- Email: [email protected]
Polymer interface changes in electrophoretic deposition
Progress in Organic Coatings 47 (2003) 324–330
Polymer interface changes in electrophoretic deposition Ras B. Pandey∗ Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406-5046, USA
Abstract A Monte Carlo computer simulation model for the electrophoretic deposition of polymer chains on a discrete lattice is used to study the polymer density profile, interface growth, and its dependence on field, temperature, and molecular weight. The interface width (W) decreases W ∼ E−1/2 on increasing the field (E). Width (W) depends non-monotonically on the temperature (T): a power-law decay is followed by a power-law increase on raising the temperature. Monotonic decay of the interface width with the molecular weight is possibly a stretched exponential. Conformation and dynamics of a tracer chain is used to probe its characteristics in interface to bulk region. The root mean square (rms) displacement of the center of mass of the tracer chain shows an ultra-slow motion, R ∼ t ν (ν ∼ 0.1–0.01 at E = 0.1–1.0) as the driven chain moves deeper from interface to bulk. Longitudinal compression of the radius of gyration (Rg ) of the chain increases with the field; transverse components (Rgx , Rgy ) are larger than the longitudinal component (Rgz ). The transverse component (Rgx (y)) becomes oscillatory due to periodic squeezing at high fields as the field competes with the polymer barriers. © 2003 Elsevier B.V. All rights reserved. Keywords: Monte Carlo simulation; Power-law decay; Radius of gyration
1. Introduction Coating via deposition involves directed release of constituents, their equilibration/relaxation, and nano-fabrication (i.e. bonding, entanglement, etc.) on desirable substrates. Functionality of the coating depends on the characteristics of constituents, their density distribution, interface width, and roughness which could be modified by monitoring and controlling the deposition process and equilibration methods. Interface growth and roughness via deposition of particles on substrates have been extensively investigated in last two decades by theories, computer simulations, and laboratory experiments [1,2]. Theory based on Kardar–Parisi–Zhang (KPZ) model [3], an extension of Edward–Wilkinson (EW) model [4] provided an impetus for explosive interest and growth in investigation of interface growth in past two decades. Numerous computer simulation models and methods (involving various details and algorithms) are proposed to describe deposition processes such as, solid-on-solid, physical and chemical vapor depositions, and growth by molecular beam epitaxy (MBE) in order to understand the interface dynamics in a variety of contexts including materials design, thin films, coating, fracture, and flow [1,2]. A number of procedures are considered in many of these models including rate of release of particles, their trajectories ∗ Tel.: +1-601-266-4485; fax: +1-601-266-5149. E-mail address: [email protected] (R.B. Pandey).
0300-9440/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0300-9440(03)00146-2
(i.e. diffusion, ballistic, etc.) during deposition. Depending on systems of interests, different relaxation and adhesion procedures (i.e. irreversible growth, restricted sticking/bonding, mobility, and diffusion) after particles reach the growing surface are also implemented. Thus, the growth of the interface width and scaling of the roughness particularly with the substrate length has been well understood in particle deposition. Unexpected results are, however, still reported in many such systems experimentally (e.g., Ag(1 0 0) homoepitaxy [5], Si/bulk Si interface [6], polycrystalline Si1−x Gex [7]) as well as by computer simulations [5,8]. Understanding the growth of polymer thin films, coating, and their modifications [9–13] is much more difficult than the corresponding interface growth phenomena by particles deposition [1,2]. Chains dynamics, conformation, and its relaxation [14–16] add considerable complexities at various stages of deposition with desirable variables such as driving field, rate of deposition, temperature, molecular weight, etc. In this article, we consider electrophoretic deposition of polymer chains on an adsorbing substrate by a computer simulation model on a discrete lattice [17–20]. As the polymer chains are driven by an electric field or pressure gradient toward an impenetrable substrate, conformations of chains undergo enormous changes from the driven flow region (dilute solution) to deposition at the substrate. The polymer density grows and the interface develops. The longitudinal fluctuation in the density profile in the growth region defines the interface width [1,2]. Enormous efforts have been
R.B. Pandey / Progress in Organic Coatings 47 (2003) 324–330
recently devoted toward understanding the morphological details, i.e. conformation and density profiles, of polymer chains at and near surfaces; we refrain from citing the vast list of growing literature, except few textbooks and collected articles [9–13] here for example. Systematic studies of the interface growth and roughness, conformation, and density profile for electrophoretic deposition of polymers by computer simulations are, however, limited. We consider a discrete polymer chain model with fixed bond length (equal to lattice constant) with various segmental dynamics such as kink-jump, crank-shaft, slithering-snake (reptation), end-moves, etc. [16]. Apart from simplicity, the advantage of this model includes the flexibility in implementing dynamics at different length scales by appropriate combination of segmental moves [19]. The interface dynamics, a collective phenomenon of cooperative properties of chains, is a complex issue, and therefore, it is desirable for us to constrain to relatively simple yet flexible model for constituent chains. Over the years, we have studied interface growth, roughness, conformation, and density profile at various stages of the growth with various segmental dynamics as a function of temperature, field, and molecular weight [17–20]. We have found many interesting results some of which will be reported here. In the following section, we introduce the model followed by results and discussion of the interface width in Section 3. Analysis of dynamics and conformation of chains is presented in Section 4 with a conclusion at the end.
