Estimation of the available surface and the jamming coverage in the Random Sequential Adsorption of a binary mixture of disks

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1 Colloids and Surfaces A: Physicochem. Eng. Aspects ) 1 10 Estimation of the available surface and the jamming coverage in the Random Sequential Adsorption of a binary mixture of disks Marian Manciu, Eli Ruckenstein Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 11 August 2003; accepted 2 October 2003 Abstract A simple analytical expression is proposed for the area available to a disk on a surface for a Random Sequential Adsorption RSA) of binary mixtures of disks. The expression was obtained by combining the low-order terms of the density expansion of the available area with the asymptotic behaviour of the surface coverage near the jamming point. Comparison with Monte Carlo simulations shows that this approach provides a fair estimation of the jamming coverage for both kinds of disks Elsevier B.V. All rights reserved. Keywords: Random Sequential Adsorption; Jamming; Disk mixture 1. Introduction Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1]. However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface the particle insertion method) [3]. The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this Corresponding author. Tel.: /2214; fax: address: feaeliru@acsu.buffalo.edu E. Ruckenstein). low-density expansion are known [4]; they do not provide sufficient information for the high density behaviour. A typical example is a fortunate continuation of the virial expansion of a hard sphere gas, the Carnahan Starling equation of state [5], which predicts that the divergence of the compressibility occurs when the total volume of spheres is equal to the total volume available, while at close packing the spheres occupy only about 74% of the total volume. Since the density at which the compressibility diverges is known for monodisperse rigid spheres, it can be incorporated a posteriori in the equation of state [6], or one can test which one of the Padé analytic continuations of the virial series provides a better agreement [7]. However, there is no reliable procedure to predict a priori the compressibility divergence using only the first virial coefficients. As a consequence, there is no procedure which can predict even roughly the divergence phase transition) for a binary mixture of hard spheres. While accurate empirical equations of state are known at low densities for binary mixtures [8], they predict divergence of the compressibility only when the total volume is occupied. The situation is similar for a 2D binary mixture of hard disks. The non-equilibrium problem is even more complicated. The large particles can have surface binding energy much larger than kt and in this case they neither diffuse nor desorb from the surface. The Random Sequential Adsorption RSA) model [9] assumes that a particle, which arrives at a random location on a surface, is adsorbed only if there is /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /j.colsurfa

2 2 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) 1 10 sufficient room for it around that particular location that is, it does not overlap with any other adsorbed particles). Only one continuum RSA model, the 1D random parking of cars, received an analytical solution [10]. This problem implies that points are selected randomly along a line of length L. If there is enough space around the point to fit a car of length l, the car is parked; otherwise, another random point is selected, and so on. For a L line, when the density of cars is about N RSA l/l = 0.76 there is no more room to place a car. This result can be contrasted to the equilibrium problem of parking cars, for which cars can be parked until close compaction N E l/l = 1 the 1D gas of hard rods does not have a phase transition [11]). Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a new particle Φθ) can be written as a power series in particle density θ = Nπr 2 /A, where N is the number of adsorbed particles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of θ for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for θ C = 0.76, and 0.38 for the 1D segments on a line), 2D disks on a surface) and 3D spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at θ = 1, 0.91 and 0.74, respectively. The RSA model received renewed attention after Feder [12] observed that the adsorption on the surface of apoferritin molecules large iron-storage proteins with a diameter of about 10 nm), which adsorb irreversibly, reached saturation at a coverage θ C = Monte Carlo simulations of Random Sequential Adsorption of disks on a surface last prohibitively long in the vicinity of the jamming point; however