% The paper is in plain TeX and it can be % TeXed directly as provided. No input files, % etc., are needed. % % The appended postscript figures each begin with the line "%!P...". % There are four separate files. The postscript text is after the % closing line of the TeX file: "\bye", and it will therefore % not interfere with TeXing. If you strip off the postscript parts, % they each can be sent directly to a postscript laser printer to get % Figures 1 through 4 of this preprint. \def\NI{\noindent} \long\def\UN#1{$\underline{{\vphantom{\hbox{#1}}}\smash{\hbox{#1}}}$} \magnification=\magstep 1 \overfullrule=0pt \hfuzz=16pt \voffset=0.0 true in \vsize=8.8 true in \def\NP{\vfil\eject} \baselineskip 20pt \parskip 6pt \hoffset=0.1 true in \hsize=6.3 true in \nopagenumbers \pageno=1 \footline={\hfil -- {\folio} -- \hfil} \headline={\ifnum\pageno=1 \hfill October 1992 \fi} \hphantom{AA} \hphantom{AA} \centerline{\UN{\bf Collective Effects in Random Sequential}} \centerline{\UN{\bf Adsorption of Diffusing Hard Squares}} \vskip 0.4in \centerline{\bf Jian--Sheng Wang} \centerline{\sl Department of Physics, Hong Kong Baptist College, 224 Waterloo Road, Kowloon, Hong Kong} \centerline{\bf Peter Nielaba$^{\bf *}$} \centerline{\sl Institut f\"ur Physik, Universit\"at Mainz, Staudingerweg 7, D--6500 Mainz, Germany} \centerline{\bf Vladimir Privman} \centerline{\sl Department of Physics, Clarkson University, Potsdam, New York 13699--5820, USA} \vskip 0.6in \centerline{\bf ABSTRACT} We study by Monte Carlo computer simulations random sequential adsorption (RSA) with diffusional relaxation, of lattice hard squares in two dimensions. While for RSA without diffusion the coverage approaches its maximum jamming value (large-time fractional coverage) exponentially, added diffusion allows the deposition process to proceed to the full coverage. The approach to the full coverage is consistent with the $\sim t^{-1/2}$ power law reminiscent of the equilibrium cluster coarsening in models with nonconserved order-parameter dynamics. \vskip 0.3in \NI {\bf PACS numbers:}$\;$ 68.10.Jy, 02.50.+s, 82.65.-i \NP Random sequential adsorption (RSA) models have been studied extensively due to their relevance to deposition processes on surfaces [1]. The depositing particles are represented by hard-core extended objects; they are not allowed to overlap. In monolayer deposition of colloidal particles and macromolecules [2] one can further assume that the adhesion process is irreversible. Once a particle is in place its relaxation on the surface proceeds on time scales much larger than the deposition process. However, recent experiments on protein adhesion at surfaces [3] indicate that in these systems effects of surface relaxation, presumably due to diffusional rearrangement of particles, are observable on time scales of the deposition process. The resulting large-time coverage is denser than in fully irreversible RSA and in fact it is experimentally comparable to the fully packed (i.e., locally semi-crystalline) particle arrangement. Studies of RSA with diffusional relaxation by analytical means encounter several difficulties associated with possible collective effects in hard-core particle systems at high densities (such as, for instance, phase separation), and with the possibility, in certain lattice models, of locally ``gridlocked'' vacant sites. The latter effect may actually prevent full coverage in some models; this matter remains an open problem at this time. Both difficulties are not present