Biomacromolecules 2000, 1, 407-412
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Electrophoretic Deposition of Polymer Chains: a Monte Carlo Study of Density Profile and Conformation Grace M. Foo† and R. B. Pandey*,‡ Supercomputing and Visualization Unit, Computer Center, National University of Singapore, Singapore 119260; and Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, Mississippi 39406-5046 Received January 12, 2000; Revised Manuscript Received July 5, 2000
Effects of molecular weight (Lc) and field strength (E) on conformation of polymer chains and their density are investigated at the surface and bulk of a driven chain system in three dimensions with a Monte Carlo simulation. As the polymer chains deposit on the surface, a linear density gradient develops in low field. While the gradient becomes steeper on increasing the field in low Lc, onset of oscillation in the polymer density profile appears with higher Lc, above a characteristic value of field (Ec) which decreases with increasing molecular weight. The substrate coverage (θj) exhibits a rapid increase on increasing the field and tends to decline at high fields. In the bulk, the polymer density decreases rapidly with the molecular weight while the substrate coverage (θj) decays exponentially, θj ∼ e-βLc. The radius of gyration (Rg) of chains undergoes a crossover from SAW to rodlike conformation in the bulk on increasing the field. On the surface, chains are longitudinally compressed with a sub-SAW conformation, i.e., the scaling exponent V ∼ 0.50 - 0.55(Rg ∼ Lc-V). Nonmonotonic response with opposite trend is observed in the variation of the longitudinal (minimum) and transverse (maximum) components of the radius of gyration with the field. 1. Introduction Adsorption, permeation, and spreading of polymer on substrates/surfaces and their properties such as adhesion, friction, and lubrication of polymers on surfaces depend on the shape and packing of the polymer chains.1-4 Therefore, studying the conformation of polymers and their packing density at surfaces and interfaces is important in understanding the basic and applied issues in diverse areas. Some examples include biomolecular adsorption at cellular membranes, their structural stability and mechanical response, manufacturing of semiconductors, designing of composite materials, development of coatings, thin films, etc.1-8 While the characteristics of chains at surfaces is hard to study analytically,9,10 one may use computer simulation experiments11-13 to study appropriate models of varying complexity depending on resources. There have been numerous computer simulation studies of polymer at/near surfaces14-18. However, the conformation and density profiles of driven polymer chains toward an impenetrable surface is relatively less understood.19 For example, when the polymer chains are driven by a field as in DNA electrophoresis,20-22 it is not clear how the polymer density and their conformational profile evolve at the gel boundaries/wall. In our model, an external field is used to drive the polymer chains toward an attractive wall. We use kink-jump moves in addition to reptation dynamics23 to move the chain nodes. Attempts are made to address the * Corresponding author. † National University of Singapore. ‡ University of Southern Mississippi.
effect of the field strength and molecular weight on the conformation of polymer and their density at the substrate/ surface and bulk developed from the deposition of polymer chains. In an earlier communication,23 we studied the growth of the interface width and scaling of the roughness using a Monte Carlo simulation of driven polymer chains on a discrete lattice. Using the same model, we examine the conformation and density profile of the polymer chains in the bulk and at the wall/substrate. The model is described in section 2; the simulation results are discussed in section 3 and summarized in section 4. 2. Method and Model We consider discrete three-dimensional lattices of size Lx × L × L with comparatively large Lx, typically Lx ) 100, 200 and L ) 20-80. Polymer chains of length Lc (with Lc + 1 nodes) connected by constant bonds of unit length (in units of lattice constant) are modeled as self-avoiding walks (SAWs), with kink-jump and reptation dynamics.11-13 A small number of chains (1-8) are released at constant intervals (every 200 MCS for a total of 80 000-100 000 MCS) from near the source end (i.e., x ) 1 to x ) 0.2 × Lx) of the lattice. They move toward an impenetrable attractive wall (yz plane) at the opposite end (x ) Lx) in the presence of an external field of strength E biased in the longitudinal, positive x direction. In addition to excluded volume, nearest neighbor polymerpolymer repulsive and polymer-wall attractive (adsorbing) interactions are considered. To implement such interactions
10.1021/bm000289c CCC: $19.00 © 2000 American Chemical Society Published on Web 08/12/2000
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Figure 1. Snapshots of chains: length Lc ) 4 on a L ) 20, Lx ) 100 lattice after 80 000 MCS, at field E ) 0.1 (a) and E ) 0.5 (b); length Lc ) 40 chains on L ) 40, Lx ) 200 lattice at E ) 0.5 (c) and E ) 5.0 (d).
