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Modeling the Adsorption Behavior of Linear End-Functionalized Poly(ethylene glycol) on an Ionic Substrate by a Coarse-Grained Monte Carlo Approach Stefano Elli,*,† Lidia Eusebio,‡ Paolo Gronchi,‡ Fabio Ganazzoli,*,‡ and Marco Goisis§ †
Institute for Biochemical Research ‘‘G. Ronzoni’’, via G. Colombo 81, 20133 Milan, Italy, ‡Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘G. Natta’’, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy, and §Italcementi - Innovation Department, via Camozzi 124, 24121 Bergamo, Italy Received July 26, 2010. Revised Manuscript Received September 9, 2010
The rheology of cement pastes can be controlled by polymeric dispersants such as branched polyelectrolytes that adsorb on the surfaces of silicate particles. In the present work, we analyze the adsorption behavior of ad hoc-prepared end-carboxylated poly(ethylene glycol), or PEG, on CaCO3 particles as a model of cement in an early hydration stage. The experimental adsorption isotherms form the base of a theoretical study aimed at unraveling polymer conformational aspects of adsorption. The study was carried out with Monte Carlo simulations using a coarse-grained bead-andspring model of linear end-charged polymer chains adsorbing on a flat, continuous, uniformly charged surface. The adsorption driving force was introduced by a Debye-H€uckel electrostatic screened potential to describe the interaction between the negatively charged end group of PEG and the positively charged CaCO3 surface empirically. With a suitable length-scale conversion between real PEG and the coarse-grained model, the calculated and experimental adsorption isotherms can be semiquantitatively compared. The theoretical results reproduce the fundamental aspects of polymer adsorption, in essential agreement with analytical approaches relating the isotherm shape to the polymer conformational properties. The conformational transition mushroom-brush of the adsorbed polymer is located on the isotherm and is related to the molecular shape. The solvent quality effect and the solution ionic strength are also considered, and their implications on the isotherms are discussed.
Introduction Superplasticizers (SPs) are polymer-based admixtures that are among the most used additives in concrete to avoid cement particle flocculation, to preserve the mixture workability over time, and to reduce the water/cement ratio. Understanding the interaction mechanism between admixtures and mineral phases that create hybrid organic-inorganic systems and their effect on the kinetics of cement setting and hardening is still a huge research problem. A cement mixture is a highly subdivided multiphase system. From a physicochemical viewpoint, its structures and properties are dominated by the surface interactions among the different phases. The interfacial phenomena are physically related to diffusion and the nonspecific adsorption of material among the phases and to complex surface chemical reactions such as hydration. For these reasons, it is useful to start a comprehensive and systematic study with well-known, “simple”, and chemically inert mineral phases1,2 in an early hydration stage. Typical phases used for these studies are CaCO3, MgO, SiO2, and Al2O3.3-5 SPs are surface modifiers that adsorb on dispersed particles, changing the forces acting among them and altering the thermodynamic stability of the whole dispersed system. SPs *Corresponding authors. E-mail:
[email protected], fabio.ganazzoli@ polimi.it. (1) Jolicoeur, C.; Nkinamubanzi, P. C.; Simard, M. A.; Piotte, M. 4th CANMET/ACI International Conference on Superplasticizers and Other Chemical Admixtures; ACI: Detroit, MI, 1994. (2) Mikanovic, N.; Khayat, K.; Page, M.; Jolicoeur, C. Colloids Surf., A 2006, 291, 202–211. (3) Houst, Y. F.; Bowen, P.; Perche, F. 12th International Congress on the Chemistry of Cement; Montreal, Quebec, Canada, 2007. (4) Jolicoeur, C.; Simard, M. A. Cem. Concr. Compos. 1998, 20, 87–101. (5) Eusebio, L.; Fumagalli, D.; Gronchi, P. J. Therm. Anal. Calorim. 2009, 97, 33–37.
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for concrete are divided into two classes:6,7 (i) SPs based on sulfonated naphthalene formaldehyde and sulfonated melamine formaldehyde condensates, acting mainly by electrostatic repulsion, and (ii) SPs based on comblike polymers with a polyacrylic (PA) or polymethacrylic (PMA) backbone and poly(ethylene glycol) (PEG) side chains. The latter polymers act by steric hindrance, reducing the short-range attractive forces acting among the dispersed particles. In this work, we analyze the interaction between SP-based polymers and the surface cations of CaCO3 particles dispersed in water, elucidating by computer simulations and experimental measurements how the conformational properties of the chains can affect their adsorption behavior. To simplify the scenario of the complex adsorption mechanism of commercial branched polymers, we considered only the repetitive unit of a PA or PMA-grafted PEG comb polymer (Figure 1), where the negative end provides the interaction with the substrate and a long chain contributes to the dispersing activity. Two such linear polymers were synthesized, with weight-average molecular weights of 1000 (PEG1K) and 4000 (PEG4K) Da, corresponding to average degrees of polymerization of about 22 and 90, respectively. In the literature, the study of polymer confined at an interface was carried out using theoretical, numerical simulation, and experimental approaches. Polymer brushes obtained by grafting one end of linear chains to a surface have been used as models to investigate the conformational behavior of the chains in that (6) Ramachandran, V. S. Concrete Admixtures Handbook: Properties, Science, and Technology, 2nd ed.; Noyes Publications: Park Ridge, NJ, 1995. (7) Spiratos, N.; Page, M.; Mailvaganam, N. P.; Malhotra, V. M.; Jolicoeur, C. Superplasticizers for Concrete: Fundamentals, Technology, and Practice; NRC Publications: Ottawa, Canada, 2003.
