D) and higher di-mensional [e.g., two-dimensional (2D)] models have pre-viously been numerically studied [3, 4]. The 2D RSAmodel has been particularly widely used for simulationof the adsorption of colloids and proteins [5].The 1D self-assembly continuously attracts growingtheoretical interest and technological demand. The at-tractive properties of low-dimensional functional materi-als were intensively studied in different fields of modernmaterial science and nanotechnology. The 1D compos-ite nanomaterials can be used in electronic and opticalnanodevices, chemical and biological sensors, and envi-ronment, energy, and biomedical fields [6–8]. The differ-ent strategies were applied to prepare the 1D crystallinenanoarrays [9]. The 1D assembly of metal nanoparticles(so-called nanochains) were fabricated for tunable surfaceplasmon resonance properties [10]. Insertion of organicdyes into the 1D channels of zeolite L allowed producing ∗ Corresponding author: [email protected] † Corresponding author: [email protected] the functional composites with intriguing optoelectronicand photochemical properties [11–13]. The properties ofsuch composites significantly depend on orientations andarrangements of particles in 1D channels [11]. The incor-poration of bioorganic systems (protein-containing waternanodroplets) in a porous inside 1D silica nanochannelswas studied [14]. The 1D confined supramolecular ar-chitectures of chromophores can be used for solar energyharvesting and storage [15].For 1D RSA packing onto a line (the so-called car park-ing problem), an analytical description of the processescan be obtained in many cases [16, 17]. For example,the kinetics of the RSA deposition of equal disks (contin-uum RSA problem) is described by the following equa-tion [18, 19] ϕ ( t ) = (cid:90) t exp (cid:20) − (cid:90) x y − (1 − e − y ) d y (cid:21) d x, (1)where ϕ ( t ) is the packing density (coverage). When ap-proaching the jamming limit ( t → ∞ ), the terminal pack-ing coverage ϕ ( t ) demonstrates the algebraic time depen-dence [20, 21]: ϕ ( t ) = ϕ j − exp( − γ ) t − ν , (2)where γ = 0 . . . . is Euler’s constant and ϕ j = C R = 0 . . . . is the famous R´enyi’s parking con-stant [18]. ν = 1 /d f , where d f is related to the numberof degree of freedom for a deposited object. d f coincideswith the dimensionality of the system when isotropic par-ticles are deposited in continuous media.An analytical expression of the density pair distribu-tion function g ( r ) for 1D RSA packing has also been de-rived [22], and the result compared with the well-knownFrenkel’s result for the continuum equilibrium problemof 1D fluid [23, 24]. a r X i v : . [ c ond - m a t . d i s - nn ] J un At jamming concentration (i.e., at ϕ = ϕ j = C R ),a comparison of the pair distribution functions derivedfor the RSA [ g ( r )] and equilibrium fluid [ g e ( r )] prob-lems revealed the influence of the irreversibility on thenear-neighbor correlations [22]. The equilibrium func-tion g e ( r ) displayed a relatively large correlated region(large oscillations), whereas the RSA function g ( r ) wasshort-ranged and close to unity at r > ε (thelength-to-diameter ratio) the jamming (maximum) cov-erage increased from the value ϕ j = 0 . ε = 1), went through a maximum (cusp) at some valueof ε m ≈ .
5, and then decreased at higher aspect ra-tios [25]. Such behavior has been observed in the pack-ing of randomly oriented ellipses [25] and later on forellipses ( ε m ≈ . ϕ j = 0 . ± . ε m ≈ . ϕ j = 0 . ± . ε m ≈ . ϕ j = 0 . ± . ε m was explained by excludedvolume effects [25]. Universality in the jamming limit forelongated particles (e.g., needles, rectangles, ellipses) in1D systems has also been discussed [28]. In particular,the dependence of the universality class on the object’sshape was demonstrated.Similar ϕ j ( ε ) behavior was also observed in the RSApacking of elongated particles in 2D and 3D systems [25].For the “Paris car parking problem”, when approach-ing the jamming limit ( t → ∞ ), an algebraic time de-pendence ϕ ( t ) ∝ t − ν was observed [26]. The exponent ν = 1 /d f was dependent on the particle shape and thenumber of orientational degrees of freedom of each par-ticle. For ellipses, the simulated empirical exponent was d f = 1 .