phenomenologically, we assign a positive charge or interaction density (F ) 1) to each chain node and a negative charge (F ) -1) at each lattice position on the wall (yz plane at x ) Lx). An external field E couples with the movement of the chain nodes via interaction energy, -E dx, where dx () 0 or (1) is the displacement step along the x direction. Thus, the interaction energy of the system is described by Eint )
∑ij FiFj + E ∑i Fi dxi
(1)
where the first summation is restricted to nearest neighbor sites, and dxi ) 0, (1 is the displacement of the ith chain node along the x direction. The Metropolis Monte Carlo algorithm is used to move randomly selected nodes from the growing pool of chains. One MCS (Monte Carlo step) is defined as N ()Lx × L × L) attempts to move randomly selected chain nodes. A kink-jump move is attempted if the selected node is an internal one while reptation or endflip moves are attempted (with equal probability) if end nodes are selected. A periodic boundary condition is used to move chains across the y and z boundaries. Simulations are performed with a range of chain lengths (Lc ) 4, 20, 40, 60, and 80), field strengths (E ) 0.1, 0.3, 0.5, 1, and 5), and system sizes (L ) 20, 30, 40, 50, 60, and 80) at a fixed temperature T ) 1. Units for the interaction energy (eq 1), field strength (E), and temperature (T) are arbitrary and normalized by kB, the Boltzmann constant. For
each set of parameters, 1-7 simulations were performed to obtain reliable estimates of the properties. 3. Results Snapshots of the polymer chains provide physical insight into the polymer density, chain’s shape, and their packing. Figure 1 shows typical snapshots for short (Lc ) 4) chains under different field strengths. Visual inspections reveal that, at a low field E ) 0.1 (Figure 1a), the chains are less densely packed (at the wall and in the bulk, i.e., between the wall and the growing surface) than at higher field E ) 0.5 (Figure 1b). The higher field tends to align the chains along the field direction, especially for longer chains compared with shorter ones at the same magnitude of field (Figure 1c). Layering and stacking of these longer chains is observed at very high field (Figure 1d). 3.1. Density Profiles. For a view of the distribution of the chains, we examine their monomer density profile, i.e., variation of the yz-planar monomer density with the position (x) along the lattice. In Figure 2a, monomer density profile for short chains (Lc ) 4) at different values of field is presented. The density increases toward the wall, and the approach of its magnitude, i.e., the shape of the density gradient, to the maximum value at the wall depends on the field strength. For example, at a low field (E ) 0.1), the monomer density rises gradually and nearly linearly while there is a steeper rise at higher fields (E ) 1.0, 5.0). These
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Table 1. Monomer Density near/at Wall for Chains of Size L ) 80 and Lx ) 100 Lattice at Different Field Magnitudes with Error Bar ∼ 0.03 monomer density
Figure 2. Monomer density profile of Lc ) 4 chains on L ) 20, Lx ) 100 lattice (a), and Lc 40 chains on L ) 40, Lx ) 200 lattice (b), at various field magnitudes.
profiles are consistent with the snapshots of the chains. For longer (Lc ) 40) chains (Figure 2b), the evolution of the polymer density toward the substrate is still linear at low field (E ) 0.1). At higher value of the field (E ) 0.5), the growth of the density profile is much larger with a sign of oscillation. On further increasing the field (E ) 1.0, 5.0), we find that the oscillation in the four-density profile becomes more pronounced and better developed. These characteristics signal a layering/stacking of chains at higher fields as visually observed in the snapshots. Note the contrast in density profiles: while no oscillations are observed for short chains over the range of field strength explored, oscillations emerge in the profiles of longer chains (Lc ) 40) (Figure 2b). For longer chains (Lc ) 80), oscillations are observed even at lower field E ) 0.5. Thus, there is a threshold or characteristic value of field (Ec) for the onset of oscillation such that no oscillation occurs at E < Ec and oscillation (i.e., layering) occurs at E g Ec. We find that the threshold value, Ec, decreases on increasing the chain length. Further, we note that, the higher the field and longer the chains, the larger the period of oscillation as expected. At higher field, chains are elongated (rodlike conformations) with larger radius of gyration (Rg) and endto-end distance which is a measure of the periodicity in the oscillation of the planer density. Our earlier simulations14 showed similar oscillatory behavior. A close examination of the polymer density at/near the wall (see Table 1) suggests that, on increasing the field, the polymer density increases
distance X
field E
Lc ) 4
Lc ) 20
Lc ) 40
Lc ) 80
98 99 100
0.1 0.1 0.1
0.43 0.41 0.64
0.36 0.33 0.59
0.30 0.27 0.56
0.25 0.23 0.52
98 99 100
0.3 0.3 0.3
0.65 0.65 0.86
0.52 0.50 0.75
0.45 0.41 0.68
0.38 0.33 0.63
98 99 100
0.5 0.5 0.5
0.78 0.80 0.95
0.62 0.61 0.83
0.51 0.48 0.75
