Jammed systems of oriented needles always percolate on square lattices
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Jammed systems of oriented needles always percolate on square lattices
Grzegorz Kondrat, ∗ Zbigniew Koza, and Piotr Brzeski
Faculty of Physics and Astronomy, University of Wroc law, 50-204 Wroc law, Poland (Dated: May 10, 2018)Random sequential adsorption (RSA) is a standard method of modeling adsorption of largemolecules at the liquid-solid interface. Several studies have recently conjectured that in the RSA ofrectangular needles, or k -mers, on a square lattice the percolation is impossible if the needles aresufficiently long ( k of order of several thousand). We refute these claims and present a strict proofthat in any jammed configuration of nonoverlapping, fixed-length, horizontal or vertical needles ona square lattice, all clusters are percolating clusters. PACS numbers: 05.50.+q 64.60.A-
I. INTRODUCTION
Adsorption of large molecules like polymers,biomolecules or nanotubes at the liquid-solid interface isan important part of various natural and technologicalprocesses, including those found in industrial bioreactors[1], water purification [2], or production of conductingnanocomposites [3]. In many cases the adsorptionphenomenon is essentially an irreversible and localizedprocess in which the adsorbed molecules eventually forma monolayer on the target surface [4]. It is thereforequite common to investigate such phenomena using therandom sequential adsorption (RSA) model. The mainidea behind it is simple [4–6]: starting with an emptysubstrate, one tries to put on it a sequence of somegeometric objects, e.g. disks or rectangles, each at arandom position. An attempt is successful if the newobject does not overlap with the ones already depositedon the surface, otherwise a new attempt is made at adifferent, randomly chosen location. Once the object isattached to the surface, it stays there motionless forever,reducing a chance for the subsequent molecules to beadsorbed in its neighborhood. The dynamics graduallyslows down and finally no unoccupied room remainson the substrate that could accommodate the nextobject—the process stops, the system has reached the socalled jamming limit (a discussion of various aspects ofjammed configurations can be found in the review [7]).Numerous extensions of the basic RSA model havebeen studied so far, including imperfect substrates [8, 9],various object shapes (e.g., disks [4], spheres [10], sphero-cylinders [11], infinitely thin needles [12], squares [13, 14],ellipses [15], and rectangles [8, 9, 15–19]), polydispersity[20–22], shape flexibility [23, 24], post-adsorption dynam-ics (e.g., desorption [25] and diffusion [26]), and partial[27] or full object overlapping [28]. In each RSA pro-cess, basic or extended one, as the molecules are beingdeposited onto the surface, they may touch each otherand form larger clusters of connected (or “neighboring”)objects, and if such a cluster spans the opposite sides of ∗ [email protected] the system, we have to do with a percolating cluster.Consequently, there are two basic quantities charac-terizing RSA processes. The first one is the jammingthreshold 0 < c j ≤ A ad ) to the totalsurface area ( A ) in the jammed state. The second oneis the percolation threshold c p , a quantity similar to thejamming threshold in that it is also defined as A ad /A ,except that A ad must be now determined at the momentwhen the adsorbed molecules start to form a percolat-ing cluster [29]. While c j characterizes any RSA pro-cess, c p is well defined only for some of them. For exam-ple, in models where nonoverlapping objects, e.g. circles,are randomly deposited on a continuous substrate, e.g. alarger square, no object can actually touch another oneand a percolating cluster cannot be formed. However,the information whether (or under which conditions) anRSA process leads to percolation or not is a fundamentalcharacteristic of this process.In 2000 Vandewalle et al. [17] advanced a hypothesisthat in the RSA of needle-like rectangles (also known as k -mers) on a square lattice the ratio c p /c j is constant forall needle sizes. If correct, this hypothesis would indi-cate existence of a deep relation between jamming andpercolation. However, more elaborate studies refuted thisclaim. Instead, as will be discussed in detail in Section II,several researchers came to the conclusion that for suffi-ciently long needles the system does not percolate. How-ever, this striking conjecture is based on extrapolationof numerical results obtained for relatively short needlesand no physical mechanism responsible for such a behav-ior is known.Thus, analysis of the above-mentioned reports raisesthe question of percolation breakdown. Does percola-tion really break down for very long needles or is it anartifact brought about by using an incorrect fitting func-tion to the numerical data? There seems to be two waystowards the solution. The first one is to carry out a di-rect numerical examination of percolation for extremelylong needles. However, this would require using so hugeamounts of computer resources (memory and computa-tional time) that such simulations have not been endeav-ored yet [8]. The other option is to prove or disprove theconjecture mathematically. The second option is moreattractive, especially since exact arguments in percola-tion theory are relatively rare. Here we present such astrict proof that any cluster in a jammed configuration offixed-length needles on a square lattice is a percolatingcluster. Consequently, the RSA of fixed-length needlesalways percolates on a square lattice. II. CONTRIBUTIONS TO THE HYPOTHESISOF THE PERCOLATION BREAKDOWN
In their study of RSA of needles on a square lattice,Kondrat et al. [18] noticed a peculiar dependence of theratio of the percolation threshold to the jamming thresh-old ( c p /c j ) on the needle length ( k ), c p /c j ≈ a + b log k, (1)with a = 0 .
