Langmuir 2007, 23, 5557-5562
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Particle Assembly on Surface Features (Patterned Surfaces) Zbigniew Adamczyk,* Jakub Barbasz, and Maria Zembala Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, 30-239 Krako´ w, Niezapominajek 8, Poland ReceiVed December 20, 2006. In Final Form: February 8, 2007
Irreversible adsorption (deposition) of spherical particles on surface features of various shapes (collectors) was studied using the random sequential adsorption (RSA) model. The collectors in the form of linear line segments, semicircles, and circles were considered. Numerical simulation of the Monte Carlo type enabled one to determine particle configurations, the jamming coverage, and the end to end length of particle monolayers for various collector length (L) to particle size (d) ratio L h ) L/d. It was revealed that the jamming coverage for linear collectors Θ′∞ increases for L h > 2 according to a linear dependence with respect to 1/L h . For 2 > L h > 1, a parabolic dependence of Θ′∞ on 1/L h was predicted, characterized by the maximum value of Θ′∞ ) 1.125 for L h ) 4/3. These dependencies allowed one to formulate an equation determining the length of nanostructures on surfaces if the averaged number of adsorbed particles is known. It was also predicted that the end to end length of the monolayer on a linear collector 〈Le〉/L increased linearly with 1/L h for L h > 2. For 2 > L h > 1 the dependence of 〈Le〉/L on L h was approximated by a polynomial expression, exhibiting a maximum of 〈Le〉/L ) 1.17 for L h ) 1.45. In the case of circular collectors, the jamming coverage was found to be substantially smaller for the same value of 1/ L h . It was demonstrated that the theoretical results are in agreement with our preliminary experimental data obtained for latex particles adsorbing on polyelectrolyte modified mica and on patterned surfaces obtained by a polymer-on-polymer stamping technique of gold covered silicon (Zheng et al. Langmuir 2002, 18, 4505).
I. Introduction Adsorption and deposition (irreversible adsorption) of colloids and bioparticles on solid/liquid interfaces is of great significance for many practical and natural processes such as protein, bacteria, or enzyme immobilization and separation, removal of pathological cells, immunological assays, biofouling of transplants and artificial organs, etc. Adsorbed molecules or deposited particles, depending on their coverage degree and characteristics, modify initially homogeneous surfaces. Cationic polyelectrolytes are used to increase retention of filler particles (e.g., titania) in paper making processes,1 to promote particle deposition,2-3 or selective selfassembling of colloid particles or cells at patterned surfaces.4-7 Thus, it is of special importance to learn more about adsorbing species (in particular their sizes and shapes) and to find methods allowing characterization of heterogeneous or patterned surfaces. It is interesting to observe that rigid polyelectrolyte molecules, like poly(sodium 4-styrene sulfonate) (PSS) for example, assumes an extended, rod-like shape,8-9 especially at lower ionic strength. In the case of more flexible and amphoteric polyelectrolytes, like poly(allylamine hydrochloride) (PAH) or poly(acrylic acid) (PAA), their shape can be regulated by the pH and ionic strength. This allows one to obtain molecule shapes resembling semicircles * Corresponding author. E-mail:
[email protected]. (1) Baluk, M. Y.; van de Ven, T. G. M. Colloids Surf. A 1990, 46, 157-176. (2) Serizawa, T.; Kamimura, S.; Akashi, M. Colloids Surf. A 2000, 164, 237245. (3) Adamczyk, Z.; Zembala, M.; Michna, A. J. Colloid Interface Sci. 2006, 303, 353-364. (4) Zheng, H.; Rubner, M. F.; Hammond, P. T. Langmuir 2002, 18, 45054510. (5) Zheng, H.; Berg, M. C.; Rubner, M. F.; Hammond, P. T. Langmuir 2004, 20, 7215-7222. (6) Karakurt, I.; Leiderer, P.; Bonberg, J. Langmuir 2006, 22, 2415-2417. (7) Krueger, C.; Jonas, U. J. Colloid Interface Sci. 2002, 252, 331-338. (8) Adamczyk, Z.; Zembala, M.; Warszyn´ski P.; Jachimska, B. Langmuir 2004, 20, 10517-10525. (9) Stoll, S. In Colloidal Biomolecules Biomaterials and Biomedical Applications; Elaissari, A., Ed.; Marcel Dekker: New York, 2004; p 211.