0.36 0.33 0.64
98 99 100
1.0 1.0 1.0
0.94 0.96 1.0
0.75 0.75 0.93
0.51 0.52 0.79
0.31 0.34 0.66
98 99 100
5.0 5.0 5.0
1.0 1.0 1.0
0.60 0.60 0.84
0.27 0.27 0.50
0.11 0.11 0.15
but declines above a characteristic value for the longer chainssthe nonmonotonic dependence of polymer density with the field is more pronounced with longer chains. 3.2. Density in Bulk and Coverage at Wall. In this section, we take a closer view of the monomer density in the bulk and at the wall and study the dependence of the density on the field magnitude and chain length. The bulk evolves as the polymer chains continue to deposit. The region adjacent to the wall/surface where the polymer density becomes stable in MCS time is considered bulk. To evaluate the bulk density, we average polymer density from chains found within a representative section along the lattice not too close to the wall where there are lots of chains. This distance varies from x ) 90-94 on a Lx ) 100 lattice for short chains (Lc ) 4) to x ) 155-180 on a Lx ) 200 lattice for long chains (Lc ) 80). The monomer density at/near the wall or the surface coverage (θj) is obtained by averaging chains found within a small distance from the wall, typically five units in a Lx ) 200 unit lattice. Since the density profiles (above) show variation with chain length, we expect the behavior of the coverage and bulk density to vary with chain length. In response of the bulk monomer density to the field of short chains (Lc ) 4, 10), we see a rapid monotonic rise of density with E which saturates at high field (E g 1.0) (Figure 3). A similar increasing trend of the monomer density is noted but with progressively smaller magnitudes of saturated density as the chain length increases. Eventually for the longest chains (Lc ) 80), there is little variation of the monomer density with field. This is perhaps a result of steric hindrance effects which generally increases with chain length. The polymer density at the wall/substrate is defined as the surface coverage (θj).24 The surface coverage of short chains increases monotonically with the field strength. For longer chains, on the other hand, there is a nonmonotonic variation of the surface coverage with the field as was noted in our examination of the snapshots above. Increase in the surface coverage is followed by a decline on increasing the
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Figure 3. Monomer density vs field magnitude E in the bulk (open symbols) and at the wall (filled symbols) on L ) 80, Lx ) 100 lattice, for different length chains.
field, with a maximum coverage at a threshold field value (E e 1.0). Thus, at high field, coverage is high (maximum) for short chains while it is low (minimum) for the longest chains. Low coverage for long chains at high field was also noted in our earlier simulations.19 We find that the monomer density at the wall is generally higher over that in the bulk and the difference in densities decreases as chain length and field increases. Thus, to have a homogeneous distribution of polymer density (in the bulk and at the surface), it is desirable to choose polymers with higher molecular weight in high field. It is ironic that the longer chains lead to heterogeneity that is higher at the local scale but lower at the global scale. The surface coverage and the bulk polymer density show consistently lower values as the chain length increases (see Figure 3). A plot of the bulk density against chain length (Figure 4a) shows a rapid decay in a short chain regime followed by a slow approach to a constant value at long chains on increasing the chain length. Our current data are not of sufficiently good quality to provide a reliable fit even though it may be tempting to guess an exponential decay. The variation of the surface coverage with the chain length, on the other hand, seems to show an exponential decay, θj ∼ e-βLc (Figure 4b). The exponent β (Table 2) increases with field magnitude. This contrasts with the power-law decay observed in our earlier simulations with a similar but coarser model.19 Within the range of fluctuations, the polymer density seems independent of the lattice size. 3.3. Radius of Gyration. We study the conformational profile and examine the radius of gyration of chains (Rg) and its longitudinal (Rgx) and transverse (Rgt) components in the bulk and at the wall. In the bulk, the radius of gyration increases monotonically with the field before saturating at high field, as Figure 5 shows for the longer chains. It is apparent that there is a more rapid response in increase of Rg for the longer chains at low fields and nearly no response in the change of Rg on increasing the field for short chains. The rate of response of Rg to field (E) depends on the molecular weight (Lc). Small chains respond rather quickly even at lower fields. Therefore, we are unable to note appreciable change in Rg even at the low fields we have considered. The longitudinal component (Rgx) dominates the response of the radius of gyration (Figure 5). The transverse
Figure 4. Monomer density vs chain length Lc in the bulk (a) and at the wall (b) on L ) 80, Lx ) 100 lattice, at various field magnitudes.