5, while, for discorectangles and rectangles, theywere in the range 1 . (cid:54) d f (cid:54)
2. More recently, it hasbeen demonstrated that d f = 1 . .
0, respectively,for packings built of ellipses and rectangles [27]. For dis-corectangles of moderate aspect ratio, ε , a continuoustransition between these two values was observed, from d f = 1 . ε values to d f = 2 at large ε values ( ε (cid:29) II. COMPUTATIONAL MODEL
RSA packing of discorectangles were generated using asaturated packing algorithm similar to that developed forjammed packings of non-oriented anisotropic objects ona 2D plane [29] or a 1D line [27]. It is based on tracingregions where subsequent particles can be added. Thecenters of the particles were placed on the 1D line andperiodic boundary conditions were used to minimize anyfinite-size effects.The aspect ratio (length-to-width ratio) of discorect-angles was defined as ε = l/d (Fig. 1). Both infinitelythin particles (sticks) with ε = ∞ and discorectangleswith ε ∈ [1; 100] were considered.The orientation of the particles was characterized bythe mean order parameter defined as S = (cid:104) cos 2 θ (cid:105) , (3)where (cid:104)·(cid:105) denotes the average, θ is the angle between thelong axis of the particle and the director n , which givesthe direction of the preferred orientation of the particles.Note, that S = 1 and S = − dLla θ r FIG. 1. A description of the RSA packing of discorectangleson a line. Intersections of the particles are forbidden. Eachdeposited particle covers a distance a on the line. Here L isthe total length of the line, and r is the distance between theparticle centers. To simulate the general case, a model of anisotropicrandom-orientation distribution was used [30]. For thismodel, the orientations of the deposited particles are se-lected to be uniformly distributed within some intervalsuch that − θ m (cid:54) θ (cid:54) θ m , where θ m (cid:54) π/
2. For thismodel, the preassigned order parameter can be evaluatedas [31] S = sin 2 θ m θ m , (4)and for the isotropic case ( θ m = π/ S =0. During the deposition, some particle orientations maybe rejected and the real order parameter in the deposit, S , differs from the value of S . The situation is similar tothat observed in the RSA deposition of partially orientedelongated particles ( k -mers) on square lattice [32].All distances were measured in units of particle length,while time was measured using dimensionless time units, t = n/L , where n is the number of deposition attempts,and L is the total length of the line. Each depositedparticle covers a distance a on the line, thus, the averagecoverage (packing density) was defined as ϕ = L − (cid:80) i a i ,where the summation goes over all particles.The density and orientational pair correlations werecharacterized using the distribution functions g ( r ) = C ( r ) /C ( ∞ ) , (5) s ( r ) = S ( r ) /S ( ∞ ) , (6)where C ( r ) = (cid:104) ρ (0) ρ ( r ) (cid:105) , (7) S ( r ) = (cid:104) cos { θ (0) − θ ( r )] }(cid:105) , (8) ρ ( r ) is the local number density and r is the distance be-tween the particle centers (Fig. 1). The non-zero asymp-totic value of S ( ∞ ) suggests a long-range nematic order.The average number density was calculated as ρ = N/L , where N is the total number of deposited par-ticles. To determine the effects of system size, finitesize scaling analysis for L in the interval L ∈ [2 ; 2 ]( L ∈ [4096; 32768]) was performed. Typically, the lengthof the line was taken as L = 2 = 32768. The jam-ming coverage was assumed to be reached after at least L × unsuccessful attempts to place a new particle onthe line. For each given value of ε and S , the computerexperiments were repeated using from 10 to 20 indepen-dent runs. The error bars in the figures correspond tothe standard deviation of the mean. When not shownexplicitly, they are of the order of the marker size. III. RESULTS AND DISCUSSIONA. Sticks