in $1D$: there are no equilibrium phase transitions, traces of which might manifest themselves as collective effects in $D>1$ deposition with diffusion, and furthermore diffusional relaxation leads to simple hopping-diffusion interpretation of the motion of vacant sites in $1D$ which recombine to form larger open voids accessible to deposition attempts. Thus, both extensive numerical studies and their analytical interpretation were possible in $1D$\ \ [4]. For higher-$D$ models however, no results were reported in the literature, to our knowledge. In this work we report extensive numerical simulations of the RSA process with diffusional relaxation, for the lattice hard-square model [5], i.e., the square-lattice hard-core model with nearest-neighbor exclusion. This model is well studied for its equilibrium phase transition [5] which is second-order with disordered phase at low densities and two coexisting ordered phases, corresponding to two different sublattice particle coverage arrangements, at high densities. Another simplifying feature of the hard-square model is that the only possible gridlocked (locally frozen) vacancies are parts of domain walls (see further below). As a result the coverage reaches the full crystalline limit at large times, by a process of diffusional domain wall motion leading to cluster growth reminiscent of quenched binary alloys and fluids at low temperatures [6]. The approach to the full coverage is numerically consistent with the $\sim t^{-1/2}$ law which in turn can be related to the domain size growth with time as $t^{1/2}$ as expected for order-parameter nonconserving dynamics. In the remainder of this work we first report computational details and numerical results. We then describe the domain-growth interpretation of the dynamics at high densities, as well as discuss some other collective effects observed in our simulations. In each Monte Carlo trial of our simulation on a $L\times L$ square lattice with periodic boundary conditions, a site is chosen at random. Then with probability $r$ we attempt to deposit a particle and with probability $(1-r)$ we try diffusion. In the case of deposition, we check if the chosen site and its four nearest-neighbor sites are all empty. If indeed they are empty the deposition is performed. The chosen site is marked as occupied. If however any of the five sites are already occupied, then the deposition attempt is rejected and the configuration remains unchanged. In the case of diffusion, which is of course possible only if there is a particle at the selected site, we choose at random with equal probability a direction (up, down, left, or right) and try to move this particle by one lattice spacing. A move is made if the targeted new site and its three nearest neighbors different from the ``source'' site, are all empty. If the attempted move is not possible, the particle stays at its original position. A unit Monte Carlo time step is defined such that each lattice site is checked once on average. This corresponds to $L^2$ trials as described earlier. This time scale, $T$, is conveniently related to the physically more interesting time $t$ defined to have fixed deposition attempt rate per site, with varying relative diffusion attempt rate proportional to $(1-r)/r$, $$ t=rT \; . \eqno(1) $$ We implement the dynamics in two different ways. The first one is a straightforward simulation of the model. However, it is inefficient at late stages, when most trials are rejected. Thus we have implemented