2. Model and method We consider a lattice of size L × L × Lz with a large aspect ratio Lz /L. A polymer chain of length Lc is modeled by a constrained self-avoiding walk, i.e. (Lc + 1) consecutive nodes connected by bonds on the trail of a random walk with excluded volume constraints. Chains are released from one end (z = 1) and are driven by an external field E toward the substrate (impenetrable wall) at z = Lz . The field is coupled with the displacement Z of the chain node via change in energy U = −E Z. Additionally, we consider nearest-neighbor polymer–polymer repulsive and polymer–wall attractive interactions given by ρi ρj , U=J ij
where J = 1, ρi = 1 if the site i is occupied by the polymer node, 0 if the site i is empty, and −1 if the empty site i is on the substrate. Sufficient amount of chains (i.e. to achieve the polymer concentration, the fraction of sites occupied by polymer, p = 0.3) are deposited and equilibrated. For example, in some of our simulations (presented below), chains are moved stochastically after inserting a chain. The Metropolis algorithm [16] is used to move chains by a combination
325
of segmental movements, i.e. kink-jump, crank-shaft, and slithering-snake (reptation) dynamics. Each move is accepted with a Boltzmann factor exp{−β E}, where E is the change in energy for the move and β = 1/kB T with the Boltzmann constant kB and temperature T. Each segmental move, if possible, is implemented according to its local configuration. For example, if the randomly selected node is the inner chain node, then kink-jump or crank-shaft is attempted if such a local move is possible. On the other hand, if one of the end nodes of the chain is selected, then the end-move and reptation are attempted with equal probability. In all cases, change in energy via Boltzmann factor is used to accept or reject the moves. Periodic boundary condition is used along the transverse (X, Y) directions while open boundary conditions along the longitudinal (Z) direction. Attempt to move each chain node once defines one Monte Carlo step (MCS). The simulation is performed for a sufficiently large number of time steps with ample number of independent samples to obtain a reliable estimate of the averaged physical quantities. As mentioned before, we use different segmental dynamics [16] to move chains. Kink-jump dynamics and end-moves involve movement of only one node which are relatively slow and localized. Therefore, it takes much longer to relax the polymer chains only with the kink-jump and end movements. Crank-shaft dynamics, though still localized to small part of chain’s segment, is faster than kink-jump, as it involves movements of two nodes simultaneously. Slithering-snake (reptation) dynamics [14–16], on the other hand, involves global motion of the whole chain by altering conformations at the end segments. A combination of these and other modes of segmental movements are, perhaps, desirable in many polymer simulations. We will, however, concentrate here with combinations of kink-jump, crank-shaft, end-moves, and reptation. Steady-state profiles (density and conformation) and growth of the interface are reached if we continue to deposit polymer chains. In order to relax the interface width, we stop releasing new chains after a desirable growth while allowing chains to continue to execute their segmental moves [19,20]. In this paper, we present the interface changes due to field, temperature, and molecular weight in relaxed data for the density profile and conformation. Chains are relaxed before deposition at particular field and temperature; large aspect ratio is to facilitate relaxation. Note that all the variables, i.e. time steps, temperature, field, and energy are in arbitrary units. The empirical trends in variation of our measured quantities such as interface width, density, radius of gyration, etc. with these variables should be compared with the corresponding trend in laboratory measurements. Quantitative comparison to laboratory measurements is possible only after an appropriate calibration. Thus, if a coating is performed with an appropriate pressure, dependence of the interface width on pressure should be useful to compare with appropriate polymer constituents, molecular weight, and temperature. To our knowledge, such systematic experimental data are hard to find. Therefore, we resort here to empirical scaling of our observed quantities.
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3. Results and discussion Simulations are performed on various lattice sizes L × L × Lz with a large Lz /L; typically Lz = 100–200, L = 20, 40, 60. Molecular weight (i.e. the chain length Lc ) is varied with Lc = 100 for most of our data generated here. Temperature (T) and field (E) are varied. All the quantities are measured in arbitrary units. Ten to twenty independent samples are used to calculate the average quantities. We study density profile, interface width (W), radius of gyration (Rg ) and end-to-end distance (Re ) of chains, and variation of the rms displacement (R) of the center of mass of the tracer chain and that of its nodes with the time steps. The interface width (W) is defined as hij W2 = (hij − h)2 , h = , Ns ij
ij
where hij is the surface height from the substrate located at (i, j) and h the mean surface height averaged over the substrate with Ns sites. In many theoretical models and experiments, particularly with the particle deposition [1,2], the interface width (W) is found to grow with time (t) W ∼ tβ ,
t Lz ,
where β is the growth exponent and z the dynamical scaling exponent. Note that the height–height correlation length is much smaller than the substrate length L in small time limit where the above scaling for the interface growth is valid. Growth of the interface width saturates in the asymptotic time limit when the height–height correlation length exceeds the substrate length L. The saturated width (W) is found to obey an scaling relation with the substrate length (at least for particle deposition [1,2]) W ∼ Lα ,
t Lz ,
where α is the roughness exponent. The dynamical scaling exponent z = α/β. In the following we first discuss some of our results on the dependence of the interface width on field, temperature, and the molecular weight and then we comment on the dynamics of chains and their conformation. 3.1. Density profile and interface width Fig. 1 shows a few snapshots to get an idea of the density profile and roughness. A typical density profile (i.e. the variation of polymer density with z) is presented in Fig. 2 for different fields. The substrate is at z = Lz . At low fields
Fig. 1. Snapshots of polymer chains deposition.