Figure 2 shows examples of the packing patterns of in-finitely thin particles (sticks) with ε = ∞ at different val-ues of the preassigned order parameter S . At negativevalues of S , the formation of dense stacks of sticks ori-ented in the vertical direction was observed. Even at rel-atively large orientational ordering (e.g., at S = − . S (cid:62) S , and mean number density, ρ , at a zero valueof the preassigned order parameter, S = 0, and differ-ent values of the line length, L . The conventional RSA HolesStacks S =+0.0 S =-0.9 S =+0.5 S =+0.9 FIG. 2. Examples of packing patterns of infinitely thin parti-cles (sticks) with ε = ∞ at different values of the preassignedorder parameter S . Length of the line is L = 2 and time is t = 10 . model does not allow preservation of the preassigned or-der parameter S . In this model, the line substrate withpreviously deposited particles “selects” a newcomer stickwith appropriate orientation, so this would result in a de-viation of the preassigned order parameter, S , from theactually obtained one, S . The situation is rather similarto that observed for RSA deposition of partially orientedelongated particles on a square lattice [32]. Analysis ofthe deposition patterns presented in Fig. 2 evidences thatthe particles deposited in an almost horizontal direction(inclined along the line) form holes and block the furtherdeposition along the line. However, particles deposited inthe almost vertical directions promote further depositioninside stacks and serve as attractors for almost verticaldeposition. Therefore, the line substrate can serve as afilter for particles with appropriated orientation.The actual value of S gradually decreased with increas-ing time, t , approaching the value S f in the limit of aninfinitely large time, t → ∞ . This reflects the formationof stacks of particles arranged perpendicularly to the de-position line (Fig. 2, S = 0). Analysis of scaling inthe coordinates S f versus 1 /L allows estimation of thevalue of the order parameter for infinitely large system, S f = − . ± .
009 (Fig. 3). This value is only slightlydifferent from the value S f = − . ± .
002 obtainedat L = 2 . The similar behavior was also observed forfinite values of ε . For this reason, in this work, the stud-ies were mainly performed at L = 2 without additionalscaling analysis.The mean number density, ρ , continuously increaseswith packing time, t . The time dependence ρ ( t ) demon-strated clear crossover behavior. For loose packings atinitial times, below t c ≈
1, almost linear dependencies ρ ≈ t were observed. However, at t > t c , the data can bewell fitted by the power law dependence ρ = At α , where A = 0 . ± . α = 0 . ± . ≈ / R = 0 . -1 -0.7-0.6-0.5-0.4-0.3-0.2-0.10 10 -1 S t ρ ρ =at α ρ =t t c S f L L= FIG. 3. The order parameter, S , and mean number density, ρ , versus the deposition time, t , at a zero value of the preas-signed order parameter, S = 0. The data are presented fordeposition of sticks ( ε = ∞ ) on the line with length L . Insetpresents the limiting order parameter S f (at t → ∞ ) versusthe inverse length of the line 1 /L . the transition time can be estimated as t c = A − α ≈ .
93. Figure 4 shows the order parameter, S , (a) andmean number density, ρ , (b) versus the deposition time, t , at different preassigned order parameters, S . The dataare presented for sticks ( ε = ∞ ) and the line length is L = 2 . The inset in Fig. 4b presents an enlarged por-tion of the ρ ( t ) dependencies. Across the studied rangeof S ( − (cid:54) S (cid:54) S gradually decreasedwith increasing time, t , and, finally, in the limit of aninfinitely large time, they reached values of S f smallerthan the preassigned values, S (Fig. 4a). This signifi-cant dropping of order parameters began after the for-mation of stacks could be observed at intermediate de-position times. It can be explained by a manifestation ofthe filtering properties of line substrates. The crossoverbehavior in ρ ( t ) was observed at different values of thepreassigned order parameters, S (Fig. 4b). At long de-position times, the scaling exponents were approximatelythe same for different values of S ( S ( − (cid:54) S (cid:54) α ≈ /
5. However, the transition time t c was dependentupon the value of S and t c → ∞ in the limit of S → − g ( r ) (a) andorientation s ( r ) (b) pair correlation functions for differ-ent values of the preassigned order parameter, S . Boththe functions g ( r ) and s ( r ) exhibit a strong divergencein the limit of r →
0. This corresponds to the formationof stacks with parallel arrangements of sticks (Fig. 1).Moreover, at S (cid:62)
0, the density pair correlation func-tions g ( r ) demonstrate oscillations that asymptoticallyapproach g ( r ) = 1 at large distances. The pronouncedminimums observed in the range of r ∈ [0 .