or circles as demonstrated in refs 8 and 10. Similarly, many biopolymers, especially DNA, under certain conditions may assume the shape of a curved line segment, often resembling semicircles or circles.11 There are both theoretical9 and experimental12 evidences that these shapes are preserved upon molecule adsorption on solid substrates. Particle adsorption on such surface features, which is of interest in the silver staining technique, for example, can be treated as adsorption on curvilinear line segments of finite length and of negligible thickness. This is so because the diameter of a typical polyelectrolyte chain is much smaller than colloid or globular protein size, whereas its length is often larger than particle dimension. A similar problem, in the micrometer rather than nanometer range scale, arises in the case of large colloid particle deposition on ring-patterned surfaces like these produced on various substrates by a polymer-on-polymer stamping technique on gold covered silicon.4 Besides practical importance, irreversible adsorption of particles on line segments, represents one of the simplest cases of an interesting mapping problem and has a significance for basic science. Knowing the mapping function, i.e., the correspondence between the density and distribution of adsorbed particles of micrometer size and the shape and dimensions of objects attached to surfaces, one can gain quantitative information on the nanosized surface features, inaccessible for direct experimental measurements. Despite the physical and practical significance, the problem of particle adsorption on curvilinear line segments of finite length was not analyzed quantitatively in the literature. Theoretical results are available for linear segments of finite length.13-14 A recursion (10) Adamczyk, Z.; Bratek, A.; Jachimska, B.; Jasin´ski, T.; Warszyn´ski, P. J. Phys. Chem. B. 2006, 110, 22426-22435. (11) Wu, A.; Li, Z.; Zhou, H.; Wang, E. Superlattices Microstruct. 2005, 37, 151-161. (12) Pope, L. H.; Davies, M. C.; Laughton, C. A.; Roberts, C. J.; Tendler, S. J. B.; Williams, P. M. Anal. Chim. Acta 1999, 400, 27-32.
10.1021/la063677u CCC: $37.00 © 2007 American Chemical Society Published on Web 04/05/2007
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that the adsorption probability is continuous over the entire collector. The dimensionless adsorption time is defined as
τ ) Nattd/L ) Natt/L h
(1)
h ) L/d is the where Natt is the overall number of attempts and L collector length to particle size dimensionless parameter. The averaged 1D density (coverage) of particles is defined as
Θ ) Npd/L ) Np/ L h
Figure 1. Schematic representation of particle adsorption on a curvilinear segment of the length L.
formula has been derived, which allows one to calculate the jamming limit (maximum coverage) of disk-shaped particles of the diameter d as a function of the L/d ) L h parameter (where L is the length of the line) for L/d < 3. Numerical results have also been presented for 1 < L/d < 10. It was demonstrated that, in the case of large L h , the extrapolated jamming coverage approaches the asymptotic value of 0.7476 predicted for infinitely long lines.15-17 However, no results were presented, that allow one to calculate the fluctuations in the effective length of particle monolayers formed on the line, which determine the quality of mapping. This is important from an experimental point of view. Apparently, there exist no theoretical results for the case of curved segments (semicircles or circles). Thus, the goal of this work is to develop a quantitative description of particle deposition on line segments of various shapes, in particular to derive expressions for calculating the “capacity” of surface features to accommodate colloid particles. This kind of information is of significance in processes involving microcontact printing on patterned surfaces or imaging of macromolecular structures at surfaces. The theoretical results will be compared with some experimental data derived for latex particles adsorbing on surface features produced on mica and gold covered silicon.