R.B. Pandey / Progress in Organic Coatings 47 (2003) 324–330
327
1.00 0.1 0.2 0.3 0.5 1.0 1.5
0.80
d
W
0.60
0.40
0.20
40
50
60
z
70
80
90
1 3 10
100
Fig. 2. Polymer density profile at different driving field (E = 0.1–1.5). Sample size 40 × 40 × 100, polymer chain length Lc = 100, t = 104 time steps, 10 independent samples, temperature T = 1.
(E = 0.1, 0.2), the polymer density decays rather slowly from substrate to the surface. Change in density becomes sharper, i.e. the interface width becomes smaller (see below) on increasing the magnitude of the field. Growth of an interface width with time step is presented in Fig. 3 at the temperature T = 4. Note that the interface width grows initially and saturates to a steady-state value (after 104 time steps) as the polymer chains are continued to deposit. As soon as the addition of chains stops, the interface value drops rather rapidly from its saturated value (Ws ) to an equilibrium (relaxed) value W. The large value of the steady-state interface width (Ws ) seems to be dominated by the incoming chains just arriving at the growing surface. Fig. 4 shows the relaxation of the interface width corresponding to density profile presented in Fig. 2. We see that the interface widths at all fields have reached the equilibrium and show a systematic dependence on the field. A log–log plot of the equilibrium width versus field (E) shows a power-law decay, W ∼ E−1/2 (Fig. 5), consistent with our previous observations [19]. We have studied the dependence of the relaxed interface width with the temperature as shown in Fig. 6. The interface width (W) decreases and then increases on increasing the temperature. The decay of the interface width with the temperature can be described by a power-law with an exponent 1/4. This implies that, in the low temperatures, de-roughening of the interface occurs due to temperature. The roughness increases with the temperature on increasing the temperature beyond a characteristic value. Such a non-monotonic dependence is due to temperature limited de-roughening at low temperatures and entanglement (constrained entropy) limited at high temperatures. It is worth pointing that such a non-monotonic dependence
4
5
10 t
10
Fig. 3. Typical growth of the interface width W with time steps. Sample size 40×40×100, Lc = 40, T = 4, E = 0.5 with 10 independent samples.
of the interface width on temperature is also observed in system with particle deposition [5–8]. Variation of the interface width with the molecular weight is presented in Fig. 7. It is hard to find a good straight line fit on a log–log or semi-log scales. While at low molecular weight, decay of W is exponential, the dependence at larger molecular weight is not as clear. We therefore speculate a stretched exponential decay of the width with the molecular weight.
0.1 0.2 0.3 0.5 1.0 1.5
1
10
W
0.00 30
0
10
3
4
10
10 t
Fig. 4. Equilibration of the interface width at different field of corresponding density profiles shown in Fig. 2.
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R.B. Pandey / Progress in Organic Coatings 47 (2003) 324–330
2.10
-1/2
W = 1.3657 E
W
2.05
W
2.00
1.95
1.90
1
2
10
10 Lc
1 0.1
Fig. 7. Interface width versus chain length at E = 0.4, T = 1, sample size 40 × 40 × 100 with 10 independent samples.
1 E
Fig. 5. Interface width versus field. Data statistics are the same as in Fig. 2.
3.2. Conformation and dynamics Conformational (i.e. radius of gyration of chains) profile, from substrate to interface has been investigated in detail [17–20]. Several interesting observations are made on the differences in the radius of gyration (Rg ) of chains in different regions, i.e. substrate, bulk, and interface. Particularly, we find that the radius of gyration of chains is anisotropic which differ from one region to another as a func-
8.1573 -0.25
W = 1.79 T 0.37 W = 0.91 T
W
4.8268
2.8561
1.69
1 0.25
1
4
16
T Fig. 6. Interface width versus temperature. Sample 40×40×100, Lc = 40, E = 0.5, 10 independent samples.
tion of field—longitudinal component of Rg is larger than the transverse component in the bulk while opposite trend is observed at the substrate [17]. However, data for the conformational profiles are much more fluctuating than that for the density profile. Some polymer chains, while mobile, may also span over different regions which causes fluctuations in data points. Dynamics of chains is very difficult to track in different dynamic regions with different steric constraints. Therefore, we resort to keeping track of a tracer chain in order to investigate the conformation and dynamics as the chain moves into the interface region. Fig. 8 shows the rms displacement of the center of mass of the tracer chain with time steps for different driving fields. We can point out few observations immediately. When the chain is released and driven, it drifts, i.e. R ∼ t toward the growing surface. As the chain hits the growing surface and enters the interface region, and moves into the bulk of the polymer, it slows down drastically. This ultra-slow motion in the long time (asymptotic) regime is presented in Fig. 9. The rms displacements can still be described by an asymptotic power-law R ∼ t ν , with the ultra-slow exponent ν (∼0.1–0.01) depending on the field. On increasing the field, the polymer density increases, as a result the motion of each chain becomes highly constrained due to their neighboring chains segments and field. Movement of nodes is consistent with that of their center of mass. Conformation of chains also changes dramatically from the dilute region (i.e. after releasing the chain to just before hitting the growing surface) to just after deposition. Chain then relaxes as it moves from interface region towards the substrate. Variations of the longitudinal (z) and a transverse (y) component of the radius of gyration are presented in Fig. 10 for different values of the field. Note that the magnitude of the transverse component of gyration seems nearly
R.B. Pandey / Progress in Organic Coatings 47 (2003) 324–330
329
2
Rgy, Rgz
10
1
R
10
E=0.1, cm E=0.2, cm E=0.3, cm E=0.5, cm E=1.0, cm E=1.5, cm
0
10 0.87
R = 0.073 t 0.94 R = 0.167 t
0
10
E=0.1, y E=0.2, y E=0.3, y E=0.5, y E=1.0, y E=1.5, y E=0.1, z E=0.2, z E=0.3, z E=0.5, z E=1.0, z E=1.5, z 3
1
10
2
3
10
10
4
10
4
10 t
10
t Fig. 8. Root mean square (rms) displacement of the center of mass of the tracer chain versus time steps for chain length Lc = 100 at different driving fields E = 0.1–1.5. T = 1, sample size 40 × 40 × 100 with 10 independent samples.