5; 1 .
1] cor- -1 -0.999-0.9-0.500.9 -1 -1-0.8-0.6-0.4-0.200.20.40.60.81 a)b) S ρ o tt S = S = ρ o t S = -0.999-0.9-0.50.0, +0.9 FIG. 4. Order parameter, S , (a) and mean number density, ρ , (b) versus the deposition time, t , at different preassignedorder parameters, S . The data are presented for the deposi-tion of sticks ( ε = ∞ ) on a line with length L = 2 . Inset in(b) presents the enlarged portion of the ρ ( t ) dependencies. responds to the correlations between places filled withstacks and holes.To estimate the inhomogeneities of the number densi-ties for packing of the sticks, the line was divided into L cells; the local number density in each cell was evaluatedand the values of the differential distribution function f ( ρ ) were estimated. For systems with strong orderingalong the vertical axis (e.g., for S = − . ρ ∗ = ρ/ρ ≈ S = 0, the distribution function became rather morebroad, moreover for particles oriented along the horizon-tal direction (i.e., S >
0) pronounced peaks located at ρ ∗ < . -0.95-0.50.0+0.8+0.9 -0.95-0.50.0+0.8+0.9 a)b) g s rr S =S = t FIG. 5. The density g ( r ) (a) and orientation s ( r ) (b) paircorrelation functions for different values of the preassignedorder parameter, S . The data are presented for the deposi-tion of sticks ( ε = ∞ ) on a line with length L = 2 , and adeposition time t = 10 . B. Discorectangles
Figure 7 shows examples of the order parameter, S ,versus the deposition time, t , for disordered RSA pack-ings ( S = 0) at different aspect ratios, ε , of discorect-angles. Similarly to the case of sticks (Fig. 3), the orderparameters, S , gradually decreased with increasing time, t , approaching the value S f in the limit of an infinitelylarge time, t → ∞ .The changes in order parameter during the depositioncan reflect the filtering properties of the RSA deposits.A similar effect was observed for the deposition of sticks(Fig. 3). Similar filtering properties were also observed atother values of S and were more significant for elongated -0.9-0.500.50.99 f ρ * S = StacksHoles
FIG. 6. Differential distribution functions of number density, f , versus the reduced number density in cells, ρ ∗ = ρ/ρ at different preassigned order parameters, S . The data arepresented for the deposition of sticks ( ε = ∞ ) on a line withlength L = 2 . -0.5-0.4-0.3-0.2-0.10 ε = t S S f FIG. 7. Order parameter, S , versus the deposition time, t , fordisordered RSA packings ( S = 0). The data are presentedfor the deposition of discorectangles with aspect ratio ε on aline with length L = 2 . particles with large values of ε . For example, the limit-ing order parameter in the jamming state S f ( t → ∞ )was dependent on the values of the preassigned order pa-rameter S and the aspect ratio, ε (Fig. 8). Note, thatin limiting cases of ideal ordering, i.e., at | S | = 1, thevalues S f were unchanged, and in other cases the valuesof S f decreased with increasing ε (Fig. 8a). For preas-signed ordering in the horizontal direction along a line,i.e., at S >
0, the S f ( S ) dependencies were almost lin-ear, whereas for ordering in the vertical direction, i.e., at S < -1 -0.5 0 0.5 1-1-0.500.51 -1-0.500.51 S f S f ε S S =0.990.50.0-0.3-0.5-0.9 ε = ∞ a)b) FIG. 8. Final order parameter in the jamming state, S f , ver-sus the aspect ratio, ε , at different values of the preassignedorder parameter, S , (a) and S f versus S at different valuesof ε . The data are presented for the deposition of discorect-angles on a line with length L = 2 . Figure 9 shows examples of the packing density, ϕ , ver-sus the deposition time, t , for disordered RSA packings( S = 0) at different values of the aspect ratio, ε . Thepacking density, ϕ , gradually increased with increasingtime, t , approaching the jamming value ϕ j at t → ∞ .The time derivatives d ϕ/ d log t were also calculatedto evaluatethe inflections at the time dependencies of ϕ (Fig. 9). These inflections were used to estimate the char-acteristic deposition times, τ . At relatively small valuesof ε ( ε (cid:54)