Published on Web 09/24/2010
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Figure 1. Scheme of the synthesis of the end-functionalized succinic acid monoester of poly(ethylene glycol).
condition. de Gennes8-11 applied theoretical scaling methods to predict the conformation of polymer brushes in a good solvent, showing that the surface grafting density and chain length are the main variables. At low grafting densities (the so-called mushroom regime), the chains are in a coiled state and do not interact. A brush regime appears when the distance among the grafted chains is less than their size so that each coil perturbs the neighboring ones and the chains are stretched. Under this condition, de Gennes predicted an almost constant monomer density proportional to Fa2/3 inside the polymer brush, with Fa being the grafting density, and the thickness of the grafted layer scales as h NFa1/3, with N being the degree of polymerization. Murat and Grest12 modeled the behavior of brush polymers by molecular dynamics simulations using a coarse-grained model under good solvent conditions with a small to medium grafting density. The monomer density profile was found to be parabolically decreasing in shape, in agreement with the self-consistent field (SCF) approach of Milner et al.13 and Cosgrove et al.14 but different from the uniform distribution predicted by de Gennes, which seems to be a limiting behavior only reached at a large enough graft density. According to Grest, the height of the brush layer scales as h NFa1/3 for a large enough graft density and/or sufficiently long chains, in agreement with de Gennes’ conclusions. With decreasing solvent quality from good to poor, the brush height decreased significantly15,16 and the monomer density increased, eventually producing a steplike function at high surface coverage. Ligoure and Leibler17 extended the SCF theory of Milner et al. to a system formed by end-functionalized linear chains adsorbing on a flat surface and calculated the adsorption isotherms for different chain sizes and adsorption energies. A coarse-grained lattice Monte Carlo scheme was also adopted to describe the adsorption dynamics of linear end-sticking chains on a surface,18 considering the brush formation kinetics, the distribution function of the monomers at the interface,19 and the desorption kinetics when a solvent, shorter chains, or more strongly interacting chains were added to the system. Moreover, the equilibrium structure of the adsorbed polymer layer showed a monomer distribution function with a Gaussian shape, different from the parabolic profile found in polymer brushes, possibly due to free chains that penetrate the outer adsorption layer.18 A lattice (8) de Gennes, P. G. Macromolecules 1980, 13, 1069–1075. (9) Flory, P. J. Statistical Mechanics of Chain Molecules; Wiley Interscience: New York, 1969. (10) Grest, G. S.; Murat, M. Computer Simulation of Tethered Chains. In Monte Carlo and Molecular Simulations in Polymer Science; Binder, K., Ed.; Oxford University Press: Oxford, U.K., 1995; Chapter 9. (11) Yamakawa, H. Modern Theory of Polymer Solution; Harper & Row: New York, 1971. (12) Murat, M.; Grest, G. S. Macromolecules 1989, 22, 4054–4059. (13) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610–2619. (14) Cosgrove, T.; Heath, T.; van Lent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20, 1692–1696. (15) Zhulina, E. B.; Borisov, O. V.; Pryamitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140–149. (16) Grest, G. S.; Murat, M. Macromolecules 1993, 26, 3108–3117. (17) Ligoure, C.; Leibler, L. J. Phys. (Paris) 1990, 51, 1313–1328. (18) Zajac, R.; Chakrabarti, A. Phys. Rev. E 1994, 49, 3069–3078. (19) Lai, P. Y. J. Chem. Phys. 1993, 98, 669–673.
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Monte Carlo method was also employed to study the kinetic and equilibrium adsorption behavior of diblock copolymers, where monomers of one block are attracted to an ideal, continuous surface and the others are repelled from it.20,21 By considering different relative lengths of the two blocks, general agreement with a Langmuir isotherm was found when the attractive block was smaller than the repulsive one, and the adsorption kinetics was in agreement with previous results.18,19 However, to the best of our knowledge very few attempts have been made to correlate the experimental and calculated adsorption isotherms on the basis of an equilibrium description of an adsorbed polymer layer and the bulk phase, giving a picture of the effect of the polymer conformation on the adsorption behavior. In this article, we first discuss our results obtained with a coarse-grained Monte Carlo (MC) simulation using a bead-and-spring model, and the adsorbing CaCO3 substrate is described by a flat, continuous, uniformly charged surface. A Debye-H€uckel screened electrostatic potential is used to describe the adsorbing driving force between the ionic surface and the end-charged linear chains empirically, considering the screening effect due to the ionic strength of the solution. The polymer mushroom-brush conformational transition is characterized, and the polymer shape and local stiffness are studied for different concentrations, degrees of polymerization, solvent quality, and average adsorption strength, with the latter being implicitly related to the ionic strength of the solution. The simulated adsorption isotherms are then compared to the experimental ones. By defining an appropriate length scale between the model and the real polymer, a semiquantitative comparison between the simulated and experimental adsorption isotherms can be made. Our results show that a coarse-grained description is useful in unraveling fundamental aspects of polymer adsorption that are difficult or even impossible to observe experimentally. This method can easily be extended to polymers or polyelectrolytes of any topology, such as branched polymers that nowadays are not well characterized at an interface.
Experimental Section Materials. The PEG1K and PEG4K polymers were synthesized by selective esterification following the scheme of Figure 1 using succinic anhydride, pyridine, and monomethoxy-terminated poly(ethylene glycol) (Mw = 1000 and 4000 for PEG1K and PEG4K, respectively), which were purchased from Aldrich and used as received. CaCO3 was purchased from Fluka and characterized by X-ray diffraction analysis (Philips PW 1130) and particle size distribution analysis (measured by CILAS 1180 liquid). The CaCO3 average particle size was 26.4 μm, and the specific surface measured by the BET (Brunauer-Emmet-Teller) adsorption isotherm22 was 0.6310 m2 g-1. The measured isoelectric point of a CaCO3 dispersion gave a value of pH 10 (i.e., the CaCO3 surface is mainly positively charged).
Synthesis of Poly(ethylene glycol) Methyl Ether Succinate (PEG1K and PEG4K). Monomethyl ether poly(ethylene glycol) was dried by azeotropic distillation with toluene before use and dissolved in chloroform. Succinic anhydride and pyridine were added to the chloroform polymer solution, and the mixture was reacted at 60 C for 75 h.23 The solution was cooled at room temperature and concentrated to dryness by rotary evaporation. The obtained wax was dissolved in water and acidified with 1 M HCl to pH 1. The water phase was washed with diethyl ether, extracted with chloroform, and dried with anhydrous sodium sulfate. The solution was filtered and concentrated to dryness. The polydispersity index Mw/Mn of the obtained polymers was in the (20) (21) (22) 319. (23)
Zan, Y.; Mattice, W. L.; Napper, D. H. J. Chem. Phys. 1993, 98, 7502–7506. Zan, Y.; Mattice, W. L.; Napper, D. H. J. Chem. Phys. 1993, 98, 7508–7514. Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309– Fu, J.; Fiegel, J.; Hanes, J. Macromolecules 2004, 37, 7174–7180.