Fig. 10. Variation of the longitudinal (z) and transverse (y) component of the radius of gyration of the trace chain with time steps for different fields. Statistics are the same as in Fig. 2.
4. Conclusions
R
unaffected by the field while the longitudinal component shows a systematic dependence on the field. In fact, our data shows that the longitudinal component of the gyration, decreases with the field. An oscillation seems to set in at high values of the field (E = 1.0, 1.5) in the transverse component of the radius of gyration.
E=0.1, R = 16.2 t E=0.2, R = 26.4 t E=0.3, R = 30.9 t E=0.5, R = 42.0 t E=1.0, R = 54.1 t
0.101 0.075 0.071 0.041 0.014
4
10 t Fig. 9. Asymptotic region of Fig. 8.
Using a Monte Carlo computer simulation model on a discrete lattice for electrophoretic deposition of polymer chains on an impenetrable substrate, we study the evolution of density profile, growth of the interface, and scaling of the interface width with the field, temperature, and molecular weight. Deposition under driven field leads to distinct regions characterized by the polymer density: dilute solution (regions of polymer release) followed by the interface, bulk, and the substrate [20]. Growth of the interface width, described by the exponent β is known to depend on the field, temperature, and molecular weight [18–20]. We find that the relaxed (equilibrium) interface width decays with a power-law, W ∼ E−1/2 , on increasing the field (E). The temperature dependence of the interface width is found to be non-monotonic—power-law decay of the width is followed by a power-law increase on raising the temperature. The width seems to decrease with the molecular weight possibly with an stretched exponential. To probe characteristics of chains at the interface to bulk region of the polymer grown, we monitor the conformation and dynamics of a tracer polymer chain. Examination of the rms displacement of the center of mass of the chain and its nodes reveal that the chain drifts before it slows down at the interface. The global motion of the chain becomes ultra-slow as it moves deeper into the polymer matrix. The asymptotic behavior of the rms displacement is still a power-law, R ∼ t ν , with a non-universal exponent ν (∼0.1–0.01) which depends on the field. Slowing down of the chain motion is expected as the field competes with the barriers due to surrounding
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chain nodes—higher polymer density at larger field results in higher barriers density which enhances the slowing down. Conformational evolution of the trace chain shows that the longitudinal component of the radius of gyration decreases with the field. The transverse components, larger than corresponding longitudinal components do not show as clear dependence on the field except an onset of oscillation at high field (E = 1.0, 1.5). Periodic squeezing of the compressed driven chain in a confined environment of the polymer matrix is a characteristic of driven chain in a dense medium. By varying the molecular weight of the tracer chain, we hope to gain more insight into the morphological details of the interface and the bulk as our on going studies.
Acknowledgements Various parts of this work are done in collaboration with Jun Xie, Frank Bentrem, Ray Seyfarth, and Grace Foo. This work was supported in part by the MRSEC program of the National Science Foundation under the award number DMR-0213883, a DOE-EPSCoR grant, and NSF-EPSCoR grant (EPS-0132618). References [1] A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. [2] F. Family, T. Vicsek (Eds.), Dynamics of Fractal Surfaces, World Scientific, Singapore, 1991. [3] M. Kardar, G. Parisi, Y.C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 58 (9) (1986) 889–892. [4] S.F. Edwards, D.R. Wilkinson, The surface statistics of a granular aggregate, Proc. R. Soc. London A 381 (1982) 17–31. [5] C.R. Stoldt, K.J. Caspersen, B.C. Bartelt, C.J. Jenks, J.W. Evans, P.A. Thiel, Using temperature to tune film roughness: nonintuitive behavior in a simple system, Phys. Rev. Lett. 85 (4) (2000) 800–803.