15) only one inflection point was observed.However, for elongated particles with ε >
20 two inflec-tion points could be seen (at τ and τ s ) and this mayreflect the development of fast and slow deposition pro-cesses (Fig. 9).Figure 10 demonstrates examples of the behavior ofcharacteristic deposition times at different aspect ratios, ε , and preassigned order parameters, S . For example,at S = 0 (Fig. 10a), the value of τ gradually grows -2 ϕ d ϕ / d l og t ε = τ τ s t FIG. 9. Packing density, ϕ , versus the deposition time, t ,for disordered RSA packings ( S = 0). The time derivativesd ϕ/ d log t were calculated to estimate the characteristic de-position times τ and τ s . The data are presented for the de-position of discorectangles with aspect ratio ε on a line withlength L = 2 . as ε approaches the limit of τ ≈ . ε → ∞ . Thecharacteristic time of the fast process, τ , may correspondto the transition from loose uncorrelated packing at theinitial time to the more dense correlated packing at longertime. For relatively short particles with ε (cid:54)
15 onlythis transition was observed (Fig. 10a). However, forlong particles with ε (cid:62)
20, the second inflection pointat t = τ s may reflect the deposition of particles insidestacks of previously deposited particles. Therefore, thefirst inflection point is related to standard uncorrelatedadsorption. The second inflection point at larger timeswas only observed at relatively large aspect ratios ( ε >
10) and it is related to the fact that all possible holes arefilled and now adsorption can only happen in the stacks.At this stage, for deposition times in the vicinity of thesecond inflection point, the voids in stacks can be filled bythe particles with some specific orientations, and finallythe adsorption is slowing down.For the fast process, the characteristic time τ displayeddifferent dependencies on the preassigned order parame-ter at S < S >
0. These dependencies for partic-ular value of ε = 10 are presented in Fig. 10b. In the limitof | S | = 1, the value of τ tends towards ετ ( ε = 1) and τ ( ε = 1) for the ideal ordering of particles with S = − S = +1, respectively.Figure 11 shows the dependences of the packing den-sity at the jamming state, ϕ j , versus the aspect ratio, ε , at different values of the preassigned order parame-ters, S , (a) and enlarged portions of the same depen-dencies for the interval 1 (cid:54) ε (cid:54) ϕ j ( ε ) dependencies were observed. For dis-ordered RSA packing (i.e., at S = 0 a well-defined max-imum ϕ j = 0 . ± .
004 at ε ≈ .
46 was observed. a)b) ττ ε ⏐ S ⏐ S < ττ s S > ετ ( ε =1) τ ( ε =1) FIG. 10. Characteristic RSA deposition times τ and τ s ver-sus the aspect ratio of discorectangles, ε , for disordered RSApackings ( S = 0) (a), τ versus S at fixed value of ε = 10 (b).The data are presented for the deposition of discorectangleswith aspect ratios ε on a line with length L = 2 . Such behavior was in good correspondence with previousdata [25, 27]. For example, the obtained values of ϕ j forthe range of 1 (cid:54) ε (cid:54) ε was explained by the excluded volume effects [25]. Thesesupplementary degrees of freedom (absent for disks at ε = 1) also significantly affected the algebraic time de-pendence of the approach of ϕ j to jamming [26].The preassigned order parameter, S , noticeably influ-enced the character of the ϕ j ( ε ) dependencies and loca-tions of the cusps. At S >
0, the increase in S resulted -0.9-0.5-0.30.00.50.90.99 Cie la et.al., 2020 ϕ j ϕ j εε Cie ś la et al., 2020 S =0.99-0.90.50-0.3-0.5 0.9 S =0.90.50.0-0.5-0.3-0.9 a)b) FIG. 11. Packing density at jamming state, ϕ j , versus thediscorectangle aspect ratio, ε , at different values of the pre-assigned order parameter, S , (a) and enlarged portion ofthe ϕ j ( ε ) dependencies for 1 (cid:54) ε (cid:54) S = 0, themaximum ϕ j = 0 . ± .