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Article 1.0-1.1 range for PEG1K and PEG4K (Figure 1), whereas 1H NMR (DMSO) showed the following signals: δ 4.2 (m, 2H, -COOCH2-), 3.7-3.3 (m, n*4H, (-CH2-O-CH2-)n of PEG), 3.2 (s, 3H, -OCH3), 2.48 (m,4H, (-CH2)2 of succinic acid). Isotherm Determination. The adsorption isotherms were obtained at room temperature 24 h after the beginning of the hydration reaction. To build up the isotherm curve, polymer solutions with an increasing amount of dispersant/substrate ratio (SP/CaCO3) were prepared at 0.00, 0.05, 0.1, 0.3, 0.8, 1.00, 1.5, and 2.0 wt % concentrations. The water/CaCO3 weight ratio was 2 for CaCO3-based paste. The adsorbed polymer on CaCO3 was evaluated by the chemical oxygen demand (COD) technique, and the difference between the polymer in solution before and after adsorption was calculated. The separation between the supernatant solution and dispersed CaCO3 after adsorption was carried out by centrifugation. The procedure is based on the oxidation of organic and inorganic compounds, which are present in the interstitial aqueous solution, with an oxidative reactant (a solution of Cr2O72-). The dichromate excess was neutralized by Fe2þ. The concentrations of the oxidizable organic and inorganic compounds were evaluated by absorption spectrophotometry. Simulation Methodology. We adopted a bead-and-spring coarse-grained model for the polymer, and the CaCO3 surface was modeled as a perfectly smooth, flat, continuous surface with a uniform charge density. The surface charge was calculated from the CaCO3 lattice geometry (Aragonite24) by assuming that the lattice exposes a full Ca2þ ion plane by taking planes defined by cell edges a and b of the elementary cell. The surface charge density corresponds to σ = 0.13 e A˚-2. This is a rough description of the ionic adsorbing substrate, but it is consistent with the polymer model. We believe that choosing a different set of planes for the exposed surface definition should give a slightly different value of σ, which does not significantly affect the results shown in this work. The Hamiltonian function in kBT units is X X σ 1 X zi kij rij 2 þ Vðrij Þ þ - qi De exp H ¼ 2 2l ij 2ε De id where d is the bead diameter. This potential was used for the athermal solvent limit of good solvent conditions. The second nonbonded potential is a dispersive Lennard-Jones (LJ) term with a repulsive hard-core term, which in kBT units is written as 8 > r < þ" ¥, # 12 6 VðrÞ ¼ ð3Þ d d > > , r>d :ξ r r The parameters are the particle diameter d and the attraction energy ξ in kBT units. This potential was used to model the (24) de Villiers, J. P. R. Am. Mineral. 1971, 56, 758–766.
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Elli et al. interactions among the beads when the solvent quality becomes poorer. In the following text, we use the reduced units kBT = 1 and l = 1 and take d = 1 as a convenient choice. The third term in eq 1 is a Debye-H€ uckel electrostatic screened potential25,26 describing the attractive interactions between the charged surface and the charged monomer (indicated by the starred superscript) of the end-functionalized chains. Each term within the sum gives the electrostatic energy contribution of a point charge qi* of the carboxyl charged group (COO-) of a chain at a distance zi* from the surface of charge density σ in a medium having a dielectric constant of ε (taken as ε = 78 for pure water at room temperature), and the Debye length De is the screening parameter. This screened potential is a long-range potential that can be tuned in its action range through the De parameter to model the effect of a ionic double layer near the CaCO3 surface and the ions in solution. We also assume that this approach can (implicitly) describe the repulsive effect that negatively endcharged chains interacting with the plane of the Ca2þ ions experience from the CO32- counterions in the layer below it and above it as local defects, thus being exposed to the solution. These interactions are taken into account to some extent in terms of an effective screening affecting the De value. In general, a large ionic strength corresponds to a small De so that the electrostatic potential is significantly screened by the ionic double layer surrounding the ions and the surface. The constant σqi*/2ε was kept fixed at 5.0 kBT A˚-1 during all simulations, and it was estimated by considering a point charge e near a flat surface of charge density σ in a medium with a dielectric constant ε. Finally, the fourth term in eq 1 is the electrostatic screened potential acting between charged end monomers i* and j* of two different chains. The constant qi* qj* /4πε was fixed to the value of 7.0 kBT A˚, estimated as shown previously. It is important to note that the values of the constants of the last two terms in eq 1 were roughly estimated for illustrative purposes; in fact, they would strictly apply to the real system if the model unit length l were equal to 1 A˚ in a first approximation, whereas in practice it takes a value about twice as large (as discussed later). The adopted simulation cell was orthogonal with edges of Lx = Ly = 20.0 and Lz = 40.0 and was chosen so that the largest chain size was less than one-half the shortest edge. In some cases, a larger simulation box with Lx=Ly=Lz=40.0 was used, for instance, with long chains of 30 beads and also in some cases for chains of 20 beads to check for possible finite size effects. The adsorption surface was perfectly flat, continuous, and impenetrable without specific adsorption sites. It was placed at z = 0, and periodic boundary conditions were applied in the x and y directions and a purely repulsive hard wall was located at z = 40.0. In the following text, N is the bead number of each chain and Mtot is the (constant) total number of chains inside the simulation box. We employed the Monte Carlo method in continuous space with the standard Metropolis algorithm.27 The adopted procedure involves random local moves of a randomly selected bead with a minimum displacement of 0.2 adjusted to achieve an acceptance ratio of 0.5 (a commonly accepted value) so as to avoid nonergodicity problems in the phase-space sampling. Unfortunately, the efficiency of the Metropolis algorithm is limited for uniformly dense systems. Nevertheless, in practice this procedure became inefficient for volume fractions larger than 25-30%, whereas in the simulations reported here the volume fraction never exceeded 12%. All simulations were conducted starting from arbitrary initial configurations where the linear chains were stretched in a direction orthogonal to the adsorption surface with the charged monomers far from it. For systems with a higher polymer chain length (N = 20, 30) to improve the sampling as much as possible, the random seed was changed every few tens of sampled geometries (20 or 30) over the entire (25) Debye, P.; H€uckel, E. Phys. Z. 1923, 24, 185–206. (26) Stevens, M. J.; Kremer, K. J. Phys. II 1996, 6, 1607–1613. (27) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H. J. Chem. Phys. 1953, 21, 1087–1092.