II. The Model Since there are no analytical results pertaining to adsorption of particles on segments of various shapes and of finite length, we applied Monte-Carlo simulations to derive information on the jamming coverage and distribution of particles. These simulations have been carried out according to the random sequential adsorption (RSA) model in one dimension (1D).16-18 The 1D RSA process consists of placing a linear object of the dimension l on the curved line of the length L. For the sake of convenience, the line segment will be referred hereafter as the collector. We assume that the adsorbing objects are spheres of radius a and diameter d ) l ) 2a (see Figure 1). If there is an empty space (line segment between previously adsorbed particles) large enough to accommodate the new particle, it is irreversibly adsorbed, otherwise, an adsorption attempt is repeated. The main feature of this RSA process is that the new adsorption attempt is totally uncorrelated with previous adsorption attempts and (13) Bafaluy, F. J.; Choi, H. S.; Senger, B.; Talbot, J. Phys. ReV. E 1995, 51, 5985-5993. (14) Burridge, D. J.; Mao, Y. Phys. ReV. E. 2004, 69, 037102-1-037102-4. (15) Renyi, A. Publ. Math. Inst. Hung. Acad. Sci. 1958, 3, 109. (16) Pomeau, Y. J. Phys. A: Math. Gen. 1980, 13, L193-L196. (17) Hinrichsen, E. L.; Feder, J.; Jossang, T. J. Stat. Phys. 1986, 44, 793-827. (18) Evans, J. W. ReV. Mol. Phys. 1993, 65, 1281-1329. (19) Zembala, M.; Barbasz, J.; Adamczyk, Z. J. Colloid Interface Sci. to be published.
(2)
where Np is the total number of adsorbed particles over the segment. Note that the definition of coverage given by eq 2 is valid for collectors of an arbitrary shape. In the case of linear collectors of finite length, a recursion formula has been derived,13-14 which expresses the averaged number of particles over the collector under the jamming state in the form
h )〉 ) 1 + 〈Np(L
2 L h
∫0Lh Np(L′) dL′
(3)
where 〈Np(L h )〉 is the function of L h , which can be found recursively, if an initial value is known. Because, from simple geometry 〈Np(L h )〉 ) 1 for 0 < L h < 1, one can predict that
{
〈Np(L h )〉 ) 3 -
for 1 < L hL h > 1.
Figure 2. Distribution of particles under the jamming state adsorbed on line segments, semicircles, and circles derived from simulations and from experiments, (a) L h ) 5, (b) L h ) 10, (c) L h ) 20, (d) L h) 18, and (e) L h ) 35.5. The results for circles have been taken from the work of Zheng et al.4
(i) An adsorbing (virtual) particle was created by choosing at random its position on the collector of negligible thickness, P(xV,yV), where xV,yV are the Cartesian coordinates, measured relative to a convenient origin. (ii) Next, an overlapping test was carried out for the nearest neighbors of the virtual particle, having the coordinates xi,yi, which consisted in checking if the inequality (xV - xi)2 + (yV - yi)2 > d2 is fulfilled. (iii) If this criterion was met, the particle was irreversibly adsorbed with unit probability at the point P(xV,yV), (iv) If this criterion was violated, a new adsorption attempt was made, uncorrelated with previous attempts. The simulations according to the above scheme have been repeated for about 10 000 collectors, at a fixed value of the L h parameter, with the total number of adsorbed particles larger than 105. This assures a relative error of the simulations of about 0.3%. The jamming coverage of particles (defined as the state where no additional particles could be adsorbed) was determined as a function of 1/L h by extrapolating the dependencies of the coverage Θ on the inverse of the dimensionless adsorption time τ-1.