Feder [12] noted that in the vicinity of the jamming coverage, θ has a power-law dependence on time: θ C θ 1 t. 1) By extrapolating this scaling law, an accurate value for the jamming coverage, θ C = was obtained. The power-law 1) was later demonstrated theoretically, and it was shown that it is accurate not only in the immediate vicinity of jamming, but in a broader θ range [13,14]. Schaaf and Talbot [15] continued Widom s analytical approach by calculating the available area for the RSA of disks and obtained the coefficient of θ 3, which is different in RSA and equilibrium models. The next coefficient was obtained independently by Dickman et al. [16] and Given [17]. The first five terms of the series are not, however, very useful for the calculation of the jamming point θ C. Indeed, using the five known terms, there is no jamming, because Φθ) = 0 has no solution for 0 <θ<1. Furthermore, almost all analytical continuations based on Padé approximants P[i, j] provided the same conclusion, that no jamming occurs except P[1, 4], which predicted an unsatisfactory value θ C 0.4). A modality to employ the scaling law 4) in the analytical continuation of the series was proposed by Dickman et al. [16]. They calculated the first terms of the time expansion of the coverage θt), transformed it to a new variable with an appropriate asymptotic behavior y = 1 1/ 1 + bt) and found, when the Padé [3,2] approximant was employed, an excellent agreement for the jamming coverage within about 0.15%) by selecting b = 3. The use of a different value for b or of another Padé approximant deteriorated, however, dramatically the agreement. Later, Wang [18] suggested to estimate b from the convergence of the jamming coverages predicted by various Padé approximants. While the method seems to work well for the RSA of oriented squares for which the first nine terms of the low-density expansion are known), Wang concluded than there are not sufficient known terms in the RSA expansion for disks to reach an accurate prediction [18]. The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12]. However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [19,20], the available area vanishes only for the unphysical total coverage θ = θ S +θ L = 1 where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming. Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21]. The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22]. However, there are no such calculations for the RSA model. Let us first briefly review the recent conclusions about the jamming in a RSA model of binary mixtures. For the 1D model, an exact solution for the random parking of cars of two different sizes predicted that the coverage is always larger for a binary mixture than for unisized cars [23].However, another solution for the 1D model predicted a smaller coverage for binary than for unisized cars in a continuum model, but a larger coverage for binary in a lattice model [24]. For the 2D model, a Monte Carlo simulation showed that a binary mixture of disks always covered the surface better than monodisperse disks [25]. However, another Monte Carlo simulation of deposition of larger spheres on a surface randomly precovered with smaller ones indicated that the total coverage is always smaller for binary mixtures than for unisized disks [26]. These puzzling results can be understood qualitatively. In a mixture of large and small disks, if the large disks are adsorbed first, they can cover up to a fraction θ C of the surface. The jamming then is reached for the large disks,

3 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) but the small disks can still be adsorbed, and the binary mixture covers a larger fraction of the surface than the monodisperse disks. In the limit of very large and very small disks, the total coverage is clearly θ C + 1 θ C )θ C However, if some small disks are adsorbed first without reaching jamming, there is enough room for the small disks, but might not be for the larger ones adsorbed later. In this case, the coverage is lower for the binary mixture. Therefore, the available area and the jamming depend not only on the concentration of the particle already adsorbed on the surface, θ S and θ L, but also on the order of their adsorption. The only approximate analytical solution for the RSA of a binary mixture of hard disks was proposed by Talbot and Schaaf [27]. Their theory is exact in the limit of vanishing small disks radius r S 0, but fails when the ratio γ = r L /r S of the two kinds of disk radii is less than 3.3; its accuracy for intermediate values is not known. Later, Talbot et al. [28] observed that an approximate expression for the available area derived from the equilibrium Scaled Particle Theory