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Figure 2. Calculated adsorption isotherms plotted as the surface concentration of adsorbed chains Mads Sxy-1 vs the total concentration MtotSxy-1 for chains with N = 5, 10, 20, and 30 beads showing purely repulsive nonbonded interactions and a fixed Debye length of De = 2.0 (solid lines with error bars) on linear scales (a) and on a doubly logarithmic plot (b). Lines are a guide for the eye. (a) The inset shows an expanded view of the adsorption isotherms at a low concentration. For a chain with N = 20 beads, we also show the simulation results obtained with a small periodic box (edges Lx = Ly = 20.0, continuous lines) and with a large periodic box, denoted as “Big Box” (edges Lx = Ly = 40.0, filled circles). The error bars in the latter case are smaller than the symbol size. Simulations with chains of N = 30 beads are done in a large periodic box, and all others are obtained in small periodic boxes. (b) For the chain with N = 5 beads, a qualitative estimation of the mushroom-brush transition is shown as the intersection point of the two straight lines that fit the data points before and after the change in slope. (See the text.) Horizontal lines correspond to the surface concentrations that localize this transition, estimated as just mentioned for the five-beads chains. simulation time, and the corresponding acceptance ratio was monitored. The system relaxation was monitored through the potential energy and the number of adsorbed chains for every snapshot that was saved during the sampling. As a further check, most simulations for the densest systems were replicated, starting with a different initial random number to assess the reproducibility of the results and to improve the statistics. For each configuration, the number of adsorbed chains is defined as the number of chains Mads having their charged end monomers in an adsorption layer with coordinates of z < z0, where z0 is arbitrarily chosen to be equal to 1.0. In fact, we checked that an increase in z0 of up to a few units affects only the number of adsorbed chains, whereas the shape of the isotherms does not change at all. A detailed statistical correlation analysis was applied for a good estimation of errors on the average values, as previously described in earlier work.28,29
Results and Discussion Shape of the Adsorption Isotherms in an Athermal Solvent. The modeled adsorption isotherms were calculated at first with the hard-sphere repulsive nonbonded potential of eq 2, with chain lengths of N = 5, 10, 20, 30 beads. Parameter σqi*/2ε for the electrostatic energy between the charged chain ends and the surface in eq 1 is held fixed throughout this article. In the present paragraph, some fundamental aspects of the adsorption isotherms are presented and discussed, fixing the Debye length at the arbitrary value of De = 2.0 to ensure a good exchange of chains between the surface and the bulk phase within manageable simulation times. The effect of a different De value shall be discussed in a later section. The adsorption isotherms are shown in Figure 2 both on a linear and on a doubly logarithmic scale. In the Figure, we plot the surface concentration of adsorbed chains, defined as the number of adsorbed chains per unit surface area MadsSxy-1 (where Sxy = Lx 3 Ly is the surface area available for the adsorption in the simulation box) as a function of the total concentration expressed as Mtot Sxy-1 (i.e., through the total number of chains per unit surface area). As shown in Figure 2a, the calculated adsorption isotherms have a monotonic convex shape similar to the SCF results of Ligoure and Leibler,17 (28) Elli, S.; Ganazzoli, F.; Timoshenko, E. G.; Kuznetsov, Yu. A.; Connolly, R. J. Chem. Phys. 2004, 120, 6257–6267. (29) Allen, M. P.; Tildeslay, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, U.K., 1987.
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even though a plateau is not reached with the present concentration. It is interesting to recall that the initial slope of the isotherms is related to the value of the σqi* De/2ε parameter of the electrostatic adsorption energy30-32 for a system approaching true equilibrium, as can be seen starting from the simple Langmuir equation33 that involves small molecules adsorbing on a surface and forming a monolayer to its extension to adsorbing macromolecules involving a self-consistent theory approach.32 In fact, all of the calculated isotherms in Figure 2 display the same initial slope, as best seen in the log-log plot of Figure 2b. In the present case, the adsorption isotherms are the result of three opposite driving forces that change their relative weight as the surface concentration changes: (i) the electrostatic attractive interaction between the charged monomers and the surface, (ii) the steric and electrostatic repulsive interactions among the polymer chains, and (iii) the steric repulsive interactions among the polymer chains and the surface. The first contribution drives the polymer adsorption, and the second and third ones favor the desorption process. At a very low surface concentration, each adsorbed chain is in a coil conformation (mushroom regime), far from its nearest neighbors. Therefore, most of the surface is still accessible, and the number of adsorbed chains increases linearly with concentration. This regime corresponds to the linear Henry’s region32 of the isotherms and is realized for a surface concentration of adsorbed chains MadsSxy-1 that is lower than about 0.040, 0.030, 0.020, and 0.015 for chains with 5, 10, 20, and 30 beads, respectively, as is best seen in the inset of Figure 2a. It is interesting that for chains of 20 and 30 beads these surface concentrations correspond to the largest values for the mushroom regime with the chains in a self-avoiding walk (SAW) conformation,9 whereas for larger concentrations the chains begin to interact sterically. Conversely, for the shorter chains the mushroom regime estimated from the coil size would be present only for surface concentrations lower than 0.10 and 0.043 (chains with 5 and 10 beads, in that order), which are values that exceed those (30) Rodgers, S. D.; Santore, M. M. Macromolecules 1996, 29, 3579–3582. (31) Mosquet, M.; Chevalier, Y.; Brunel, S.; Guicquero, J. P.; Le Perchec., P. J. Appl. Polym. Sci. 1997, 65, 2545–2555. (32) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1998. (33) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848–1906.