III. Results and Discussion Some typical configurations of particles adsorbed on lines, semicircles, and circles derived from simulations are shown in Figure 2. The theoretical results for linear collectors are compared with experimental micrographs of latex particles adsorbed on surface features produced on mica the using polyelectrolyte patterning method. The experimental procedure will be described in more detail in our future work.17 In the case of circular collectors, the simulated “monolayers” are compared with the
experimental data obtained by Zheng et al.4 for latex particles adsorbing on ring-patterned surfaces (particle diameter 1 µm, ring diameter 11.3 µm, which gives for L h the value of 28). As can be observed, there appear well pronounced similarities between the theoretical and experimental data. The presence of aggregates in the experimental picture is probably caused by the drying procedure used before microscopic observations. A quantitative comparison can be made by exploiting the numerical results showing the averaged number of particles per collector 〈Np〉 as a function of the L h or the 1/L h parameter. These data obtained for the linear collector are shown in Figure 3. It was found that the numerical results for 1 < L h < 2 are in a good accordance with the analytical results predicted by eq 4. On the other hand, for L h > 2, the numerical results can be well interpolated by the simple expression
〈Np〉 ) 0.7476L h + 0.505
(6)
The results shown in Figure 3 suggest that the length of linear nanostructures adsorbed at solid surfaces (e.g., polymer chains) invisible under optical microscope can be determined by measuring experimentally 〈Np〉 for colloid particles of known size 1/L h . Then, the length of the structure can be calculated by inverting eqs 4 or 6. In the latter case, one obtains a simple relationship valid for 〈Np〉 > 4
L ) d[1.337〈Np〉 - 0.6755]
(7)
The proposed method seems advantageous over the AFM technique, which is particularly effective for compact polymeric material, such as polymeric dendrimers.20 In the case of elongated polymeric chains of very low thickness, the AFM method may become less suitable because of the necessity of drying up the monolayer prior observations and the tip convolution effects. The dependence of the jamming coverage for linear collectors on the L h and 1/L h parameter derived from simulations is shown in Figure 4. As can be seen, for L h > 2, the jamming coverage (20) Pericet-Camara, R.; Papastavrou, G.; Borkovec, M. Langmuir 2004, 20, 3264-3270.
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Figure 4. Dependence of the jamming coverage of particles Θ′∞ on the 1/L h parameter derived from simulations (triangles). The dashed h+ line shows the linear interpolation function (i.e., Θ′∞ ) 0.505/L 0.7476 valid for L h > 2). The dotted line shows the analytical results derived from the equation Θ′∞ ) -2(1/L h - 0.5)2 + (1/L h - 0.5) + 1. The squares show our experimental results derived for latex particles on polyelectrolyte modified mica.
of particles increases monotonically with 1/L h , which can be well fitted by the linear dependence
Θ′∞ )
0.505 + 0.7476 L h
(8)
It is interesting to observe (see Figure 4) that for L h ) 2 (the particle size two times smaller than the collector lengths) Θ′∞ ) 1, which can be predicted from simple geometrical considerations because there can be two and only two particles at the jammed state on the collector of the length L ) 2d. The jamming coverage for 2 > L h > 1 is not a monotonic function of L h , exhibiting a maximum at L h ) 4/3. This is in accordance with the theoretical predictions, cnf. Equation 4, which can be expressed as
(
Θ′∞ ) -2
) (
)
2 1 1 - 0.5 + - 0.5 + 1 ) L h L h 2 1 1 -2 - 0.5 + + 0.5 (9) L h L h
(
)
From eq 9, one can predict that the maximum value of Θ′∞ equals 1.125 occurring at L h ) 4/3. As can be seen in Figure 4, the theoretical predictions are in a good agreement with experimental results obtained for latex particles adsorbing irreversibly on mica patterned by polyelectrolytes.19 Another parameter having a major practical significance (e.g., for predicting the precision of mapping of the surface features by adsorbed particles) is the characteristic length scale of the monolayer. For sake of convenience we define this length scale 〈Le〉 as the distance between outermost points of the adsorbed particles (see Figure 1) averaged over the ensemble of collectors characterized by the same value of L h
〈Le〉 )
1
Nc
∑ Len
Nc n)1
(10)
where Len is the end to end length for each collector and Nc is the total number of collectors considered. It is useful to present
Figure 5. Dependence of the reduced end to end length 〈Le〉/L on the 1/L h parameter derived from numerical simulations (triangles). The dashed line shows the linear fit of the numerical results (i.e., 〈Le〉/L ) 0.239/L h + 1 valid for L h > 2) and the dashed-dotted line shows the polynomial fit (i.e., 〈Le〉/L ) -1.54L h -2 + 2.04L h -1 + 0.486 valid for 2 > L h > 1).