50 and b = 0 .
13. Since c p /c j cannot be greaterthan 1, it was clear that this relation must break for nee-dles of length larger than some characteristic length k ∗ ,which in this case can be estimated as 10 (1 − a ) /b ≈ k ≤
45, which raised the question: is k ∗ a real physical parameter and if so, what happens to c p /c j as k approaches and then exceeds k ∗ ?This problem was tackled by Tarasevich et al. [19],who studied the RSA of partially ordered needles. Inthe isotropic case they confirmed relation (1) for muchlonger needles ( k ≤ a = 0 . b = 0 . k ∗ = 12 400(3700). As the logarithmicformula was verified for really long needles, the problemof what happens close to and beyond k ∗ became moreinteresting. Assuming that (1) is valid up to k ∗ , theyformulated the hypothesis that for k > ∼ k ∗ the systemdoes not percolate and the ratio c p /c j simply becomesundefined, which would solve the paradox.This rather surprising conclusion was confirmed in astudy of random sequential adsorption of needles in im-perfect systems [8]. Two extensions of the original modelwere considered: either the needles have some imperfect(nonconducting) segments (a so called K model) or thelattice has some sites forbidden for adsorption (L model).It turned out that to each needle length k correspondsa critical level of impurities d K ∗ or d L ∗ (for models K andL, respectively), above which no percolation can be ob-served. They were found to verify the following phe-nomenological relations d L ∗ ( k ) = a L ( k L ∗ ) α − k α b + k α , (2)where a L , α , b , and k L ∗ are some fitting parameters (modelL of imperfect lattice), and d K ∗ ( k ) = a K log ( k K ∗ /k ) , (3) with some fitting parameters a K and k K ∗ (imperfect nee-dles). For k > k L ∗ (imperfect lattice) or k > k K ∗ (imper-fect needles) these formulas predict an unphysical, neg-ative value of the critical impurity level. Moreover, theobtained values of k L ∗ = 5900(500) and k K ∗ = 4700(1000)are consistent with the previous estimates of k ∗ . There-fore Tarasevich et al. concluded that their data confirmthe hypothesis of percolation breakdown for sufficientlylong needles.Another method of introducing imperfections to theoriginal model of random adsorption of needles was re-cently investigated by Centres et al. [9]. To model ad-sorption on surfaces of amorphous solids, they assumedthat all sites of the lattice are ready for adsorption, buta fraction ρ of the bonds has been disabled before theadsorption begins. They found a similar phenomenonto that found for lattice or needle impurities: there ex-ists a certain critical, k -dependent concentration of dis-abled bonds, ρ ∗ , above which percolation does not occur.Moreover, their data fitted well to (2) with the criticallength k ∗ = 5518(500), a value consistent with previousreports. III. THEOREM AND ITS PROOF
We will prove the following theorem:
Every jammedconfiguration of fixed-length nonoverlapping horizontal orvertical needles on a finite square lattice contains a con-nected cluster spanning two opposite edges of the lattice.