the same dynamics with an event-driven method [7]. We keep a list of all the possible moves, deposition and diffusion. Then we pick a move according to proper probability and always carry it out. The list is then updated if necessary. The Monte Carlo time is incremented according to $\Delta T = - (\ln x)/R$, where $x$ is a uniform random number between 0 and~1, while $R$ is the rate of the system configuration changes in the original dynamics. Suppose our list contains $N_{\rm depo}$ sites available for deposition and $N_{\rm diff}$ possible diffusional moves. Each occupied lattice site contributes $0,1,2,3$ or 4 possible diffusional moves, while each empty lattice site contributes 0 or 1 deposition counts, all depending on the nearest-neighbor site configuration. Then $R$ is calculated as $$ R = r N_{\rm depo} + {1\over 4} (1-r)N_{\rm diff}\; . \eqno(2) $$ Numerical estimates were obtained for the following quantities. The coverage, $\theta(T)$, was defined as the total number of sites occupied divided by $L^2/2$. The deposition process always began with empty substrate so that the coverage $\theta$ increased from 0 to 1 at full saturation. The ``susceptibility'' measuring fluctuations of the magnetization, was defined by $$\chi = L^2 \bigl[ \langle m^2 \rangle - \langle |m| \rangle^2 \bigr] \; , \eqno(3) $$ where the average $\langle\ \rangle$ is over independent runs. The magnetization or order parameter was defined as usual [5] by assigning ``spin'' values $+1$ to particles on one of the sublattices and $-1$ on another sublattice. Empty sites on both sublattices were not counted (effectively having spin values 0). Thus the magnetization magnitude $|m|$ is defined as the difference of the number of particles deposited on two sublattices, normalized by the lattice size $L^2$. The values of $|m|$ are thus from 0 to $1\over 2$. The effective domain size, $\ell (T)$, was defined as in equilibrium-model studies of cluster coarsening [6], by $$\ell = 2L \sqrt{\langle m^2\rangle} \; . \eqno(4) $$ Here the normalization is such that a uniform state (single-domain) gives the size of the system, $L$. Series of snapshots of the coverage buildup are shown in Fig.~1. As is usually done for the equilibrium hard-square system [5], particles are represented by squares of size $\sqrt{2} \times \sqrt{2}$, rotated 45$^\circ$ with respect to the original square lattice on which the particle centers are deposited. Particles on the even and odd sublattices (the sum of the $x$ and $y$ coordinates even or odd) are shown in different shades. The time-dependence of the coverage is illustrated in Fig.~2. The general features are similar to those found in the $1D$ studies [4]. For fixed deposition rate, corresponding to the time scale $t$ defined in (1), added diffusion [rate $\sim (1-r)/r$] always speeds up coverage growth. For high coverage, the $r<1$ plots are reminiscent of domain growth in phase-separation models [6]. This similarity with domain growth dynamics can be made more quantitative. Let us first consider the coverage $\theta(T)$ for large times. In the case of deposition rate $r=1$ (no diffusion), the approach to the jamming coverage $\theta (\infty) \simeq 0.728 < 1$\ \ [8] is generally exponentially fast for lattice models [9]. With diffusion, one can always reach the full coverage $\theta( \infty ) = 1$. However, the approach to the full coverage is slow, power-law, as indicated in Fig.