004 was observed at ε ≈ . ϕ j = C R = 0 . . . . at ε ≈ . L = 2 . in a shift of the maximum position toward to the largervalues of ε . For example, at S = 0 . ϕ j ≈ .
785 was observed at ε ≈ .
89, and at S = 0 . ϕ j ≈ .
767 was observed at ε ≈ .
72. Thelimit of S → ε = 1 (dashed lines in Fig. 9a,b).At S <
0, even more complicated non-monotonic ϕ j ( ε )behaviors were observed (Fig. 11b). At small values of ε (close to ε = 1) noticeable drops in the ϕ j values wereobserved, and then the curves went through their max-imums. For elongated particles, at large aspect ratiosthe excluded volume effects dominated and the density a) b) g s g s g s g s Holes Stacks g s g s s g Holes Stacks HolesStacks r rr r c) d)
FIG. 12. Examples of the density g ( r ) (a) and orientation s ( r ) (b) pair correlation functions with fragments of illustrativeRSA disordered packings of discorectangles ( S = 0 .
0) at the jamming concentrations for different aspect ratios ε = 2 (a), ε = 5(b), ε = 20 (c), and ε = 100 (d). The data are presented for deposition on a line with length L = 2 . ϕ j continuously decreased.Figure 12 presents examples of the density g ( r ) (a)and orientation s ( r ) (b) pair correlation functions atjamming states for different values of the aspect ratio, ε , and a fixed preassigned order parameter, S = 0. Atrelatively small aspect ratios (Fig. 12a,b) both the func-tions g ( r ) and s ( r ) exhibited rather complicated andlarge oscillations at r (cid:47)
2, but these became small at r >
3. These oscillations reflect the size of the correlatedregions in the RSA packing of the discorectangles. Note,that the location of the first peak in the density corre-lation function g ( r ) corresponds to the mean distancebetween the particles. This should be when the distanceis in the order of 1 or ε − for ideal horizontal ( S = 1)or vertical ( S = −
1) RSA packings, respectively. For ε = 2, the location of the first peak of the g ( r ) function was r ≈ .
75 and that reflects the noticeable contribu-tion of particles with horizontal orientations, but the lo-cation of the first peak of the s ( r ) function was close to r ≈ .
6, corresponding to the closest distance between in-clined particles. For ε = 5, the location of the first peakof the g ( r ) function was r ≈ .
23 which is very closeto ε − = 0 .
2. It reflects the contribution from particlein parallel stacks clearly visible in fragments of the illus-trative RSA packings (Fig. 12). However, the particleswith almost horizontal orientations make an importantcontribution to the second peak located at r ≈ .
6. Thispeak corresponds to the correlation of alternative regions:dense regions filled with parallel stacks and rarefied re-gions (holes) containing particles with almost horizontalorientations. For this particular case of ε = 5 the behav-ior of the g ( r ) and s ( r ) functions was approximatelyanti-bat. At larger values of ε the first peaks in g ( r ) and s ( r ) became located at r ≈ ε − reflecting the dominantcontribution from particles in parallel stacks (see corre-sponding fragments of the illustrative RSA packings inFig. 12c,d). The contribution from holes (correlation ofalternative regions) became less important with increasedvalues of ε . IV. CONCLUSION
Numerical studies of two-dimensional RSA depositionof infinitely thin particles (sticks) and discorectangles ona one-dimensional line were performed. The packing ki-netics and properties of the packs were significantly in-fluenced by the values of the preassigned order param-eter, S , and the aspect ratio, ε . The deposition wasgoverned by the formation of rarefied holes (containingparticles oriented along the line) surrounded by compar-atively dense stacks (filled by almost parallel particlesoriented in the vertical direction). This resulted in sig-nificant deviation of the actual order parameter S in thedeposit and of the preassigned order parameter S . In fact, the unsaturated packing acted as a filter for the ad-sorption of particles with appropriated orientations. Forthe RSA packing of discorectangles, the filtering prop-erties of the RSA deposits were more significant at rel-atively large aspect ratios. For elongated particles with ε (cid:62)
20, the development of fast and slow deposition pro-cesses could be observed. However, the introduction ofpreferential ordering also influenced the behavior of thecusps observed in the ϕ j ( ε ) dependencies in the inter-val 1 (cid:54) ε (cid:54)
5. The observed effects can be explainedby the impact of the partial ordering on the competi-tion between the orientational degrees of freedom of eachparticle and by excluded volume effects [25].