SPT) [19] provided a reasonable approximation for the available area for a non-equilibrium RSA model, up to the vicinity of the jamming coverage. While this expression can be used to calculate accurately the initial kinetics of adsorption, it invariably predicts that the abundant particles will be adsorbed on the surface until θ = 1, because the Scaled Particle Theory cannot predict jamming. The evaluations of the jamming coverage and of the available area for a binary mixture is of interest in many practical applications, such as protein and cell separation affinity chromatography), biocompatibility of biomaterials and separation of toner and ink particles [29]. A simple approach to estimate the available area to a disk and the jamming points for binary mixtures of disks will be proposed in what follows. The approach combines the power-law dependence of the available area in the vicinity of the jamming point with the known dependence at low coverage densities. For monodisperse disks, an excellent approximate expression can be constructed in this manner by employing information about the behaviour near the jamming point provided by Monte Carlo simulations. However, reasonable results can be obtained without using the latter information. To calculate the area available to a large disk on a surface for binary mixtures Φ L θ S, θ L ), it will be assumed that the small disks are deposited first until θ S is reached and then the large disks are adsorbed until jamming. To calculate the area available to small disks, large and small should be interchanged. The available area Φ L θ S, 0) can be calculated using one of the available approximations, that are accurate at low densities. The approximations for Φ L θ S, 0) are in most cases accurate, even when the same approximation for Φ L θ S, θ L ) fails. The available area Φ L θ S, θ) is a function of θ alone and can be continued analytically taking into account its asymptotic behavior, which is the same as that corresponding to monodisperse disks. The predictions of the jamming points, obtained from Φ L θ S,θ L )θ S )) C ) 0 are compared with the Monte Carlo simulations available in [25,26]. 2. RSA of monodisperse disks The rate of adsorption of monodisperse disks on a surface, dθ/dt, is proportional to the total surface available to a disk, Φ: dθ = Φθ) 2) dt where the surface coverage θ = Nπr 2 /A, N being the number of adsorbed disks, r the radius of a disk, A the total area and t a dimensionless time. Monte Carlo simulations indicated that, in the vicinity of jamming, θ = θ C K t 3) where θ C = and K = [12]. From Eq. 3) one obtains that dθ dt = K 2 t 3 which combined with Eqs. 2) and 3), leads for the available area in the vicinity of the jamming point to the expression: Φ F θ) = 1 2K 2 θ C θ) 3 4) The first terms of the series expansion of the available area for low surface coverages are also known [15 17]: Φθ) = i=0 c i θ i = 1 4θ θ θ θ 4 An approximate expression, which can interpolate between Eqs. 4) and 5), has the form [15]: Φ i θ) = θ C θ) 3 a 1 + a 2 θ + a 3 θ 2 + a 4 θ 3 + ) 6) where the coefficients a i are determined from the conditions to obey the limit law 4) near θ C : a 1 + a 2 θ C + a 3 θc 2 + a 4θC 3 + = 1 2K 2 6a) and to match the terms of the θ expansion 5): θc 3 a 1 = 1 6b) θ 3 C a 2 3θ 2 C a 1 = 4 θ 3 C a 3 3θ 2 C a 2 + 3θ C a 1 = 3.308,... 5) 6c) 6d) The best agreement with the limiting laws Eqs. 4) and 5) is obtained using only four terms in Eq. 6). Then, Eq. 6a) 6d) lead to a 1 = 1 θc 3 = a)

4 4 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) 1 10 a 2 = 3 4θ C θ 4 C a 3 = 3.308θ2 C 12θ C + 6 θ 5 C = b) = c) a 4 = 1/2K2 ) 3.308θ 2 C + 16θ C 10 θ C = d) Only the first three terms of expansion 5) were employed in Eq. 7a) 7d). These terms coincide to those of the equilibrium expansion. The use of the next coefficients does not improve, however, the accuracy, a situation that is not uncommon in the analytical continuation of a virial expansion [4]. The interpolating approximation 6) is compared with the limiting laws, namely the series expansion Eq. 5) and Feder law Eq. 4) in Fig. 1a. The accuracy of the interpolation can be better seen in Fig. 1b, where the ratios between Eq. 6) and the limiting laws are plotted. The interpolating function 6) satisfies both limiting laws in their ranges of validity) within a few percents, practically representing the Monte Carlo simulations within their numerical error. Fig. 1. a) The area available to a new disk as a function of surface coverage. b) The ratios of the interpolating function, Eq. 6) to the limiting laws Eqs. 4) and 5)) as functions of surface coverage. The limiting law 4) is valid at high surface coverages near jamming), while Eq. 5) is valid at low coverages.