~2. In $1D$, the power-law behavior was related to the coverage growth at large times by the process of hopping and recombination (opening up deposition sites) of small empty regions [4]. The coverage growth mechanism for large times, in the $2D$ hard-square model is instead due to interfacial dynamics. As illustrated in Fig.~1, the void space at late times consists of domain walls separating spin-up and spin-down ordered regions. Since a typical domain has area $\sim \ell^2 (T)$ and boundary $\sim \ell(T)$, we anticipate that for large times $$ 1 - \theta (T) \propto \ell^{-1} (T) \; . \eqno(5) $$ Indeed, $1-\theta$ is just the void area fraction. The large-time behavior of the domain size in models with nonconserved order-parameter dynamics is typically diffusional $\sim T^{1/2}$. For $\ell (T)$ we report the direct numerical verification later. For the coverage, we found that the data roughly fit the power law, $$ 1- \theta(T) \propto T^{-1/2} \; , \eqno(6) $$ for $T > 10^3$ in typical runs such as illustrated in Fig.~2. Thus, the RSA quantity $1-\theta(T)$ behaves analogously to the energy excess in equilibrium domain growth problems. However, on careful examination we noted that the slopes obtained by least-square fits of the straight-line portions of the curves for large $T$ were not exactly $-0.5$ but rather ranged from about $-0.48$ to about $-0.52$ when the rate parameter $r$ was decreased from $0.8$ to $0.01$, corresponding to accelerating the relative diffusion rate. These differences could not be fully attributed to statistical errors. Presumably, they represent effects of corrections to the leading power law behavior. The ``susceptibility'' $\chi$ for a given finite size $L$ has a peak and then decreases to zero, indicating long-range order for large $T$; see Fig.~3. The peak location seems size-dependent, at $T_{\rm peak} \propto L^2$; thus, it is difficult to observe for large system sizes. Since finite-size effects set in for $\ell (T) \sim L$, which given the ``bulk'' power law $\ell (T) \sim T^{1/2}$ leads precisely to the criterion $T \sim L^2$, we expect this maximum in fluctuations to be a manifestation of the ordering process at high densities. For equilibrium hard squares [5], the critical-point density corresponds to coverage $\theta \simeq 0.742$. However, in all our simulations, even with the fastest relative diffusion rate (the case $r=0.01$), we found no interesting features in $\chi (T)$ for times for which the coverage was near the critical value. Since numerical effort to reach a given coverage increases with increasing the diffusion rate at the expense of deposition attempts, there still remains a numerical challenge to observe the buildup of rounded critical-point fluctuations in $\chi$ or other quantity, for RSA with diffusion. Our present simulations turn out to be sensitive only to those collective effects which are associated with ordering at higher than critical-point coverages. In Fig.~4, the effective domain size $\ell(T)$ is plotted vs.~time for $r=0.1$. The large-time asymptotic law is well fitted by $T^{1/2}$. As already mentioned, the late-stage dynamics of our model is similar to the kinetics of ordering by quenching an equilibrium system into a two-phase region. In fact, our model can be mapped approximately to the Ising model at zero temperature. Note that the straight-line, diagonal sections of the domain boundaries separating two phases, can not move. Only the corners can evolve due to diffusion, just like corners of interfaces in the Ising models at zero temperature evolve by order-parameter