ACKNOWLEDGMENTS
We acknowledge funding from the National Academyof Sciences of Ukraine, Project Nos. 0117U004046 and0120U100226 (7/9/3-f-4-1230-2020) (N.I.L., N.V.V.),and the Russian Foundation for Basic Research, ProjectNo. 18-07-00343 (Yu.Yu.T.). [1] James W. Evans, “Random and cooperative sequentialadsorption,” Rev. Mod. Phys. , 1281 (1993).[2] Salvatore Torquato and Frank H. Stillinger, “Jammedhard-particle packings: From Kepler to Bernal and be-yond,” Rev. Mod. Phys. , 2633 (2010).[3] A. Cadilhe, N. A. M. Ara´ujo, and Vladimir Privman,“Random sequential adsorption: From continuum to lat-tice and pre-patterned substrates,” J. Phys.: Condens.Matter , 065124 (2007).[4] Vladimir Privman and Han Yan, “Random sequentialadsorption on imprecise lattice,” J. Chem. Phys. ,244704 (2016).[5] Zbigniew Adamczyk, “Modeling adsorption of colloidsand proteins,” Curr. Opin. Colloid Interface Sci. , 173–186 (2012).[6] Xiaofeng Lu, Ce Wang, and Yen Wei, “One-dimensionalcomposite nanomaterials: Synthesis by electrospinningand their applications,” Small , 2349–2370 (2009).[7] Rupesh S. Devan, Ranjit A. Patil, Jin-Han Lin, andYuan-Ron Ma, “One-dimensional metal-oxide nanostruc-tures: Recent developments in synthesis, characteriza-tion, and applications,” Adv. Funct. Mater. , 3326–3370 (2012).[8] Tianyou Zhai and Jiannian Yao, eds., One-dimensionalnanostructures: principles and applications (John Wiley& Sons, 2012).[9] Weigang Lu, Puxian Gao, Wen Bin Jian, Zhong LinWang, and Jiye Fang, “Perfect orientation ordered in-situ one-dimensional self-assembly of Mn-doped PbSenanocrystals,” J. Am. Chem. Soc. , 14816–14821(2004).[10] Yong Yang, Shigemasha Matsubara, Masayuki Nogami,Jianlin Shi, and Weiming Huang, “One-dimensional self-assembly of gold nanoparticles for tunable surface plas- mon resonance properties,” Nanotechnology , 2821–2827 (2006).[11] Ettore Fois, Gloria Tabacchi, and Gion Calzaferri,“Orientation and order of xanthene dyes in the one-dimensional channels of zeolite L: Bridging the gap be-tween experimental data and molecular behavior,” J.Phys. Chem. C , 16784–16799 (2012).[12] Gloria Tabacchi, Gion Calzaferri, and Ettore Fois, “One-dimensional self-assembly of perylene-diimide dyes byunidirectional transit of zeolite channel openings,” Chem.Commun. , 11195–11198 (2016).[13] Rebeca Sola-Llano, Leire Gartzia-Rivero, Ainhoa Oliden-Sanchez, Jorge Ba˜nuelos, I˜nigo L´opez Arbeloa, and Vir-ginia Mart´ınez-Mart´ınez, “Chapter 13 - dye encapsula-tion into one-dimensional zeolitic materials for opticalapplications,” in Chemistry of Silica and Zeolite-BasedMaterials , Chemical, Physical and Biological Aspects ofConfined Systems, Vol. 2, edited by Abderrazzak Douhaland Masakazu Anpo (Elsevier, 2019) pp. 229–248.[14] Lara Giussani, Gloria Tabacchi, Salvatore Coluccia, andEttore Fois, “Confining a protein-containing water nan-odroplet inside silica nanochannels,” Int. J. Mol. Sci. ,2965 (2019).