[6] S. Setzu, G. Lerondel, R. Romestain, Temperature effect on the roughness of the formation interface of p-type porous silicon, J. Appl. Phys. 84 (6) (1998) 3129–3133. [7] S.H. Lee, M.G. So, Effects of deposition temperature and pressure of the surface roughness and the grain size of polycrystalline Si1−x Gex films, J. Mater. Sci. 35 (2000) 4789–4794. [8] M. Kalke, D.V. Baxter, A kinetic Monte Carlo simulation of chemical vapor deposition: non-monotonic variation of surface roughness with growth temperature, Surf. Sci. 95 (2001) 95–101. [9] R.P. Wool, Polymer Interfaces: Structure and Strength, Hanser Gardner, 1995. [10] R.W. Richards, S.K. Pearce (Eds.), Polymer Surfaces and Interfaces III, Wiley, New York, 1999. [11] K.L. Mittal (Ed.), Polymer Surface Modification: Relevance to Adhesion, vol. 2, VSP, 2000. [12] R.A.L. Jones, R.W. Richards, Polymer at Surfaces and Interfaces, Cambridge University Press, Cambridge, 1999. [13] D.J. Lohse, T.P. Russell, L.H. Sperling (Eds.), Interfacial Aspects Multicomponent Polymer Materials, Plenum Press, New York, 1997. [14] P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. [15] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. [16] K. Binder (Ed.), Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, New York, 1995. [17] G.M. Foo, R.B. Pandey, Characteristics of driven polymer surfaces: growth and roughness, Phys. Rev. E 61 (2) (2000) 1793–1799; G.M. Foo, R.B. Pandey, Electrophoretic deposition of polymer chains on an adsorbing surface in (2 + 1) dimensions: conformational anisotropy and nonuniversal coverage, Phys. Rev. Lett. 80 (17) (1998) 3767–3770. [18] F.W. Bentrem, R.B. Pandey, F. Family, Roughening, deroughening, and nonuniversal scaling of the interface width in electrophoretic deposition of polymer chains, Phys. Rev. E 62 (1) (2000) 914– 917. [19] F.W. Bentrem, J. Xie, R.B. Pandey, Interface relaxation in electrophoretic deposition of polymer chains: Effects of segmental dynamics, molecular weight, and field, Phys. Rev. E 65 (2002) 041606-1–041606-4. [20] F.W. Bentrem, J. Xie, R.B. Pandey, Density and conformation with relaxed substrate, bulk, and interface in electrophoretic deposition of polymer chains, J. Mol. Struct. (Theochem.) 592 (2002) 95– 103.
Polymer interface changes in electrophoretic deposition Ras B. Pandey∗ Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406-5046, USA
Abstract A Monte Carlo computer simulation model for the electrophoretic deposition of polymer chains on a discrete lattice is used to study the polymer density profile, interface growth, and its dependence on field, temperature, and molecular weight. The interface width (W) decreases W ∼ E−1/2 on increasing the field (E). Width (W) depends non-monotonically on the temperature (T): a power-law decay is followed by a power-law increase on raising the temperature. Monotonic decay of the interface width with the molecular weight is possibly a stretched exponential. Conformation and dynamics of a tracer chain is used to probe its characteristics in interface to bulk region. The root mean square (rms) displacement of the center of mass of the tracer chain shows an ultra-slow motion, R ∼ t ν (ν ∼ 0.1–0.01 at E = 0.1–1.0) as the driven chain moves deeper from interface to bulk. Longitudinal compression of the radius of gyration (Rg ) of the chain increases with the field; transverse components (Rgx , Rgy ) are larger than the longitudinal component (Rgz ). The transverse component (Rgx (y)) becomes oscillatory due to periodic squeezing at high fields as the field competes with the polymer barriers. © 2003 Elsevier B.V. All rights reserved. Keywords: Monte Carlo simulation; Power-law decay; Radius of gyration
1. Introduction Coating via deposition involves directed release of constituents, their equilibration/relaxation, and nano-fabrication (i.e. bonding, entanglement, etc.) on desirable substrates. Functionality of the coating depends on the characteristics of constituents, their density distribution, interface width, and roughness which could be modified by monitoring and controlling the deposition process and equilibration methods. Interface growth and roughness via deposition of particles on substrates have been extensively investigated in last two decades by theories, computer simulations, and laboratory experiments [1,2]. Theory based on Kardar–Parisi–Zhang (KPZ) model [3], an extension of Edward–Wilkinson (EW) model [4] provided an impetus for explosive interest and growth in investigation of interface growth in past two decades. Numerous computer simulation models and methods (involving various details and algorithms) are proposed to describe deposition processes such as, solid-on-solid, physical and chemical vapor depositions, and growth by molecular beam epitaxy (MBE) in order to understand the interface dynamics in a variety of contexts including materials design, thin films, coating, fracture, and flow [1,2]. A number of procedures are considered in many of these models including rate of release of particles, their trajectories ∗ Tel.: +1-601-266-4485; fax: +1-601-266-5149. E-mail address: [email protected] (R.B. Pandey).
0300-9440/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0300-9440(03)00146-2
(i.e. diffusion, ballistic, etc.) during deposition. Depending on systems of interests, different relaxation and adhesion procedures (i.e. irreversible growth, restricted sticking/bonding, mobility, and diffusion) after particles reach the growing surface are also implemented. Thus, the growth of the interface width and scaling of the roughness particularly with the substrate length has been well understood in particle deposition. Unexpected results are, however, still reported in many such systems experimentally (e.g., Ag(1 0 0) homoepitaxy [5], Si/bulk Si interface [6], polycrystalline Si1−x Gex [7]) as well as by computer simulations [5,8]. Understanding the growth of polymer thin films, coating, and their modifications [9–13] is much more difficult than the corresponding interface growth phenomena by particles deposition [1,2]. Chains dynamics, conformation, and its relaxation [14–16] add considerable complexities at various stages of deposition with desirable variables such as driving field, rate of deposition, temperature, molecular weight, etc. In this article, we consider electrophoretic deposition of polymer chains on an adsorbing substrate by a computer simulation model on a discrete lattice [17–20]. As the polymer chains are driven by an electric field or pressure gradient toward an impenetrable substrate, conformations of chains undergo enormous changes from the driven flow region (dilute solution) to deposition at the substrate. The polymer density grows and the interface develops. The longitudinal fluctuation in the density profile in the growth region defines the interface width [1,2]. Enormous efforts have been
R.B. Pandey / Progress in Organic Coatings 47 (2003) 324–330
recently devoted toward understanding the morphological details, i.e. conformation and density profiles, of polymer chains at and near surfaces; we refrain from citing the vast list of growing literature, except few textbooks and collected articles [9–13] here for example. Systematic studies of the interface growth and roughness, conformation, and density profile for electrophoretic deposition of polymers by computer simulations are, however, limited. We consider a discrete polymer chain model with fixed bond length (equal to lattice constant) with various segmental dynamics such as kink-jump, crank-shaft, slithering-snake (reptation), end-moves, etc. [16]. Apart from simplicity, the advantage of this model includes the flexibility in implementing dynamics at different length scales by appropriate combination of segmental moves [19]. The interface dynamics, a collective phenomenon of cooperative properties of chains, is a complex issue, and therefore, it is desirable for us to constrain to relatively simple yet flexible model for constituent chains. Over the years, we have studied interface growth, roughness, conformation, and density profile at various stages of the growth with various segmental dynamics as a function of temperature, field, and molecular weight [17–20]. We have found many interesting results some of which will be reported here. In the following section, we introduce the model followed by results and discussion of the interface width in Section 3. Analysis of dynamics and conformation of chains is presented in Section 4 with a conclusion at the end.