nonconserving dynamics. It has been known that the $t^{1/2}$ law is robust, independent of the number of ordered phases, dimensionality, and details of interactions in domain growth problems [6]. In summary, we found that when diffusion is introduced in random sequential adsorption processes, the system can relax to the full coverage, consistent with experiments on protein adsorption at surfaces. The approach to the full coverage is slow (power law). The dynamics for large coverages is governed by domain growth. This research was supported in part by the Hong Kong Baptist College Faculty Research Grant and by the DFG-Sonderforschungsbereich 262/D1 (Germany). \NP \centerline{\bf REFERENCES} {\frenchspacing \item{${}^{\bf *}$} DFG Heisenberg Fellow \item{[1]} For recent reviews see M.C.~Bartelt and V.~Privman, Int. J.~Mod. Phys. B{\bf 5}, 2883 (1991); J.W.~Evans, Rev. Mod. Phys. (1992), in print. \item{[2]} J.~Feder and I.~Giaever, J.~Colloid Interface Sci. {\bf 78}, 144 (1980); G.Y.~Onoda and E.G.~Liniger, Phys. Rev. A{\bf 33}, 715 (1986); N.~Ryde, N.~Kallay and E.~Matijevi\'c, J.~Chem. Soc. Faraday Trans. {\bf 87}, 1377 (1991). \item{[3]} J.J.~Ramsden, preprint (1992); see also J.~Phys. Chem. {\bf 96}, 3388 (1992). \item{[4]} V.~Privman and P.~Nielaba, Europhys. Lett. {\bf 18}, 673 (1992); P.~Nielaba and V.~Privman, Mod. Phys. Lett. B {\bf 6}, 533 (1992); V.~Privman and M.~Barma, J.~Chem. Phys. (1992), in print. \item{[5]} L.K.~Runnels, in {\sl Phase Transitions and Critical Phenomena}, Vol.~2, p.~305, C.~Domb and M.S.~Green, eds. (Academic, London, 1972). \item{[6]} J.D.~Gunton, M.~San~Miguel, P.S.~Sahni, {\sl Phase Transitions and Critical Phenomena}, Vol.~8, p.~267, C.~Domb and J.L.~Lebowitz, eds. (Academic, London, 1983); O.G.~Mouritsen, in {\sl Kinetics and Ordering and Growth at Surfaces}, p.~1, M.G. Lagally, ed. (Plenum, NY, 1990); A.~Sadiq and K.~Binder, J.~Stat. Phys. {\bf 35}, 517 (1984). \item{[7]} K.~Binder, in {\sl Monte Carlo Methods in Statistical Physics}, 2nd. ed., p.~1, K.~Binder, ed. (Springer-Verleg, Berlin, 1986). \item{[8]} Numerical studies of the RSA process of hard squares without diffusion were reported by P.~Meakin, J.L.~Cardy, E.~Loh and D.J.~Scalapino, J.~Chem. Phys. {\bf 86}, 2380 (1987); R.~Dickman, J.--S.~Wang and I.~Jensen, J.~Chem. Phys. {\bf 94}, 8252 (1991); see also references quoted therein. \item{[9]} V.~Privman, J.--S.~Wang and P.~Nielaba, Phys. Rev. B{\bf 43}, 3366 (1991). } \NP \centerline{\bf FIGURE CAPTIONS} \ \noindent\hang{\bf Fig.~1.}\ \ Configurations of the hard-square RSA-with-diffusion model obtained with the deposition rate parameter $r=0.1$, at times $T=10$, 20, 30, and 100. Particles centered on the even sublattice are shown in grey, while those on the odd sublattice are shown in black. Lattice size was $32\times 32$. \noindent\hang{\bf Fig.~2.}\ \ Void area fraction, $1-\theta$, plotted against time $t=rT$ on a double-logarithmic scale, for the deposition rate parameter $r=1$, $0.8$, $0.1$, $0.01$, and system size 521$\times$521. The data were averaged over 800, 600, 800, and 400 runs for $r=1$, 0.8, 0.1, and 0.01, respectively. The dashed-dotted line has slope of $-{1\over 2}$. \noindent\hang{\bf Fig.~3.}\ \ Fluctuation of the magnetization, $\chi$, vs.~time $T$, for system sizes $L=$4, 8, 16, 32, and 64. The deposition rate was $r=0.1$. The data were averaged over $10^5$ to $10^6$ runs. \noindent\hang{\bf Fig.~4.}\ \ Effective domain size $\ell$, plotted vs.