In this theorem, a “needle” is a rectangle of size 1 lat-tice unit (l.u.) by k l.u., with its corners located at theunderlying lattice nodes. Two needles are connected di-rectly if they share a part of their sides of length ≥ k = 1 the problem reduces to the classi-cal site percolation. Of course, we consider only systemslarge enough to accommodate at least one needle. More-over, since the theorem is trivial for k = 1, henceforth weassume that k > A. Method I
For convenience, we start from proving the followingLemma:
Every cluster at a jammed configuration extendsto one of two consecutive edges of the system.
If onelabels the system edges using the geographical notation(N, E, S, and W for the top, right, bottom, and leftedge, respectively), the lemma states that any cluster atjamming must touch at least one edge in each of the fourpairs: (N, E), (E, S), (S, W) and (W, N).We will prove the Lemma a contrario —let us assumethat there exists a jammed configuration of fixed-length column type: H H H H H H V V E V - n ee d l e FIG. 1. An exemplary cluster with the topmost cell in eachlattice column marked with a dot. Orientation (horizontalor vertical) of the needles occupying these cells can be usedto classify the corresponding lattice columns as H or V. Ifthe rightmost column is of type V and the rightmost needle(“EV-needle”) does not touch the edge of the system, there isa room, marked with a hatched pattern, for another needle. nonoverlapping needles with a cluster that does not touchany of two consecutive system edges, say N and E.Since the system is a square lattice of size L × L ( L ≥ k ), it can be regarded as a set of L columns, each madeof L elementary lattice cells. For each column we canidentify the set of all its cells that belong to the cluster.If this set is nonempty, we can use the topmost cell fromthis set to classify the column as follows: if the column’stopmost cell belongs to a horizontal needle, the columnis said to be of type H, otherwise the cell belongs to avertical needle and the column is said to be of type V.This is illustrated in Fig. 1.Consider now the rightmost column containing thesites from our cluster. We will label it c HE because itis cluster’s easternmost column and it is of type H. Tosee the reason for the latter property, suppose this col-umn is of type V. In such a case it would contain at leastone vertical needle (marked as an EV-needle in Fig. 1).Since we assumed that the cluster does not touch sys-tem’s edge E, there exists a column to the right of theEV-needle and it contains at least k consecutive emptysites, see Fig. 1. This, however, contradicts the assump-tion that the system is jammed.There are now two possibilities: either all columns withthe sites from the cluster are of the same type H or atleast one of these columns is of type V. We will considereach of these cases separately.
1. Case A: all columns are of type H
Let r N denote the topmost row containing the cluster(see Figure 2). As all columns are assumed to be oftype H, this row must contain at least one horizontalneedle, which we will call “NH-needle”. As no cell abovethe topmost row can belong to the cluster and since wehave assumed that the cluster does not touch system edge column name: c HE column type: H H H H H H H H H H r o w n a m e : r N NH-needle
FIG. 2. If all lattice columns are of type H and the clusterdoes not extend to the top edge of the system, there is a room(marked with a hatched pattern) for another needle directlyabove any horizontal needle in the cluster’s topmost row, r N . column name: c VE c HE column type: H H H V V H H H H H H r o w n a m e : r HN r VS E V - n ee d l e FIG. 3. For clusters occupying columns of type H and V,the “EV-needle” is defined as the topmost vertical needle inthe rightmost column of type V, r VS is the lowest row occu-pied by this needle, and r HN is the topmost row containing ahorizontal needle to the right of the EV-needle. N, all cells neighboring the NH-needle from above areempty and can accommodate another needle. As thiscontradicts the assumption that the system is jammed,the proof of case A is completed.