2. Model and method We consider a lattice of size L × L × Lz with a large aspect ratio Lz /L. A polymer chain of length Lc is modeled by a constrained self-avoiding walk, i.e. (Lc + 1) consecutive nodes connected by bonds on the trail of a random walk with excluded volume constraints. Chains are released from one end (z = 1) and are driven by an external field E toward the substrate (impenetrable wall) at z = Lz . The field is coupled with the displacement Z of the chain node via change in energy U = −E Z. Additionally, we consider nearest-neighbor polymer–polymer repulsive and polymer–wall attractive interactions given by ρi ρj , U=J ij
where J = 1, ρi = 1 if the site i is occupied by the polymer node, 0 if the site i is empty, and −1 if the empty site i is on the substrate. Sufficient amount of chains (i.e. to achieve the polymer concentration, the fraction of sites occupied by polymer, p = 0.3) are deposited and equilibrated. For example, in some of our simulations (presented below), chains are moved stochastically after inserting a chain. The Metropolis algorithm [16] is used to move chains by a combination
325
of segmental movements, i.e. kink-jump, crank-shaft, and slithering-snake (reptation) dynamics. Each move is accepted with a Boltzmann factor exp{−β E}, where E is the change in energy for the move and β = 1/kB T with the Boltzmann constant kB and temperature T. Each segmental move, if possible, is implemented according to its local configuration. For example, if the randomly selected node is the inner chain node, then kink-jump or crank-shaft is attempted if such a local move is possible. On the other hand, if one of the end nodes of the chain is selected, then the end-move and reptation are attempted with equal probability. In all cases, change in energy via Boltzmann factor is used to accept or reject the moves. Periodic boundary condition is used along the transverse (X, Y) directions while open boundary conditions along the longitudinal (Z) direction. Attempt to move each chain node once defines one Monte Carlo step (MCS). The simulation is performed for a sufficiently large number of time steps with ample number of independent samples to obtain a reliable estimate of the averaged physical quantities. As mentioned before, we use different segmental dynamics [16] to move chains. Kink-jump dynamics and end-moves involve movement of only one node which are relatively slow and localized. Therefore, it takes much longer to relax the polymer chains only with the kink-jump and end movements. Crank-shaft dynamics, though still localized to small part of chain’s segment, is faster than kink-jump, as it involves movements of two nodes simultaneously. Slithering-snake (reptation) dynamics [14–16], on the other hand, involves global motion of the whole chain by altering conformations at the end segments. A combination of these and other modes of segmental movements are, perhaps, desirable in many polymer simulations. We will, however, concentrate here with combinations of kink-jump, crank-shaft, end-moves, and reptation. Steady-state profiles (density and conformation) and growth of the interface are reached if we continue to deposit polymer chains. In order to relax the interface width, we stop releasing new chains after a desirable growth while allowing chains to continue to execute their segmental moves [19,20]. In this paper, we present the interface changes due to field, temperature, and molecular weight in relaxed data for the density profile and conformation. Chains are relaxed before deposition at particular field and temperature; large aspect ratio is to facilitate relaxation. Note that all the variables, i.e. time steps, temperature, field, and energy are in arbitrary units. The empirical trends in variation of our measured quantities such as interface width, density, radius of gyration, etc. with these variables should be compared with the corresponding trend in laboratory measurements. Quantitative comparison to laboratory measurements is possible only after an appropriate calibration. Thus, if a coating is performed with an appropriate pressure, dependence of the interface width on pressure should be useful to compare with appropriate polymer constituents, molecular weight, and temperature. To our knowledge, such systematic experimental data are hard to find. Therefore, we resort here to empirical scaling of our observed quantities.