~time $T$ on a double-logarithmic scale, with $r=0.1$ and system size 256$\times$256. The data were averaged over 1500 runs. \bye %!PS %% Figure 1 %% It will take some time to print this out; be patient! %% THIS FIGURE CAN BE EXECUTED AS A POSTSCRIPT "MOVIE" BY %% THE X-windows GHOSTSCRIPT FACILITY OR OTHER PREVIEWER gsave /Helvetica findfont 8 scalefont setfont 30 40 moveto (FIG. 1 Wang et al) show grestore /L 32 def /MCstep 1 def /Rate 0.1 def /Seed 1 def /L2 L dup mul def /Lplus L 5 add def /MCstep { L2 mul } def /Rate Rate 2147483647 mul cvi def /S L2 array def /center-write { L 2 idiv -2 add -4 moveto 0 setgray } def %% procedure definitions /neighbors %% take center i, return 4 neighbors: ip,jp,im,jm { dup 1 add dup L mod 0 eq { L sub } if exch dup L add L2 mod exch dup dup 1 sub exch L mod 0 eq { L add } if exch L sub L2 add L2 mod } def /xycoordinates %% translate single number i into x,y { dup L mod exch L idiv } def /ifempty %% test if all four sites are empty, return bool { false 4 { S 3 -1 roll get or } repeat not } def /drawpath %% take a site i, draw a 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1730 l 1362 1729 l 1371 1728 l 1384 1728 l 1384 1727 l 1406 1727 l 1407 1726 l 1569 1726 l stroke 1658 806 m (0.01) show 1793 951 m (0.1) show 1827 1270 m (0.8) show 1488 1780 m (1) show 2177 1490 m (slope -1/2) show [ 25 15 5 15 ] 0 setdash 1827 1728 m 2528 1132 l stroke /Helvetica findfont 30 scalefont setfont 160 75 moveto (FIG. 2 Wang et al) show showpage %%Trailer %%BoundingBox: 50 100 634 487 %!PostScript %% Figure 3 %%BoundingBox: (atend) %%Creator: Xmgr version 2.07 - PJ Turner OGI-ESE %%EndComments /m {moveto} bind def /l {lineto} bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.25 0.25 scale 1 setlinecap 90 rotate 144 -2362 translate 0 0 0 setrgbcolor 3 setlinewidth /Times-Italic findfont 60 scalefont setfont [] 0 setdash 528 420 m 528 2029 l 2641 2029 l 2641 420 l 528 420 l stroke [] 0 setdash 528 420 m 528 430 l 1056 420 m 1056 430 l 1584 420 m 1584 430 l 2113 420 m 2113 430 l 2641 420 m 2641 430 l 528 2029 m 528 2019 l 1056 2029 m 1056 2019 l 1584 2029 m 1584 2019 l 2113 2029 m 2113 2019 l 2641 2029 m 2641 2019 l stroke 528 420 m 528 440 l 1584 420 m 1584 440 l 2641 420 m 2641 440 l 528 2029 m 528 2009 l 1584 2029 m 1584 2009 l 2641 2029 m 2641 2009 l /Times-Italic findfont 60 scalefont setfont /Helvetica findfont 60 scalefont setfont stroke 528 370 m gsave 528 370 translate 0 rotate 0 -20 m (0) CS (0) show grestore 1584 370 m gsave 1584 370 translate 0 rotate 0 -20 m (100) CS (100) show grestore 2641 370 m gsave 2641 370 translate 0 rotate 0 -20 m (200) CS (200) show grestore /Helvetica findfont 72 scalefont setfont 1584 242 m gsave 1584 242 translate 0 rotate 0 0 m (T) CS (T) show grestore 528 420 m 538 420 l 528 956 m 538 956 l 528 1493 m 538 1493 l 528 2029 m 538 2029 l 2641 420 m 2631 420 l 2641 956 m 2631 956 l 2641 1493 m 2631 1493 l 2641 2029 m 2631 2029 l stroke 528 420 m 548 420 l 528 956 m 548 956 l 528 1493 m 548 1493 l 528 2029 m 548 2029 l 2641 420 m 2621 420 l 2641 956 m 2621 956 l 2641 1493 m 2621 1493 l 2641 2029 m 2621 2029 l /Helvetica findfont 60 scalefont setfont stroke 490 420 m gsave 490 420 translate 0 rotate 0 -20 m (0) RJ (0) show grestore 490 956 m gsave 490 956 translate 0 rotate 0 -20 m (10) RJ (10) show grestore 490 1493 m gsave 490 1493 translate 0 rotate 0 -20 m (20) RJ (20) show grestore 490 2029 m gsave 490 2029 translate 0 rotate 0 -20 m (30) RJ (30) show grestore /Helvetica findfont 72 scalefont setfont 318 1224 m gsave 318 