[15] Ettore Fois and Gloria Tabacchi, “Water in zeolite L andits MOF mimic,” Zeitschrift f¨ur Kristallographie - Crys-talline Materials , 495–511 (2019).[16] J. Talbot, G. Tarjus, P. R. Van Tassel, and P. Viot,“From car parking to protein adsorption: an overview ofsequential adsorption processes,” Colloids Surf., A ,287–324 (2000).[17] Pavel L. Krapivsky, Sidney Redner, and Eli Ben-Naim, A Kinetic View of Statistical Physics (Cambridge Uni-versity Press, 2010). [18] A. R´enyi, “On a one-dimensional problem concerningrandom space filling,” Selected Translations in Math-ematical Statistics and Probability , 203–218 (1963),translation from Magyar Tud. Akad. Mat. Kutat´o Int.K¨ozl. 3, No.1–2, 109–127 (1958).[19] Jose J. Gonz´alez, P. C. Hemmer, and J. S. Høye, “Coop-erative effects in random sequential polymer reactions,”Chem. Phys. , 228–238 (1974).[20] Y. Pomeau, “Some asymptotic estimates in the randomparking problem,” J. Phys. A: Math. Gen. , L193(1980).[21] P. L. Krapivsky, “Kinetics of random sequential parkingon a line,” J. Stat. Phys. , 135–150 (1992).[22] B. Bonnier, D. Boyer, and P. Viot, “Pair correlationfunction in random sequential adsorption processes,” J.Phys. A: Math. Gen. , 3671 (1994).[23] J. Frenkel, Kinetic Theory of Liquids (Dover publicationsInc., New York, USA, 1946).[24] Zevi W. Salsburg, Robert W. Zwanzig, and John G.Kirkwood, “Molecular distribution functions in a one-dimensional fluid,” J. Chem. Phys. , 1098–1107 (1953).[25] P. M. Chaikin, Aleksandar Donev, Weining Man,Frank H. Stillinger, and Salvatore Torquato, “Some ob-servations on the random packing of hard ellipsoids,” Ind.Eng. Chem. Res. , 6960–6965 (2006). [26] Adrian Baule, “Shape universality classes in the randomsequential adsorption of nonspherical particles,” Phys.Rev. Lett. , 028003 (2017).[27] Micha(cid:32)l Cie´sla, Konrad Kozubek, Piotr Kubala, andAdrian Baule, “Kinetics of random sequential adsorptionof two-dimensional shapes on a one-dimensional line,”Phys. Rev. E , 042901 (2020).[28] Yacov Kantor and Mehran Kardar, “Universality in thejamming limit for elongated hard particles in one dimen-sion,” EPL (Europhysics Letters) , 60002 (2009).[29] Karol Haiduk, Piotr Kubala, and Micha(cid:32)l Cie´sla, “Satu-rated packings of convex anisotropic objects under ran-dom sequential adsorption protocol,” Phys. Rev. E ,063309 (2018).[30] I. Balberg and N. Binenbaum, “Computer study of thepercolation threshold in a two-dimensional anisotropicsystem of conducting sticks,” Phys. Rev. B , 3799–3812 (1983).[31] Nikolai I. Lebovka, Nikolai V. Vygornitskii, and Yuri Yu.Tarasevich, “Relaxation in two-dimensional suspensionsof rods as driven by Brownian diffusion,” Phys. Rev. E , 042139 (2019).[32] Nikolai I. Lebovka, Natalia N. Karmazina, Yuri Yu. Tara-sevich, and Valeri V. Laptev, “Random sequential ad-sorption of partially oriented linear k -mers on a squarelattice,” Phys. Rev. E84