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obtained from the isotherms of Figure 2a,b. This difference could be related to the electrostatic interactions that perturb the adsorbed coils before real contact can actually take place, and in fact it is more evident for 5-bead chains than for 10-bead chains. After the linear Henry’s region, the calculated isotherms increase with a lower and a continuously decreasing positive slope. A different adsorption regime31 begins, where the adsorbing chains must compete for space on the surface. This behavior is typical of anticooperative adsorption, where the adsorption becomes less favorable as the surface concentration becomes larger as a result of steric and electrostatic repulsive interactions. In Figure 2b, two linear regimes are recognized in the log-log plot, corresponding to a change from the linear Henry’s regime to a power law regime with an exponent smaller than unity at larger concentrations. The intersection between the linear regimes of the log-log plot corresponds to the mushroom-brush conformational transition,32 and its projection on the vertical axis estimates the surface concentration of adsorbed chains corresponding to this change. This procedure was applied to all cases (even though the fitting lines are reported only for N = 5 for clarity), and the estimated values for the mushroom-brush transition are MadsSxy-1 = 0.055, 0.035, 0.022, and 0.014 for chains with N = 5, 10, 20, and 30 beads, respectively, in good agreement with the previous values. No saturation effect can be detected because the explored concentration range is quite low, amounting to a volume fraction that is lower than 12%. It is important to point out that it is difficult to observe a real plateau in the adsorption isotherms of polymers, at least if a limited concentration range is explored. In this case, in fact, the saturation at a high concentration mainly depends on the polymer molecular weight31,32 and not on the surface coverage, unlike what happens with small molecules. We need to consider that in the present model the connected beads are linked by harmonic springs, with a minimum distance equal to 1.0 (the bead diameter) and an average distance equal to about 1.8 as a result of the thermal fluctuations. As the concentration increases, the coils can alleviate the steric interactions by conformational changes, assuming a more extended conformation, but also by decreasing the average spring length. However, the changes in this length amount to less than 2% for the most concentrated systems compared to the dilute chains, so their contribution to “softening” the adsorption isotherms is minor. It is also apparent in Figure 2 that for end-adsorbing polymers an increase in the chain length (N = 5, 10, 20, 30) at a fixed parameter σqi* De/2ε of the electrostatic adsorption energy leads to a decrease in the adsorbed amount, in agreement with previous experimental31,32,34 and theoretical17-21 results. Under this condition, shorter chains make a lower internal energy contribution to the system with a lower entropy loss as compared to longer chains because the surface can accommodate a greater number of short rather than long chains. Conformation of the Adsorbed Chains in an Athermal Solvent. The conformational properties of the chains at the interface can be described by the ratio between the mean-square perpendicular zz and parallel xx (or yy) components of the average radius of gyration tensor ÆRg,zz2æ/ÆRg,xx2æ of the adsorbed chains. This ratio is plotted in Figure 3 as a function of the surface concentration of adsorbed chains MadsSxy-1. For a spherical coil in the laboratory frame of reference, this ratio is close to 1 because ÆRg,xx2æ = ÆRg,yy2æ = ÆRg,zz2æ. However, in the present case Figure 3 shows that the ratio ÆRg,zz2æ/ÆRg,xx2æ increases with sigmoid saturation behavior as the number of adsorbed chains increases. It is interesting that the inflection points of the sigmoids
In eq 4, rk,kþ1 is the vector associated with the kth bond of the chain, with k = 1 being the end segment closest to the surface and R being the end-to-end vector, whereas the angular brackets indicate the configurational average of the adsorbed chains. Therefore, the persistence length lpers(k) is the average projection of the vector associated with the kth segment on the end-to-end vector and quantifies the local chain stiffness. lpers(k) is reported in Figure 4 as a function of the k index for the adsorbed chains with 20 and 30 beads and 2 values of the surface concentration MadsSxy-1, namely, 0.01 and 0.03, and for an isolated chain of 20 beads for comparison. The adsorbed chains in a mushroom conformation have the largest values of lpers(k) near the surface because the first segment (k = 1) is normal to the surface and roughly parallel to the end-to-end vector R. Away from the surface, for k > 1 the local stiffness decreases monotonically with a sigmoid shape, with the inflection point located in the middle of the chain. In fact, the directional correlation between the first bond and the successive ones (k > 1) decreases as k increases. A chain average stiffness is a measure of how fast directional correlations decrease with k (i.e., with the topological separation). For the 20-beads chain, the segments with k > 13 display the same local stiffness as an isolated molecule
(34) Kumacheva, E.; Klein, J.; Pincus, P.; Fetters, L. J. Macromolecules 1993, 26, 6477–6482.
(35) Connolly, R.; Bellesia, G.; Timoshenko, E. G.; Kuznetsov, Y. A.; Elli, S.; Ganazzoli, F. Macromolecules 2005, 38, 5288–5299.
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Figure 3. Ratio between the mean-square perpendicular zz and parallel xx components of the radius of gyration tensor for the adsorbed chains plotted vs the surface concentration of adsorbed chains MadsSxy-1 for chains with N = 5, 10, 20, and 30 beads. For the chain with N = 20 beads, the simulation are done both with a small periodic box of edges Lx = Ly = 20.0 (solid lines with error bars) and with a large periodic box denoted as Big Box with edges of Lx = Ly = 40.0 (b). The error bars in the latter case are smaller than the symbol size. For chains with N = 5 and 10 beads, a small periodic box is used, whereas for the longest chains (N = 30 beads) a large periodic box is considered.
are located in the ranges of MadsSxy-1 = 0.045-0.075, 0.030-0.040, 0.020-0.030, and 0.010-0.020 for chains with 5, 10, 20, and 30 beads, respectively, corresponding to the location of the previously discussed mushroom-brush transition. This finding shows another way to localize this transition through the conformational properties of the adsorbed chains, even though it cannot be easily observed experimentally. The chain stretching in a direction perpendicular to the surface may also be viewed as a stiffening induced by the neighboring chains, which is best described through the local persistence length lpers as a function of the surface concentration of adsorbed chains MadsSxy-1. An effective persistence length lpers(k) for the kth generic chain segment28,35 can be defined as lpers ðkÞ ¼
rk, k þ 1 R jrk, k þ 1 j 3
ð4Þ
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Figure 4. Local persistence length lpers(k) as a function of the k index of the segments for adsorbed chains with 20 (9) and 30 (b) beads in a large periodic box of size Lx = Ly = 40.0 and for an isolated chain of 20 beads as a comparison ()). The segment k = 1 is closest to the interface, and the segment with k = 19 is farthest from the interface. The adsorbed polymers are reported with two different surface concentrations of adsorbed chains MadsSxy-1 = 0.01 (solid symbols) and 0.03 (open symbols).