Figure 5. Radius of gyration Rg vs field magnitude E for chains in the bulk on a L ) 80 lattice with longitudinal (Rgx) and transverse (Rgt) components. Table 2. Exponents for Scaling of Coverage with Length, θj ∼ e-βLc, at the Wall of a L ) 80, Lx ) 100 Lattice at Various Field Values with Error Bar ∼ 0.001 field E
exponent β
0.1 0.3 0.5 1.0 3.0 5.0
0.006 0.006 0.009 0.014 0.023 0.028
component (Rgt), on the other hand, has a reverse trend, i.e., it decreases with increasing field. Thus, we see a conformational stretching of the chain with the field in the bulk.
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Table 3. Exponents for Scaling of Chain Radius with Length, Rg ∼ Lcv, in the Bulk and at the Wall of a L ) 80 Lattice at Various Field Valuesa bulk
wall
field E
v
vx
vt
v
vx
vt
0.1 0.3 0.5 1.0 5.0
0.60
0.61
0.59
0.59 0.56 0.56 0.55
0.41 0.41 0.42
0.62 0.59 0.59 0.58
a
1.0
Exponents slightly larger than 1 are truncated to 1. Error bar ∼ 0.03
Figure 6. Radius of gyration Rg vs field magnitude E for chains of length Lc ) 80 at the wall on a L ) 80 lattice with longitudinal (Rgx) and transverse (Rgt) components.
At the wall (transverse surface), there is a reverse trend in response of the radius of gyration, i.e., from decreasing to increasing in the magnitude of Rg as the field increases (Figure 6). The longitudinal (Rgx) and transverse components (Rgt) respond oppositely (Figure 6), with the latter dominating the former at low fields. Thus, the chains are compressed at the wall with a maximum compression at an optimum value of field which is not the highest field. As the field is increased, chains begin to compress, leading to decrease in the longitudinal component of the radius of gyration (E ) 0-1.0). In high field, chains are still compressing, but they do not have sufficient time to relax in the compressional state before being trapped (constrained sterically) by the incoming chains which move faster at higher field. In addition, there is a spatial conformational crossover, from an elongated conformation in the bulk to a longitudinally compressed conformation at the surface. In general, the magnitude of Rg in the bulk is higher than that at the surface in high field. The radius of gyration generally increases with the chain length and we find a power-law scaling over lower values of field (E e 1.0) Rg ∼ LcV
(2)
Estimates for V and the exponents Vx and Vt for the longitudinal (Rgx) and transverse (Rgt) components, respectively, for chains at the wall and in the bulk are presented in Table 3. In the bulk, the power-law scaling suggests a relatively isotropic conformation at very low E ()0.1) with exponents V ∼ 0.60, Vx ∼ 0.61, Vt ∼ 0.59. At high E ()5.0), the longitudinal exponent Vx ∼ 1.0 seems to suggest that the chains are stretched out into rodlike configurations. Therefore, in the bulk, a conformational crossover seems to occur from SAW conformation (at low field E ) 0.1) to stretchedout (rodlike) conformation (at high field E ) 5.0) as a function of field. At the wall, we see SAW-like exponents over E e 1.0 with a slight decreasing trend, V ∼ 0.59-0.55 (Figure 7). The chains appear to be compressed longitudinally
Figure 7. Radius of gyration and components vs L, on a log-log scale (Rgx with open symbols and Rgt with filled symbols), for chains on a L ) 80 lattice.