2. Case B: columns of mixed types, H and V
Consider now the case where at least one column isof type V. Let c VE denote the rightmost column of typeV. Cluster’s topmost cell in this column belongs to avertical needle, which we call the EV-needle (see Fig. 3).Its bottom cell defines a reference row, which we denoteas r VS . We also define r HN as the topmost row containinga horizontal needle in any of the columns located to theright of column c VE . The remaining part of the proofdepends on the relation between r VS and r HN . a. Case B1: r VS > r HN . In this case all cells ofthe cluster that are to the right of column c VE lie below column name: c VE c HE column type: H H H V V H H H H r o w n a m e : r VS r H(cid:0) E V - n ee d l e FIG. 4. If r VS > r HN and the cluster does not extend to theright edge of the system, the cells bordering the EV-needlefrom the right are empty and can hold another needle. column name: c VE c HE column type: V H H H H H H H H H r o w n a m e : r HN r VS NH-needle E V - n ee d l e FIG. 5. If r VS ≤ r HN , there is a room for another needledirectly above the NH-needle (the topmost horizontal needleto the right of the EV-needle). row r VS . This means that the cells located directly tothe right of the EV-needle are unoccupied and can ac-commodate another needle, see Fig. 4. This, however,contradicts the assumption that the system is jammed. b. Case B2: r VS ≤ r HN . Let “NH-needle” denotethe rightmost horizontal needle located at row r HN . Eachcell occupied by this needle lies to the right of the EV-needle (see Fig. 5). Directly above the NH-needle thereare neither vertical needles (all columns to the right ofcolumn c VE are of type H), nor horizontal ones (oth-erwise the NH-needle would not be the topmost one).Thus, the cells neighboring the NH-needle from aboveare empty and can accommodate another needle. How-ever, this contradicts the assumption that the system isjammed. This completes the proof of the Lemma.The final step is to show that the Lemma implies theTheorem. Let us assume that the there is a jammedconfiguration of needles and a cluster in it. There aretwo cases: either this cluster extends to all four edgesof the system or not. In the former case the cluster istrivially a percolating cluster. In the latter one it doesnot touch at least one system edge, say, N. However, the FIG. 6. A hull (solid line) is a polygon composed of alter-nating horizontal and vertical sides tightly surrounding theneedles forming a cluster. The solid circles and the hatchedpattern mark the lattice cells bordering the hull from the in-side and outside, respectively; by construction, the former areoccupied by needles whereas the latter are empty.
Lemma ensures that in this case it must extend to thetwo edges adjacent to N, that is, to E and W, and hencemust be a percolating cluster.
B. Method II
Suppose that it is possible to fill a finite square latticeof size L × L , L ≥ k , with nonoverlapping horizontal orvertical needles of size 1 × k in such a way that the systemis jammed and a cluster of connected needles exists suchthat it does not touch the system borders. We will showthat this assumption leads to a contradiction.Let us define the hull of a cluster as the minimal simplepolygon encompassing it, see Fig. 6. This polygon ismade of horizontal and vertical line segments of integerlength. It divides the plane into the space occupied bythe cluster together with, perhaps, some holes betweenthe needles forming it, and the remaining space.By construction, any elementary square borderingthe hull from inside must be occupied by some needle,whereas none of the squares bordering the hull from theoutside can be occupied by a needle, see Fig. 6. Thus,the length of any side of the hull is an integer less than k , otherwise one could add a needle on the squares bor-dering this side from the outside, which contradicts theassumption that the system is jammed.Each horizontal line segment of the hull is followed bya vertical one and so forth by turns, so that the numberof its vertices, N , is even and the angles at its verticesare either π/ π/