Figure 5. Calculated adsorption isotherms plotted as in Figure 2a with filled circles and error bars for chains with N = 10 beads and a different attractive interaction quantified by Lennard-Jones energy parameter ξ = 0.0, 1.0, and 2.0 (in kBT units). The lines are a guide for the eye. The experimental adsorption isotherms for endcarboxylated PEG on CaCO3 in water are also shown for PEG1K and PEG4K by crosses and empty circles, respectively.
of the same length (empty diamonds in Figure 4). Upon increasing the surface concentration of adsorbed chains MadsSxy-1 from 0.01 to 0.03, the local stiffness lpers(k) increases for the segments close to the interface (k < 13 and k < 20 for the chains with 20 and 30 beads, respectively). This feature is due to the chain stretching induced by the steric interactions with the neighboring chains for larger amounts of adsorption. Figure 4 also shows that the local stiffness lpers(k) is enhanced at all k values when the chain length is increased, as already found for linear chains and comb polymers in dilute solution.28,35 Effect of Dispersive Interactions among the Adsorbed Chains. Until now, the adsorption isotherms were calculated for chains in the athermal limit of good solvent conditions, yielding the largest swelling in dilute solution. It is also interesting to study the effect of a somewhat poorer solvent. This condition can be experimentally achieved by changing the solvent composition or the temperature. In coarse-grained models, where the solvent is implicitly described, the effect of solvent quality can be introduced through a pairwise attractive interaction term between the beads through a nonbonded LJ potential with a lower hard-sphere cutoff as shown in eq 3. In this way, the effective interactions among the beads can indirectly account for their interactions with the solvent. Considering polymer chains with 10 beads, the effect of the attractive potential on the calculated isotherms is shown in Figure 5. The Langmuir 2010, 26(20), 15814–15823
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Figure 6. Ratio between the perpendicular zz and parallel xx components of the radius of gyration tensor for the adsorbed chains vs the surface concentration of adsorbed chains MadsSxy-1 for chains with N = 10 beads with different strengths of attractive interactions quantified by the Lennard-Jones energy parameter ξ = 0.0, 1.0, and 2.0 (in kBT units). Lines are a guide for the eye.
strength of the attractive interactions among the beads is quantified by the energy parameter of the LJ potential, ξ = 0.0, 1.0, and 2.0 (in kBT units), assuming as before a Debye length of De = 2.0. A value of ξ = 2.0 kBT corresponds to strong attractive interactions, so during the simulation the chains tend to aggregate in the bulk phase, whereas a value of ξ = 1.0 kBT corresponds to much better solvent conditions that are only slightly poorer than the good solvent limit. In fact, with the latter value the apparent Flory exponents9,28 are estimated to be νR = 0.550(5) and νS = 0.542(5) for the root-meansquare end-to-end distance and radius of gyration, respectively, in the range of chain lengths of 10-100 beads, to be compared with the common asymptotic value of 0.5889,28 in the good solvent limit. Upon enhancing the interaction strength among the beads, the surface concentration of adsorbed chains significantly increases, in particular, at high concentrations, whereas at a small concentration the linear region becomes more extended (Figure 5). Two effects are involved in this behavior. The first one is the decrease of the molecular size, so that a greater number of molecules can be accommodated on the same surface area in the mushroom regime (MadsSxy-1 < 0.04). In this case, in fact, the meansquare radii of gyration of the 10 beads chains are ÆRg2æ = 7.25(9), 6.50(8), 5.39(7) for attractive interactions ξ = 0.0, 1.0, 2.0 kBT, in the order. The second effect is due to the attractive interactions, which are most important at a high surface concentration and cooperatively enhance the amount of adsorption. It is interesting to note that these interactions do not affect the electrostatic adsorption energy parameter σqi* De/2ε, so that the initial slope of the isotherms in Figure 5 remains unchanged, even though the total adsorption amount can increase at a high concentration. In Figure 6 the ratio between the mean-square perpendicular zz and parallel xx (or yy) components of the radius of gyration tensor ÆRg, zz2æ/ ÆRg, xx2æ is plotted as a function of the surface concentration of adsorbed chains MadsSxy-1 for different ξ values. This ratio shows the same sigmoid behavior as the general one of Figure 3 for ξ = 0.0 and 1.0 kBT. Moreover, the inflection points (approximately located at MadsSxy-1 = 0.03-0.04 and 0.05-0.07, in the order) show the change between the mushroom and the brush regime, in fair agreement with the values MadsSxy-1 = 0.043 and 0.049 estimated from the mushroom sizes. On the other hand, the plot has a different shape for ξ = 2.0 kBT. In fact, the ratio ÆRg.zz2æ/ ÆRg,xx2æ decreases as a function of the surface concentration, displaying a minimum in the range MadsSxy-1 = 0.06-0.09, and afterward it increases again for surface concentrations beyond the end of the linear region in the isotherm in Figure 5. In this case, the attractive interactions among the adsorbed chains and within DOI: 10.1021/la102962z
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each chain decrease the perpendicular size, while increasing the parallel component because of the contacts with the adjacent chains. At a surface concentration MadsSxy-1 > 0.09, the brush regime appears and the molecules begin to assume a stretched conformation perpendicular to the surface. The experimental adsorption isotherms for PEG1K and PEG4K on CaCO3 are also reported in Figure 5, considering the specific surface of the substrate (Experimental Section) and the approximate length scale correspondence between the model chains and real PEG, given by l = 2 A˚. This choice was made by considering that for a PEG chain in the planar zigzag conformation the distance between two consecutive oxygen atoms (three successive bonds) is about 3.6 A˚, which estimates the monomer length. In the present bead-and-spring model, the average distance between consecutive beads is equal to about 1.8l; therefore, a conversion length scale can be defined as 1.8l = 3.6 A˚, or l = 2 A˚. In principle, this choice for the model unit length would require the introduction of a corrective factor on the order of unity