5 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) Fig. 2. Predictions for the available area Φθ). The jamming point can also be evaluated without using the information provided by the Monte Carlo simulations [30]. While this approach is not useful for monodisperse disks, for which a much better interpolating expression can be obtained using the accurate values for θ C and K predicted by Monte Carlo simulations, it can be employed to estimate the available area and the jamming coverage for binary mixtures of disks, for which in general the values of θ C and K are not known. In Fig. 2, we represent the interpolating function in a natural representation, Φθ)) 1/3 versus θ, which becomes linear near the jamming point. The interpolating function can be approximated by an expression of the type ) Φ)θ)) 1/3 = a 0 θ) a i+1 θ i 8) i=0 with the coefficients provided by the matching of the first terms of the low-density expansion Eq. 5). The best approximate was obtained by using the first four coefficients, which are given by a 0 a 1 = 3 c 0 8a) a 2 a 0 a 1 = 3 c 1 c 0 a 3 a 0 a 2 = 3 c 0 c 2 c1 ) 2 ) 8b) 8c) ) a 3 = 3 2c 1 c 2 c 0 9c0 2 5c3 1 81c0 3 8d) where c i are the coefficients of the low-density expansion Eq. 5)). The approximate expression predicts a jamming at a 0 = 0.539, which is not far from the value obtained from Monte Carlo simulations, θ C = An even simpler approximation can be constructed by matching only the first three terms of expansion 5), with those of expression: Φθ)) 1/3 = b 1 b 0 θ) + b 2 b 0 θ) 2 9) whose coefficients are obtained from b 0 b 1 + b 2 b 2 0 = 3 c 0 b 1 + 2b 2 b 0 = 3 c 0 c 1 b 2 = 3 c 0 c 2 c1 ) 2 ) 9a) 9b) 9c) providing jamming for θ C = b 0 = It was recently noted by Talbot et al. [28] that an approximate equation derived for thermal equilibrium of disks from the Scaled Particle Theory, namely [19] Φ SPT θ) = 1 θ) exp 3θ ) 1 θ θ2 1 θ) 2 10) provided good agreement for the available area in the non-equilibrium RSA model, up to the vicinity of the jamming coverage. In Fig. 2, the interpolating function 6), is compared with the approximations 8) and 9), which are more accurate than the SPT prediction 10), particularly at high θ values.

6 6 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) RSA of binary mixture of disks When two kinds of disks of radii r S and r L γr S with γ>1) are deposited randomly on a surface, the problem becomes much more involved. It was shown that the Feder law holds for monodisperse disks [12 14]. In contrast, for the RSA of binary mixtures, the jamming of the large particles is reached exponentially and then the density of small particles obeys Feder s power law Eq. 4)) until their jamming [31]. However, the Monte Carlo simulations of an alternate deposition of particles, namely: first all the small ones and then the large ones, revealed that the approach to jamming by the large particles also obeys Feder s law [26]. The jamming coverage θ C and the proportionality constant K depend on the ratio γ = r L /r S and on the initial area occupied by the small particles the precovered area), θ S. To seek a reasonable accurate analytical approximation for the available area, as a function of θ S = N S πrs 2 /A and θ L = N L πrl 2 /A one should have accurate values for a reasonable number of coefficients in the low-density expansion of the binary RSA model, which is not a trivial task. Even for binary mixtures of disks at equilibrium, a problem that received much more attention than RSA, analytical expressions are known only for the first three terms of the virial expansion [21]. The values of the fourth and fifth terms, obtained using laborious numerical calculations, were reported only for a few values of γ and molar fractions of the two types of disks [22]. In the non-equilibrium RSA of binary particles, one should take into account, when calculating the higher terms of the series, not only various γ and molar fractions, but also the order of deposition of particles. Furthermore, as already noted, it is not clear whether the involved calculations needed to obtain the next unknown terms of the low-density expansion would improve much the accuracy of estimating the jamming coverage. An alternate possibility is to obtain approximations for the coefficients of the low-density expansion using analytical approximations of the available surface and to