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3. Results and discussion Simulations are performed on various lattice sizes L × L × Lz with a large Lz /L; typically Lz = 100–200, L = 20, 40, 60. Molecular weight (i.e. the chain length Lc ) is varied with Lc = 100 for most of our data generated here. Temperature (T) and field (E) are varied. All the quantities are measured in arbitrary units. Ten to twenty independent samples are used to calculate the average quantities. We study density profile, interface width (W), radius of gyration (Rg ) and end-to-end distance (Re ) of chains, and variation of the rms displacement (R) of the center of mass of the tracer chain and that of its nodes with the time steps. The interface width (W) is defined as hij W2 = (hij − h)2 , h = , Ns ij
ij
where hij is the surface height from the substrate located at (i, j) and h the mean surface height averaged over the substrate with Ns sites. In many theoretical models and experiments, particularly with the particle deposition [1,2], the interface width (W) is found to grow with time (t) W ∼ tβ ,
t Lz ,
where β is the growth exponent and z the dynamical scaling exponent. Note that the height–height correlation length is much smaller than the substrate length L in small time limit where the above scaling for the interface growth is valid. Growth of the interface width saturates in the asymptotic time limit when the height–height correlation length exceeds the substrate length L. The saturated width (W) is found to obey an scaling relation with the substrate length (at least for particle deposition [1,2]) W ∼ Lα ,
t Lz ,
where α is the roughness exponent. The dynamical scaling exponent z = α/β. In the following we first discuss some of our results on the dependence of the interface width on field, temperature, and the molecular weight and then we comment on the dynamics of chains and their conformation. 3.1. Density profile and interface width Fig. 1 shows a few snapshots to get an idea of the density profile and roughness. A typical density profile (i.e. the variation of polymer density with z) is presented in Fig. 2 for different fields. The substrate is at z = Lz . At low fields
Fig. 1. Snapshots of polymer chains deposition.
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1.00 0.1 0.2 0.3 0.5 1.0 1.5
0.80
d
W
0.60
0.40
0.20
40
50
60
z
70
80
90
1 3 10
100
Fig. 2. Polymer density profile at different driving field (E = 0.1–1.5). Sample size 40 × 40 × 100, polymer chain length Lc = 100, t = 104 time steps, 10 independent samples, temperature T = 1.
(E = 0.1, 0.2), the polymer density decays rather slowly from substrate to the surface. Change in density becomes sharper, i.e. the interface width becomes smaller (see below) on increasing the magnitude of the field. Growth of an interface width with time step is presented in Fig. 3 at the temperature T = 4. Note that the interface width grows initially and saturates to a steady-state value (after 104 time steps) as the polymer chains are continued to deposit. As soon as the addition of chains stops, the interface value drops rather rapidly from its saturated value (Ws ) to an equilibrium (relaxed) value W. The large value of the steady-state interface width (Ws ) seems to be dominated by the incoming chains just arriving at the growing surface. Fig. 4 shows the relaxation of the interface width corresponding to density profile presented in Fig. 2. We see that the interface widths at all fields have reached the equilibrium and show a systematic dependence on the field. A log–log plot of the equilibrium width versus field (E) shows a power-law decay, W ∼ E−1/2 (Fig. 5), consistent with our previous observations [19]. We have studied the dependence of the relaxed interface width with the temperature as shown in Fig. 6. The interface width (W) decreases and then increases on increasing the temperature. The decay of the interface width with the temperature can be described by a power-law with an exponent 1/4. This implies that, in the low temperatures, de-roughening of the interface occurs due to temperature. The roughness increases with the temperature on increasing the temperature beyond a characteristic value. Such a non-monotonic dependence is due to temperature limited de-roughening at low temperatures and entanglement (constrained entropy) limited at high temperatures. It is worth pointing that such a non-monotonic dependence
4
5
10 t
10
Fig. 3. Typical growth of the interface width W with time steps. Sample size 40×40×100, Lc = 40, T = 4, E = 0.5 with 10 independent samples.
of the interface width on temperature is also observed in system with particle deposition [5–8]. Variation of the interface width with the molecular weight is presented in Fig. 7. It is hard to find a good straight line fit on a log–log or semi-log scales. While at low molecular weight, decay of W is exponential, the dependence at larger molecular weight is not as clear. We therefore speculate a stretched exponential decay of the width with the molecular weight.
0.1 0.2 0.3 0.5 1.0 1.5
1
10
W
0.00 30
0
10
3
4
10
10 t
Fig. 4. Equilibration of the interface width at different field of corresponding density profiles shown in Fig. 2.
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2.10
-1/2
W = 1.3657 E
W
2.05
W
2.00
1.95
1.90
1
2
10
10 Lc
1 0.1
Fig. 7. Interface width versus chain length at E = 0.4, T = 1, sample size 40 × 40 × 100 with 10 independent samples.
1 E
Fig. 5. Interface width versus field. Data statistics are the same as in Fig. 2.
3.2. Conformation and dynamics Conformational (i.e. radius of gyration of chains) profile, from substrate to interface has been investigated in detail [17–20]. Several interesting observations are made on the differences in the radius of gyration (Rg ) of chains in different regions, i.e. substrate, bulk, and interface. Particularly, we find that the radius of gyration of chains is anisotropic which differ from one region to another as a func-
8.1573 -0.25
W = 1.79 T 0.37 W = 0.91 T
W
4.8268
2.8561
1.69
1 0.25
1
4
16
T Fig. 6. Interface width versus temperature. Sample 40×40×100, Lc = 40, E = 0.5, 10 independent samples.