1224 translate 90 rotate 0 0 m (c) CS /Symbol findfont 72 scalefont setfont (c) show grestore /Times-Roman findfont 90 scalefont setfont /Helvetica findfont 90 scalefont setfont /Helvetica findfont 60 scalefont setfont 3 setlinewidth %% curve data begin %% L = 4 549 428 m 570 435 l 591 441 l 612 444 l 633 444 l 654 443 l 675 440 l 697 438 l 718 435 l 739 432 l 760 430 l 781 428 l 802 426 l 823 425 l 845 424 l 866 423 l 887 422 l 908 422 l 929 421 l 950 421 l 971 421 l 993 420 l stroke %% L 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1888 l 1299 1910 l 1310 1931 l 1320 1951 l 1331 1972 l 1341 1994 l 1352 2015 l 1359 2029 l stroke 1200 1917 moveto (64) show 1319 1693 moveto (32) show 1376 1240 moveto (16) show 867 557 moveto (8) show 697 464 moveto (4) show /Helvetica findfont 30 scalefont setfont 160 75 moveto (FIG. 3 Wang et al) show showpage %%Trailer %%BoundingBox: 126 100 633 486 %!PostScript %% Figure 4 %%BoundingBox: (atend) %%Creator: Xmgr version 2.07 - PJ Turner OGI-ESE %%EndComments /m {moveto} bind def /l {lineto} bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.25 0.25 scale 1 setlinecap 90 rotate 144 -2362 translate 0 0 0 setrgbcolor 3 setlinewidth /Times-Italic findfont 60 scalefont setfont [] 0 setdash 528 420 m 528 2029 l 2641 2029 l 2641 420 l 528 420 l stroke [] 0 setdash 3 setlinewidth 740 420 m 740 430 l 864 420 m 864 430 l 952 420 m 952 430 l 1020 420 m 1020 430 l 1076 420 m 1076 430 l 1123 420 m 1123 430 l 1164 420 m 1164 430 l 1200 420 m 1200 430 l 1444 420 m 1444 430 l 1568 420 m 1568 430 l 1656 420 m 1656 430 l 1725 420 m 1725 430 l 1781 420 m 1781 430 l 1828 420 m 1828 430 l 1869 420 m 1869 430 l 1905 420 m 1905 430 l 2149 420 m 2149 430 l 2273 420 m 2273 430 l 2361 420 m 2361 430 l 2429 420 m 2429 430 l 2485 420 m 2485 430 l 2532 420 m 2532 430 l 2573 420 m 2573 430 l 2609 420 m 2609 430 l 740 2029 m 740 2019 l 864 2029 m 864 2019 l 952 2029 m 952 2019 l 1020 2029 m 1020 2019 l 1076 2029 m 1076 2019 l 1123 2029 m 1123 2019 l 1164 2029 m 1164 2019 l 1200 2029 m 1200 2019 l 1444 2029 m 1444 2019 l 1568 2029 m 1568 2019 l 1656 2029 m 1656 2019 l 1725 2029 m 1725 2019 l 1781 2029 m 1781 2019 l 1828 2029 m 1828 2019 l 1869 2029 m 1869 2019 l 1905 2029 m 1905 2019 l 2149 2029 m 2149 2019 l 2273 2029 m 2273 2019 l 2361 2029 m 2361 2019 l 2429 2029 m 2429 2019 l 2485 2029 m 2485 2019 l 2532 2029 m 2532 2019 l 2573 2029 m 2573 2019 l 2609 2029 m 2609 2019 l stroke 528 420 m 528 440 l 1232 420 m 1232 440 l 1937 420 m 1937 440 l 2641 420 m 2641 440 l 528 2029 m 528 2009 l 1232 2029 m 1232 2009 l 1937 2029 m 1937 2009 l 2641 2029 m 2641 2009 l /Helvetica findfont 60 scalefont setfont stroke 3 setlinewidth 528 370 m gsave 528 370 translate 0 rotate 0 -20 m (10) CS (10) show grestore 1232 370 m gsave 1232 370 translate 0 rotate 0 -20 m (100) CS (100) show grestore 1937 370 m gsave 1937 370 translate 0 rotate 0 -20 m (1000) CS (1000) show grestore 2641 370 m gsave 2641 370 translate 0 rotate 0 -20 m (10000) CS (10000) show grestore /Helvetica findfont 72 scalefont setfont 1584 242 m gsave 1584 242 translate 0 rotate 0 0 m (T) CS (T) show grestore 3 setlinewidth 528 630 m 538 630 l 528 753 m 538 753 l 528 841 m 538 841 l 528 909 m 538 909 l 528 964 m 538 964 l 528 1011 m 538 1011 l 528 1051 m 538 1051 l 528 1087 m 538 1087 l 528 1330 m 538 1330 l 528 1453 m 538 1453 l 528 1540 m 538 1540 l 528 1608 m 538 1608 l 528 1664 m 538 1664 l 528 1710 m 538 1710 l 528 1751 m 538 1751 l 528 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