extend analytically these approximations. The expressions for available area, such us those derived from Scaled Particle Theory, Φ SPT i θ S,θ L ) [19]: Φ SPT S θ S,θ L ) = 1 θ S θ L ) exp 3θ S + θ L /γ)2+1/γ)) θ S+θ L /γ)) 2 ) 1 θ S θ L 1 θ S θ L ) 2 11a) Φ SPT L θ S,θ L ) = 1 θ S θ L ) exp 3θ L + γ2 + γ)θ S θ S + γθ L ) 2 ) 1 θ S θ L 1 θ S θ L ) 2 11b) were already shown by Talbot et al. [28] to be excellent approximations at low densities. However, by predicting a total final coverage of θ S + θ L = 1, they fail in the vicinity of the jamming point. Therefore, they fail to provide both the final coverage and the final ratio of adsorbed species at jamming. In the present approach, the area available for large disks, Φ L θ S,θ L ), will be approximated as follows interchange everywhere S and L for the area available to small disks). It will be assumed as an approximation that only the small disks are deposited until the surface coverage θ S is reached, and then only the large particles are adsorbed until their jamming. Along the path θ S = 0, θ L = 0) θ S,0)it is assumed that the initial area available to the large disks, Φ L θ S, 0) can be approximated by Φ SPT L θ S, 0), providing that this it is not in the vicinity of the jamming point which implies that Φ SPT L θ S, 0) is sufficiently large). Along the path θ S, 0) θ S,θ), the SPT approximation fails at large θ, in the vicinity of the large disks jamming point. However, Φ L θ S,θ)is a function of θ alone, and obeys the asymptotic behaviour 4) with the constants θ C and K dependent on γ and θ S ). Therefore, an expression Φ L θ S,θ) of the type 8) or type 9) can be constructed starting from the first coefficients of the low-density expansion: Φ L θ S,θ)= 1 k Φ SPT L θ S,θ) k! θ k θ k 12) k=0 θ=0 The approximation Φ L θ S,θ L ) will provide an estimate for the available area for large disks on a surface upon which the adsorbed small and large disks cover the fraction θ S and θ L of the surface, respectively. While no data are available in literature for Φ L θ S,θ L ), the accuracy of the estimation Φ L θ S,θ L ) can be verified by comparing the jamming of large disks obtained analytically from the conditions Φ L θ S,θ L ) C ) = 0; Φ S θ S ) C,θ L ) = 0 13) with the jamming predicted by Monte Carlo simulations. Let us first examine whether Eq. 8) or Eq. 9) provides better results, by calculating the total jamming coverage, θ C θ S +θ L ) C as a function of θ S in the simple case γ = 1 monodisperse disks), for which accurate results are known from Monte Carlo simulations θ C = 0.547). The low-θ L expansion is given by Φ L θ S,θ)= 1 k Φθ) k! θ k θ θ S ) k 14) k=0 θ=θs were we selected for Φθ) either the interpolating function 6) or the SPT expression 10). The interpolating function 6) was preferred to expression 5) because the former is accurate over the entire θ range, while the latter only for low θ values. The jamming of the large particles estimated in this manner for various initial coverages θ S are compared in Fig. 3 with the exact result, θ C = As expected, both approximations based on Eqs. 8) and 9) are better at large θ S values, where Feder s law is obeyed, when the interpolating function is employed for the low-θ L expansion. In

7 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) Fig. 3. The total jamming coverage for a mixture of monodisperse disks γ = 1) predicted using either Eq. 8) or Eq. 9), combined with the low-θ L expansion terms calculated from either Eq.6) or Eq. 10), as a function of the small disks coverage, θ S, are compared to the accurate jamming point, θ C = contrast, when the SPT expression 10) is used in Eq. 14), the jamming point estimation deteriorates at high θ S values, mainly because Eq. 10) ceases to be a good approximation in that region. While both approximations 8) and 9) overpredict systematically the jamming, when the SPT ex- pression is employed for the low-θ expansion 14), Eq. 9) appears to be a better one and will be used in what follows. Let us compare the approach described above with the existing Monte Carlo simulations. While the values for the available area Φ S θ S, θ L ) are not available in literature, we Fig. 4. Predicted jamming point for small disks, as a function of the surface coverage by large disks, θ L, compared with the results of Monte Carlo simulations reported in [25], for γ = 2.0 and 8.0.