tion of field—longitudinal component of Rg is larger than the transverse component in the bulk while opposite trend is observed at the substrate [17]. However, data for the conformational profiles are much more fluctuating than that for the density profile. Some polymer chains, while mobile, may also span over different regions which causes fluctuations in data points. Dynamics of chains is very difficult to track in different dynamic regions with different steric constraints. Therefore, we resort to keeping track of a tracer chain in order to investigate the conformation and dynamics as the chain moves into the interface region. Fig. 8 shows the rms displacement of the center of mass of the tracer chain with time steps for different driving fields. We can point out few observations immediately. When the chain is released and driven, it drifts, i.e. R ∼ t toward the growing surface. As the chain hits the growing surface and enters the interface region, and moves into the bulk of the polymer, it slows down drastically. This ultra-slow motion in the long time (asymptotic) regime is presented in Fig. 9. The rms displacements can still be described by an asymptotic power-law R ∼ t ν , with the ultra-slow exponent ν (∼0.1–0.01) depending on the field. On increasing the field, the polymer density increases, as a result the motion of each chain becomes highly constrained due to their neighboring chains segments and field. Movement of nodes is consistent with that of their center of mass. Conformation of chains also changes dramatically from the dilute region (i.e. after releasing the chain to just before hitting the growing surface) to just after deposition. Chain then relaxes as it moves from interface region towards the substrate. Variations of the longitudinal (z) and a transverse (y) component of the radius of gyration are presented in Fig. 10 for different values of the field. Note that the magnitude of the transverse component of gyration seems nearly
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2
Rgy, Rgz
10
1
R
10
E=0.1, cm E=0.2, cm E=0.3, cm E=0.5, cm E=1.0, cm E=1.5, cm
0
10 0.87
R = 0.073 t 0.94 R = 0.167 t
0
10
E=0.1, y E=0.2, y E=0.3, y E=0.5, y E=1.0, y E=1.5, y E=0.1, z E=0.2, z E=0.3, z E=0.5, z E=1.0, z E=1.5, z 3
1
10
2
3
10
10
4
10
4
10 t
10
t Fig. 8. Root mean square (rms) displacement of the center of mass of the tracer chain versus time steps for chain length Lc = 100 at different driving fields E = 0.1–1.5. T = 1, sample size 40 × 40 × 100 with 10 independent samples.
Fig. 10. Variation of the longitudinal (z) and transverse (y) component of the radius of gyration of the trace chain with time steps for different fields. Statistics are the same as in Fig. 2.
4. Conclusions
R
unaffected by the field while the longitudinal component shows a systematic dependence on the field. In fact, our data shows that the longitudinal component of the gyration, decreases with the field. An oscillation seems to set in at high values of the field (E = 1.0, 1.5) in the transverse component of the radius of gyration.
E=0.1, R = 16.2 t E=0.2, R = 26.4 t E=0.3, R = 30.9 t E=0.5, R = 42.0 t E=1.0, R = 54.1 t
0.101 0.075 0.071 0.041 0.014
4
10 t Fig. 9. Asymptotic region of Fig. 8.
Using a Monte Carlo computer simulation model on a discrete lattice for electrophoretic deposition of polymer chains on an impenetrable substrate, we study the evolution of density profile, growth of the interface, and scaling of the interface width with the field, temperature, and molecular weight. Deposition under driven field leads to distinct regions characterized by the polymer density: dilute solution (regions of polymer release) followed by the interface, bulk, and the substrate [20]. Growth of the interface width, described by the exponent β is known to depend on the field, temperature, and molecular weight [18–20]. We find that the relaxed (equilibrium) interface width decays with a power-law, W ∼ E−1/2 , on increasing the field (E). The temperature dependence of the interface width is found to be non-monotonic—power-law decay of the width is followed by a power-law increase on raising the temperature. The width seems to decrease with the molecular weight possibly with an stretched exponential. To probe characteristics of chains at the interface to bulk region of the polymer grown, we monitor the conformation and dynamics of a tracer polymer chain. Examination of the rms displacement of the center of mass of the chain and its nodes reveal that the chain drifts before it slows down at the interface. The global motion of the chain becomes ultra-slow as it moves deeper into the polymer matrix. The asymptotic behavior of the rms displacement is still a power-law, R ∼ t ν , with a non-universal exponent ν (∼0.1–0.01) which depends on the field. Slowing down of the chain motion is expected as the field competes with the barriers due to surrounding
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chain nodes—higher polymer density at larger field results in higher barriers density which enhances the slowing down. Conformational evolution of the trace chain shows that the longitudinal component of the radius of gyration decreases with the field. The transverse components, larger than corresponding longitudinal components do not show as clear dependence on the field except an onset of oscillation at high field (E = 1.0, 1.5). Periodic squeezing of the compressed driven chain in a confined environment of the polymer matrix is a characteristic of driven chain in a dense medium. By varying the molecular weight of the tracer chain, we hope to gain more insight into the morphological details of the interface and the bulk as our on going studies.
Acknowledgements Various parts of this work are done in collaboration with Jun Xie, Frank Bentrem, Ray Seyfarth, and Grace Foo. This work was supported in part by the MRSEC program of the National Science Foundation under the award number DMR-0213883, a DOE-EPSCoR grant, and NSF-EPSCoR grant (EPS-0132618). References [1] A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. [2] F. Family, T. Vicsek (Eds.), Dynamics of Fractal Surfaces, World Scientific, Singapore, 1991. [3] M. Kardar, G. Parisi, Y.C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 58 (9) (1986) 889–892. [4] S.F. Edwards, D.R. Wilkinson, The surface statistics of a granular aggregate, Proc. R. Soc. London A 381 (1982) 17–31. [5] C.R. Stoldt, K.J. Caspersen, B.C. Bartelt, C.J. Jenks, J.W. Evans, P.A. Thiel, Using temperature to tune film roughness: nonintuitive behavior in a simple system, Phys. Rev. Lett. 85 (4) (2000) 800–803.
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