for the electrostatic constants in eq 1: σqi*/2ε and qi*qj*/4πε (more exactly, 2.0 and 0.5, respectively). Accordingly, the numerical values of these constants, reported in the Experimental Section are only approximate for the problem at hand. A precise quantitative estimate of the values of these constants and of the Debye length (see later discussion) is beyond the scope of this article, focusing more on the general behavior of the model than on the quantitative agreement with the experimental data. The experimental isotherms reported in Figure 5 show a convex shape over the whole concentration range and two-step adsorption behavior, which could be related to an adsorbed polymer conformational change on the surface31 or to a reorientation of the chains36 or possibly to the building of a multilayer.37 In fact, at a very small total concentration (MtotSxy-1 < 0.1) the experimental isotherms show a change in their slope at an adsorbed amount corresponding to about MadsSxy-1 = 0.02 for PEG1K and 0.006 for PEG4K. This behavior could be related to a change in the adsorption regime from diluted noninteracting chains (mushroom regime) to a state where the adsorbed coils begin to interact (brush regime), as was shown before on the calculated isotherms in Figure 2 at about the same adsorbed amount, MadsSxy-1 = 0.02 for chains of 20 beads and 0.015 for chains of 30 beads. For PEG in water, the radius of gyration was estimated to be 9.5 and 27.1 A˚ for molecular weights of 1000 and 4000 Da, respectively, using the observed scaling law31,38 Rg = 0.215Mw0.583 A˚. The Rg value for the lowermolecular-weight PEG is also supported by molecular dynamics simulations in explicit water at T = 300 K39 with the NAMD40,41 simulation engine and the general amber force field (GAFF).42 Provided that the adsorbing surface is completely accessible to PEG, the surface concentration of adsorbed chains for the mushroombrush transition can be estimated to be MadsSxy-1 = 0.014 and 0.002 for PEG1K and PEG4K, respectively, values that are slightly underestimated as compared to the first step for the experimental isotherms in Figure 5. This discrepancy could be related to the incomplete availability of the adsorbing surface due to its roughness (36) Schmitz, K. S. Macromolecules 2000, 33, 2284–2285. (37) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: Oxford, U.K., 2000. (38) Devandan, K.; Selser, J. C. Macromolecules 1991, 24, 5943–5947. (39) Elli, S. Unpublished results. (40) Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kale, L.; Schulten, K. J. Comput. Chem. 2005, 26, 1781–1802. (41) NAMD was developed by the Theoretical and Computational Biophysics Group at the Beckman Institute for Advanced Science and Technology at the University of Illinois at Urbana;Champaign. (42) Wang, J.; Wolf, R.; Caldwell, J.; Kollman, P.; Case, D. J. Comput. Chem. 2004, 25, 1157–1174.
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Figure 7. Calculated adsorption isotherms (filled circles with error bars) for chains with N = 10 beads with a nonbonded attractive potential (Lennard-Jones energy parameter ξ = 1.0) at different ionic strengths as quantified by Debye lengths De = 3.0 2.0, 1.0, and 0.7. Experimental adsorption isotherms for end-carboxylated PEG on CaCO3 for PEG1K () and PEG4K (O).
or to surface defects possibly involving grain boundary effects. Both experimental isotherms show a second step located at MadsSxy-1 = 0.03 and 0.09 for PEG1K and PEG4K, respectively, that appear to be more difficult to characterize. A larger number of experimental data could be useful for a more complete characterization of the shape of the isotherms at higher concentration, whereas a more accurate modelization on the atomic scale could shed additional light on these details. In any case, we stress that Figure 5 indicates that a poorer solvent quality leads to a poorer agreement between the calculated and the experimental isotherms. This observation is in keeping with the good solvent behavior of PEG in water, which implies few attractive interactions among the chains. Effect of the Ionic Strength and the Debye Length. We also studied the effect of the ionic strength of the solution, which can be modified in the simulations through the Debye length De. We considered polymers chains with N = 10 beads interacting through a LJ potential with a nonbonded attractive part characterized by an energy parameter ξ = 1.0 kBT, which leads to good solvent behavior. The adsorption isotherms calculated for these chains with different Debye lengths of De = 3.0, 2.0, 1.0, and 0.7 are plotted in Figure 7. In this way, we can also check whether the (slight) attractive interactions among the chains may affect the shape of the adsorption isotherms, for example, by creating nucleation effects on the adsorbing surface with cooperative behavior, possibly at a low De value when the electrostatic interactions are greatly screened. Figure 7 shows that the adsorbed amount decreases upon reducing De (i.e., by increasing the effective ionic strength of the solution) within the explored concentration range. It is also interesting that at a low surface concentration the slope of the adsorption isotherms decreases with a decreasing De (Figure.7) because of the change in the intrinsic strength of the electrostatic adsorption energy through the parameter σqi*De/2ε. This fact implies that the ionic strength increases, the screening effect of the double layer becomes larger, and the range of the electrostatic potential drops. In other words, a smaller number of chains in the bulk solution experience the driving force for adsorption and the repulsive electrostatic interactions between like charges on the molecules become weaker. Qualitatively, the shape of the adsorption isotherms does not change significantly as De decreases from 3.0 to 1.0 within the present total concentration range. Under the highest ionic strength condition (De = 0.7), the adsorption isotherm becomes linear in total concentration within the explored range (i.e., MtotSxy-1 = 0.04-0.42, the linear Henry’s region). Under these conditions, the Langmuir 2010, 26(20), 15814–15823
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Figure 8. Surface concentration of adsorbed chains MadsSxy-1 as a function of the ionic strength of the solution expressed through the Debye length De for a fixed total polymer concentration of MtotSxy-1 = 0.25. The polymer chain length is N = 10 beads, and the energy parameter of the Lennard-Jones potential is ξ = 1.0 kBT. Error bars are smaller than the symbols.