8 8 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) 1 10 will use the results obtained by Meakin and Jullien [25] for the RSA of a binary mixture of hard disks of different radii and different concentrations. While both kinds of disks are present in the vicinity of the surface, the large ones reach exponentially the jamming and then practically only the small ones are adsorbed until they reach their jamming point. The values of θ L and θ S ) C, which correspond to the jamming of small particles, Φ S θ S,θ L ) = 0, derived from Monte Carlo simulations [25] are compared in Fig. 4 with the estimates based on the approach described above, which employs Eqs. 9), 11a), 12) and 13). Except for the systematic overprediction of jamming by about 0.05, due mainly to the use of the SPT approximation, the estimates are in a fair agreement with the simulations. It is of interest to test the accuracy of the present approach by estimating the jamming of large disks in the presence of small ones. When both kinds of disks are present in the vicinity of the surface, the large ones reach rapidly their Fig. 5. Predicted jamming point for the large spheres as a function of the initial surface available to the large spheres, Φ L θ S,0), compared with the results of the Monte Carlo simulations reported in [26]: a) λ = 2.2; b) λ = 5.0; c) λ = 10.0.

9 M. Manciu, E. Ruckenstein / Colloids and Surfaces A: Physicochem. Eng. Aspects ) Fig. 5. Continued ). jamming, whereas the small disks continue to be adsorbed on the surface for a much longer time. Because the small disks coverage changes very much in the region in which the large disks are almost at their jamming see Fig. 2 of [25]), the value of θ S, at which the jamming of large particles occurs, cannot be determined accurately the precision being of the same order of magnitude as the estimation employed here). Accurate values could be obtained from Monte Carlo simulations of successive depositions of small and large disks. The only simulations reported in literature, however, refer to a related, but different problem, namely the RSA of large spheres adsorbed on a surface precovered with small spheres [26]. As noted by Talbot and Schaaf [27], the surface available for a large sphere of radius a L = λa S on a surface precovered with small spheres is the same, in the limit θ L 0, as the surface available to large disks of radius r L = γr S for γ = 2 λ 1. 15) Unfortunately, since the mapping of spheres into disks fails for θ L 0, the derivatives with respect to θ L cannot be calculated accurately from Eqs. 11a) and 11b). Since these values are needed to approximate Φ, this inaccuracy generates additional errors. An additional difficulty is that the SPT approximation is excellent for low surface coverages, when Φθ S,θ L ) 1), but fails at large surface coverages Φθ S,θ L ) 0). While the small particles might not occupy a large area on the surface, they can exclude most of the surface to the large ones. Since the SPT approximation fails when the available area is small, we plot in Fig. 5a c the jamming coverage of large particles, Φ L θ S,θ L ) C ) as a function of the initial area available to them, Φ L θ S, 0), to better identify the range of validity of the approximation. There is a good agreement between the Monte Carlo simulations and the present approach for not too low values of Φ L θ S, 0). 4. Conclusions Despite the attention received by the Random Sequential Adsorption model of particles, only the one-dimensional problem Renyi s car parking problem) was solved analytically. Accurate results could be obtained from Monte Carlo simulations in higher dimensions. Simple and accurate expressions for the available area can be constructed from the low-density expansion, combined with the information provided by Monte Carlo simulations for the behaviour near the jamming point. However, reasonable estimates can be made without using the Monte Carlo results. In the present paper, a procedure was suggested to estimate the available area for a Random Sequential Adsorption of binary mixture of disks, for which the Monte Carlo calculations are prohibitively long. It was assumed that first all the small large) disks were deposited until a value θ S θ L ) was reached, then the other type of disks were adsorbed until they reached their jamming. Then, the approximate Eqs. 11a) and 11b) were employed to calculate the low-density expansion coefficients of the available area as a function of θ L θ S ) alone. Finally, an approximate expression 9) with appropriate asymptotic behaviour near the jamming point was constructed. Whereas this method provided only estimates of the jamming points, they represent a significant improvement over the predictions of the Scaled Particle Theory, Eqs. 11a) and 11b).

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