〈Le〉 in the reduced form 〈Le〉/L as shown in Figure 5. As can be noted, the dependence of 〈Le〉/L on the 1/L h parameter for L h> 2 can be well fitted by
〈Le〉/L )
0.239 +1 L h
(11)
For 2 > L h > 1 the fitting function was found to be
〈Le〉/L ) -
1.54 2.04 + 0.486 + L h2 L h
(12)
From eq 12 one can predict that 〈Le〉/L attains a maximum of 1.17 for L h ) 1.45. It is also interesting to observe that because 〈Le〉/L is always larger than unity (see Figure 5) the real length of a collectors is always smaller than the length of the adsorbed particle monolayer measured experimentally. However, when L h > 5 (particle size five or more times smaller than the collector length) the deviation becomes smaller than 5%, which is rather negligible in comparison to the experimental error. As a measure of the “fuzzyness” of structures formed by spherical particles adsorbing on linear surface features, one can take the mean square deviation of Le, defined in the usual way
σ j Le )
[
1
]
Nc
∑
Nc n)1
1/2
(L - Len)2
(13)
The dependence of σ j Le calculated from eq 13 on L and 1/L h is presented in Figure 6. As can be noted, for L h < 3.33, the mean square deviation increases almost linearly with 1/L h attaining a first maximum of 0.19 for L h ) 2.38. This indicates that the collector shape is expected to be most diffuse for the particle size approximately 2.5 times smaller than its length. A further increase in 1/L h resulted in a decrease in σ j Le, which attained a minimum value of 0.17 for L h ) 2. An interesting behavior of the σ j Le vs 1/L h dependence was observed for the range of 2 > L h > 1, where a secondary maximum of the height 0.44 was observed at 1/L h ) 0.8 (see Figure 6). The dependence of σ j Le on 1/L h was
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Figure 6. Dependence of the reduced standard deviation of Le, σ j Le on the 1/L h parameter derived from simulations (triangles). The dashed line shows the polynomial fit: σ j Le ) 264L h -6 - 402L h -5 + 218L h -4 - 52.6L h -3 + 5.49L h -2 + 0.284L h -1 + 0.0008 valid for ∞ > L h > 2. The dotted line shows the polynomial fit: σ j Le ) 4.59L h -3 - 12.5L h -2 + 11.2L h -1 - 2.90, valid for 2 > L h > 1.25. The dashed-dotted line shoes the polynomial fit: σ j Le ) -9.42L h -2 + 15.3L h -1 - 5.79, valid for 1.25 > L h > 1.