(Vt ∼ 0.41-0.42) while the transverse exponents are SAWlike (Vx ∼ 0.58-0.62). We do not see much variation of Rg with lattice size L for chains in the bulk; therefore, our results and conclusions are independent of the finite size of the sample. 4. Summary and Conclusion A Monte Carlo simulation is presented to study the evolution of the polymer density and conformational characteristics of a model system of polymer chains driven by an external field toward an impenetrable adsorbing wall. Attempts are made to find how the monomer density at the wall and in the bulk depend on the field strength E and the molecular weight, i.e., the chain length Lc. In weak field, a linear density gradient sets in. For short polymer chains, the gradient increases on increasing the field with steep gradient at higher field. For relatively long chains, while a linear density gradient is present at low field, an oscillatory behavior in the polymer density profile seems to emerge in high field. The onset of oscillation in the density profile appears at a characteristic values of field (Ec) which decreases on increasing the chain length (molecular weight). The polymer density in the bulk increases monotonically as field strength increases, with short chains showing a greater response. The density at the wall, i.e., the surface coverage (θj) for longer chains, shows a nonmonotonic dependence on the field. The surface coverage increases rapidly to a maximum at characteristic values of field E < 1.0 and drops to low values at high field. The polymer density decreases
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with chain length in both the bulk and at the wall with an exponential decay (θj ∼ e-βLc) for the surface coverage. The radius of gyration and its longitudinal and transverse components are analyzed. In the bulk, the chains align (stretch) increasingly with increasing field: the longitudinal component of Rg increases while the transverse component decreases as the field increases. At the surface, the response of Rg to field exhibits nonmonotonic behavior in a longitudinally compressed conformation with opposite trends in their longitudinal and transverse components (still sub-SAW conformation). Scaling of Rg with the molecular weight shows an isotropic SAW conformation (V ∼ 0.6) in very low field to elongated (rodlike (V ∼ 1)) conformation in high field in the bulk. Acknowledgment. Simulations were performed on Beowulf clusters at the Department of Computational Science of National University of Singapore and at the University of Southern Mississippi. Partial support from a NSF-EPSCoR grant is acknowledged. This work is useful for a DOEEPSCOR grant. References and Notes (1) Dynamics of Fractal Surfaces; Family, F., Vicsek, T., Eds.; World Scientific: Singapore, 1991. (2) Barabasi, A.-L.; Stanley, H. E. Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, England, 1995. (3) Fleer, G. J.; et al. Polymers at Interfaces; Chapman and Hall: London, 1993. Eisenriegler, E. Polymers near Surfaces; World Scientific: Singapore, 1993. (4) Wool, R. P. Polymer Interfaces: Structure and Strength; Hanser Publishers, New York 1995.
Foo and Pandey (5) Situmorang, M.; Gooding, J. J.; Hibbert, D. B.; Bennet, D. Biosens. Bioelectr. 1998, 13, 953. (6) Leng, A.; Streckel, H.; Stratmann, M. Corrosion Sci. 1999, 41, 547. (7) Toko, Y.; Bai, Z. Y.; Sugiyamrna, T.; Katoh, K.; Akahane, T. Mol. Cryst. Liq. Cryst. 1997, 304, 1985. (8) Arnundson, K. R. Phys. ReV. E 1999, 59, 1808. (9) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University, Ithaca, NY, 1979. (10) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, England, 1986. (11) Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford University Press: New York, 1995. (12) Baumgaertner, A. In The Monte Carlo Methods in Condensed Matter Physics; Binder, K., Ed.; Springer-Verlag: New York 1995. (13) Computational Modeling of Polymers; Bicerano, J., Ed.; Marcel Dekker: New York, 1992. (14) Issaevitch, T. A.; Jasnow, D.; Balazs, A. C. J. Chem. Phys. 1993, 99, 8244.; Balazs, A. C. MRS Bull. 1997, 22, 13. (15) Grest, G. S.; Phys. ReV. Lett. 1996, 76, 4979. D’Ortona, U.; Coninck, J. D.; Koplik, J.; Banavar, J. R. Phys. ReV. Lett. 1995, 74, 928. (16) Zajac, R.; Chakrabarti, A. J. Chem. Phys. 1996, 104, 2418. Lai, P. Y.; J. Chem. Phys. 1995, 103, 5742. Baschnagel, J.; Binder, K. Macromolecules 1995, 28, 6808. (17) Pandey, R. B.; Milchev, A.; Binder, K. Macromolecules 1997, 30, 1194. (18) Yamamoto, T. J. Chem. Phys. 1998, 109, 4638. (19) Foo, G. M.; Pandey, R. B. Phys. ReV. Lett. 1997, 79, 2903; J. Chem. Phys. 1997, 107, 10260; Phys. ReV. Lett. 1998, 80, 3767. (20) Hoagland, D. A.; Smisek, D. L.; Chen, D. Y. Electrophoresis 1996, 17, 1151. Slater, G. W.; Noolandi, J. Biopolymers 1989, 28, 1781. Noolandi, J.; et al. Phys. ReV. Lett. 1987, 58, 2428. (21) Deutsch, J. M. Science 1988, 240, 922. Lumpkin, O. J.; Djardin, P.; Zirnrn, B. H. Biopolymers 1985, 24, 1573. (22) Quake, S. R.; Babcock, H.; Chu, S. Nature 1997, 388, 151. Perkins, T. T.; Smith, D. E.; Chu, S. Science 1997, 276, 2016. (23) Foo, G. M.; Pandey, R. B. Phys. ReV. E 2000, 61, 1793-1799. (24) Wang, J.-S.; Pandey, R. B. Phys. ReV. Lett. 1996, 77, 1773.
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