2. Let q + and q − denote the numberof π/ π/ N -sided polygon is ( N − π , thesequantities satisfy q + + q − = N and q + π/ q − (3 π/
2) =
FIG. 7. The main ideas of Method II. (a) In each hull onecan find two “caps” (parts of the hull with two consecutiveright angles separated, perhaps, by a “zigzag” (part od a hullwith consecutive external and internal right angles); (b) in ajammed state the length of each of the hull’s sides is smallerthan that of a needle (here: k = 3), which uniquely deter-mines the orientation of the needles filling in the caps; (c) theorientation of the needles touching the zigzag must be thesame as that in each cap, which leads to a contradiction, asthey are perpendicular to each other. ( N − π . Consequently, q + = N/ , q − = N/ − . (4)Let us assign to each vertex i of the hull an integer a i ∈ {− , } such that a i = 1 if the internal angle at i is π/ a i = − π/
2. The idea is that aswe walk along the hull in a clockwise direction, we keeptrack of our current orientation by adding 1 wheneverwe make a right turn and subtracting 1 for the left turn.From (4) we have N − X i =0 a i = q + − q − = 4 , (5)that is, whenever we return to the starting point, wemust have made 4 more right turns than the left ones.Let s i = ( a i + a i +1 ) / N ). Byconstruction, s i are also integers, s i ∈ {− , , } , whichmeasure the effect of making two consecutive turns. Theysatisfy N − X i =0 s i = N − X i =0 a i = 4 . (6)In the sequence s , s , . . . , s N − there are thus at leasttwo indices j < k such that s j = s k = 1 and s l = 0for all j < l < k . Therefore there exist two vertices, j and k , such that a j = a j +1 = 1 and a k = a k +1 = 1,separated, perhaps, by an alternating sequence of theform − , , . . . , −
1. Geometrically this corresponds to a“zigzag” ended by two “caps”, see Fig. 7a.Each cap uniquely determines the orientation (hori-zontal or vertical) of the needles that occupy it. This isbecause the length of the cap side connecting two con-secutive right angles is smaller than k , and so any nee-dle touching it from inside must be perpendicular to it(cf. Fig. 7b). Moreover, if the zig-zag segment exists FIG. 8. (a) The hull of a hypothetical cluster touching onlyone edge of the system. The lengths of all of its sides nottouching the edge must be smaller than the needle length, k .(b) Construction of another cluster with the hull such thatthe lengths of all its sides would be smaller than k . between the two caps, each of the caps enforces the di-rection of the needles touching the zig-zag segment to beparallel to the needles filling in that cap. However, theorientations of the two consecutive caps are orthogonalto each other. This leads to a contradiction: the needlestouching the zig-zag cannot be all both horizontal andvertical (Fig. 7c).If, however, no zig-zag part exists, then k = j + 1and a j = a j +1 = a j +2 = 1, that is, the cap has twoconsecutive, orthogonal sides of size smaller than k . Sucha region cannot be filled by needles of length k , whichagain contradicts our assumptions. Thus, either the hullof the cluster touches one of the edges of a finite system,in which case at least one side touching the edge is oflength ≥ k , or the system is not jammed.Is it possible that a cluster at a jamming state touchesonly one of the system’s edges? If such a cluster existed,it could be used to construct a cluster made of needlesof size 1 × k , whose hull is a polygon with all sides oflengths < k , see Fig. 8. However, we have just proventhat such a cluster does not exist. The construction isdefined as follows. One takes the original cluster as wellas its mirror reflection and joins them together with twoneedles sticking out with k − k elementary seg-ments. Each side of the resulting cluster would be smallerthan k . Moreover, the two extra needles cannot overlap,for the original cluster must touch the system edge atat least k ≥ × k , whose hull is a polygon with all sides of lengths < k .However, such a cluster does not exist. Consequently, anycluster at a jamming state connects two opposite sides ofthe system, horizontally or vertically, and therefore is apercolating cluster. IV. CONCLUSIONS AND OUTLOOK