fitted for various range of this parameter by the following polynomial expressions
σ j Le ) 264L h -6 - 402L h -5 + 218L h -4 - 52.5L h -3 + 5.49L h -2 + h>2 0.284L h -1 + 0.0008 for L σ j Le ) 4.59L h -3 - 12.5L h -2 + 11.3L h -1 - 2.9 for 2 > L h > 1.25 (14) and
h -2 + 15.3L h -1 - 5.79 for 1.25 > L h>1 σ j Le ) -9.42L The above results obtained for linear collectors were compared with the results obtained for curved collectors, when the averaged number of particles adsorbed 〈Np〉 becomes the most important parameter. The dependence of 〈Np〉 on the L h parameter for L h< 5 is shown in Figure 7a, whereas the data obtained for L h>5 are plotted in Figure 7b. As can be seen, the differences between linear collectors and semicircles become practically negligible for L h > 2. On the other hand, in the case of circular collectors, more pronounced differences in comparison with linear collectors appear for L h < 10. Generally, the 〈Np〉 for circles (determined for the same L h value) was smaller by approximately 0.7 unit for this range of L h . This difference is caused by the fact that, for linear collectors, the adsorbed particles can “stick-out” from the geometrical contour of collectors forming so-called overhangs predicted theoretically for the case of multilayer adsorption.18,21 Based on the results shown in Figure 7 one can draw an interesting conclusion that the curvature effects play a negligible role if the radius of adsorbing particles becomes a few times smaller than the characteristic radius of curvature of a surface feature. As a consequence, the theoretical results obtained for linear collectors, in particular the formula for the length of the surface feature, eq 7, can be used as a good approximation for curved collectors as well. (21) Talbot, J.; Tarjus, G.; van Tassel, P. R.; Viot, P. Colloids Surf. A 2000, 165, 287-324.
Figure 7. (a) Averaged number of particles adsorbed at the jamming state on collectors of various shape 〈Np〉 as a function the L h parameter derived from numerical simulations (points). 1. Linear segments: 〈Np〉 ) -0.0147L h 4 + 0.402L h 3 - 2.16L h 2 + 4.90L h - 2.10. 2. Semicircles: 〈Np〉 ) 1 for 0 < L h < π/2, 〈Np〉 ) -6.25 + 33.4L h -1 - 34.4L h -2) for π/2 < L h < 2, 〈Np〉 ) 0.747L h + 0.413 for L h > 2. 3. Circles: 〈Np〉 ) 1 for 0 < L h < π, 〈Np〉 ) 2 for π < L h < 1.15π, 〈Np〉 ) (1 - 0.302L h )/(0.249 - 0.082L h ) for 0.15π < L h < 4. (b) The same as (a) but for 5 < L h < 35: linear segments 〈Np〉 ) 0.747 L h + 0.505 (full points), semicircles 〈Np〉 ) 0.747L h + 0.414 (empty points), circles 〈Np〉 ) 0.747L h . (triangles). The square point denotes the experimental result derived from the work of Zheng at al.4
IV. Concluding Remarks Numerical results obtained in this work revealed that the averaged number of particles adsorbed on linear collectors under the jamming state increases with the L/d parameter, according to the dependence 〈Np〉 ) 0.7476(L/d) + 0.505, valid for L/d > 5. This theoretical prediction is in agreement with experimental results concerning latex particles adsorbing on surface features of linear and circular shape. This fact suggests that the extension of nanostructures adsorbed at solid surfaces (e.g., polymer chains) invisible under optical microscope can be calculated from the simple dependence L ) d[1.34〈Np〉 - 0.676] by measuring experimentally 〈Np〉 for colloid particles of known size d. Our theoretical results show also that the averaged end to end length of the particle monolayer 〈Le〉 can be described for linear collectors by the dependencies
〈Le〉/L ) 0.239L h - + 1 valid for L h>2 〈Le〉/L ) -1.54L h -2 + 2.04L h -1 + 0.486
valid for 2 > L h>1
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On the other hand, the mean square deviation of the length of the monolayer for L h < 2 was approximated by a polynomial expression, eq 14, which exhibited a maximum of 0.19 for L h) 2.38. This parameter can be interpreted as the measure of the fuzzyness of the mapping of surface features by adsorbed colloid particles of a spherical shape. The results obtained in this work could be exploited for a quantitative interpretation of experiments involving colloid
Adamczyk et al.
particle or protein adsorption on heterogeneous surfaces covered by polyelectrolyte chains or DNA fragments. Acknowledgment. This work was supported by the MEiSW Grant: N205 02331 112. LA063677U