We have proved that all jammed configurations ofnonoverlapping needles of size 1 × k ( k -mers) on a squarelattice are percolating ones. This disproves the recentconjecture [8, 9, 19] that in the random sequential ad-sorption of such needles on a square lattice the percola-tion does not occur if the needles are longer than somethreshold value k ∗ , estimated to be of order of severalthousand.While this result ensures that the percolation to jam-ming ratio ( c p /c j ) is well defined for all needle lengths,it does not bring us much closer to the understanding of how this ratio varies with k for k > ∼ ≥ k . This helps to understand why even a tinylattice impurity can preclude percolation in systems withvery long needles: one impurity per cluster hull side canbe enough to stop its growth.Another interesting point is whether our argumentscan be extended to other lattices, e.g. the triangular orcubic ones. REFERENCES [1] B. Senger, J.-C. Voegel, and P. Schaaf, Colloids and Sur-faces A: Physicochemical and Engineering Aspects ,255 (2000).[2] A. D¸abrowski, Advances in colloid and interface science , 135 (2001).[3] R. M. Mutiso and K. I. Winey, Progress in Polymer Sci-ence , 63 (2015).[4] J. Feder, Journal of Theoretical Biology , 237 (1980).[5] J. W. Evans, Rev. Mod. Phys. , 1281 (1993).[6] J. Talbot, G. Tarjus, P. R. Van Tassel, and P. Viot,Colloids and surfaces A , 287 (2000).[7] S. Torquato and F. H. Stillinger, Reviews of ModernPhysics , 2633 (2010).[8] Y. Y. Tarasevich, V. V. Laptev, N. V. Vygornitskii, andN. I. Lebovka, Phys. Rev. E , 012109 (2015).[9] P. M. Centres and A. J. Ramirez-Pastor, Journal ofStatistical Mechanics: Theory and Experiment ,P10011 (2015).[10] P. Meakin and R. Jullien, Physica A: Statistical Mechan-ics and its Applications , 475 (1992).[11] T. Schilling, M. A. Miller, and P. Van der Schoot, EPL(Europhysics Letters) , 56004 (2015).[12] N. Provatas, M. Haataja, J. Asikainen, S. Majaniemi,M. Alava, and T. Ala-Nissila, Colloids and SurfacesA: Physicochemical and Engineering Aspects , 209(2000).[13] B. J. Brosilow, R. M. Ziff, and R. D. Vigil, Phys. Rev.A , 631 (1991).[14] M. Nakamura, Journal of Physics A: Mathematical andGeneral , 2345 (1986).[15] P. Viot, G. Tarjus, S. M. Ricci, and J. Talbot,The Journal of Chemical Physics , 5212 (1992),http://dx.doi.org/10.1063/1.463820.[16] M. Porto and H. E. Roman, Physical Review E , 100 (2000).[17] N. Vandewalle, S. Galam, and M. Kramer, The Euro-pean Physical Journal B - Condensed Matter and Com-plex Systems , 407 (2000).[18] G. Kondrat and A. P¸ekalski, Phys. Rev. E , 051108(2001).[19] Y. Y. Tarasevich, N. I. Lebovka, and V. V. Laptev, Phys.Rev. E , 061116 (2012).[20] J. W. Lee, Colloids and Surfaces A: Physicochemical andEngineering Aspects , 363 (2000).[21] B. Nigro, C. Grimaldi, P. Ryser, A. P. Chatterjee, andP. Van Der Schoot, Physical Review Letters , 015701(2013).[22] A. P. Chatterjee and C. Grimaldi, Physical Review E ,032121 (2015).[23] P. Adamczyk, P. Romiszowski, and A. Sikorski, TheJournal of Chemical Physics , 154911 (2008).[24] G. Kondrat, The Journal of Chemical Physics , 6662(2002), http://dx.doi.org/10.1063/1.1505866.[25] L. Budinski-Petkovi´c and U. Kozmidis-Luburi´c, PhysicaA: Statistical Mechanics and its Applications , 174(2001).[26] N. I. Lebovka, Y. Y. Tarasevich, V. A. Gigiberiya, andN. V. Vygornitskii, Physical Review E , 052130 (2017).[27] I. Balberg, Philosophical Magazine Part B , 991(1987), http://dx.doi.org/10.1080/13642818708215336.[28] S. Torquato, J. Chem. Phys. , 054106 (2012).[29] D. Stauffer and A. Aharony, Introduction to PercolationTheory , 2nd ed. (Taylor and Francis, London, 1994).[30] V. Cornette, A. Ramirez-Pastor, and F. Nieto, Phys-ica A: Statistical Mechanics and its Applications , 71(2003).[31] Y. Y. Tarasevich and V. A. Cherkasova, The EuropeanPhysical Journal B60