adsorbed amount is lower than predicted for a mushroom-brush transition, which should be around MadsSxy-1 = 0.05 considering the size of the adsorbed molecules ÆRg2æ = 6.4(1). Therefore, the adsorbed chains are still in a mushroom conformation, weakly perturbed by the neighboring chains and by the nonadsorbed molecules of the bulk phase, as shown by the ratio ÆRg,zz2æ/ÆRg,xx2æ = 1.07(4) that is very close to unity and is calculated for the highest total concentration in Figure 7 (MtotSxy-1 = 0.42, De = 0.7). The experimental isotherms for PEG1K and PEG4K are drawn in Figure 7 as done before (Figure 5). The plot shows that the Debye length De could be empirically used to adjust the strength of the electrostatic adsorption energy (dictating the initial slope of the adsorption isotherms) for a better fit to the experimental data. From Figure 7, the best agreement between the experimental and calculated isotherms is achieved for a De value that is slightly larger than 1 and an energy parameter of the LJ attractive potential of ξ = 1.0 kBT, corresponding to good solvent conditions. However, the experimental isotherms are well reproduced only at small concentrations, but fitting the whole shape with two adsorption steps appears to be more problematic. It is interesting that in this description one segment corresponds to about two real PEG monomers, which is reasonable for PEG in water. Considering the degree of uncertainty of the previously defined length scale (l = 2 A˚), it is possible to adjust it as a fitting parameter to improve the agreement between the calculated and the experimental isotherms. Until now, we have shown how the Debye length De indirectly controls the strength of the electrostatic adsorption energy. It is also interesting to consider how the adsorbed amount of polymers is affected by the Debye length De (i.e., by the ionic strength of the solution at a fixed total concentration). In Figure 8, the surface concentration of adsorbed chains MadsSxy-1 is plotted as a function of De for a total concentration of MtotSxy-1 = 0.25. The plot clearly shows that the adsorbed amount increases monotonically with increasing De in a way that is suggestive of saturation behavior, even though a real plateau is not yet observed. This finding shows that at a fixed total polymer concentration a decrease in ionic strength smoothly enhances the amount of adsorbed polymer. In fact, increasing the electrostatic potential cutoff (i.e., increasing De) does increase the driving force for polymer adsorption, but at the same time it strengthens the repulsion between the charged chain ends. From another point of view, as De increases, more chains can be adsorbed but the polymer repulsion energy competes with the driving force for adsorption. As a result, the compromise between these contributions leads to the saturation behavior shown in Figure 8 at a high De value (i.e., at a low ionic strength). Langmuir 2010, 26(20), 15814–15823
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Figure 9. Monomer distribution function for adsorbed chains with 20 (O) and 30 () beads in the good solvent limit at a surface concentration of adsorbed chains of MadsSxy-1 = 0.01 and 0.03 plotted vs the distance z from the surface. The solid and dotted curves are the fitting results obtained by eq 5 with the parameters reported in Table 1 and respectively correspond to surface concentrations of 0.01 and 0.03. The inset show an expansion of the monomer distribution function at a short distance for chains with 20 and 30 beads with a surface concentration of 0.01.
An appropriate parametrization of constants σqi*/2ε and qi*qj*/ 4πε, empirically reproducing the effective electrostatic interaction in the implicit solvent considered in the present coarse-grained model, could be useful for tuning the adsorption driving force for a particular surface or medium. As a final observation, we note that no cooperative adsorption behavior indicated by positive curvature at a small concentration on the adsorption isotherms is apparent in the simulation results. Under this condition, the adsorption process does not take place with nucleation effects on the surface because of the main repulsive interaction between adsorbed chains, even at a high ionic strength, when the electrostatic repulsion between adsorbed chains is minor. Monomer Distribution Function. The distribution function of the distances z between the monomers and the surface calculated by considering only the adsorbed chains is plotted in Figure 9 for chains with 20 and 30 beads at surface concentrations of adsorbed chains of MadsSxy-1 = 0.01 and 0.03, respectively, below and above the mushroom-brush transition (approximately 0.02 and 0.015 for chains with 20 and 30 beads). The distribution functions of the distances were built as histograms with a bin size of 0.2 and are normalized so as to give a unit integral for the whole domain. The distribution functions F(z) are bivariate (inset of Figure 9). The first maximum is located at a short distance (z < 0.5) and is mainly due to the end-charged monomer attracted to the surface. At greater distances, F(z) shows a depletion zone, located at about z = 1.0, that is due to the connectivity constraint between monomers; afterwards, it develops a second maximum, followed by a tail. Apart from the initial peak at very small distances, the distribution functions can be empirically fitted by a Domb Gillis and Wilmers equation43 with a Gaussian tail in terms of fitting parameters a, b, and c, as shown in eq 5. FðzÞ ¼ azb expð - cz2 Þ
ð5Þ
In fact, at short distances (z f 1þ), F(z) is well described by a power law (F(z) ∼ zb), whereas the long tails have a Gaussian shape (F(z) ∼ exp(-cz2)) as previously shown by Lai.19 For the considered surface concentration and for z > 1, the distribution function F(z) is well fitted by eq 5. The fitting parameters are reported in Table 1 for chains with 20 and 30 beads and for surface (43) Domb, C.; Gillis, J.; Wilmers, G. Proc. Phys. Soc. London 1965, 85, 625– 645.
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Table 1. Fitting Results for the Monomer-Interface Distribution Function Obtained Using the Expression G(z) = azb exp(-cz2) for Chains with N = 20 and 30 Beads and for Surface-Adsorbed Concentrations of 0.01 and 0.03a MadsSxy-1
a
c
R2
0.0231(2) 0.0176(1)
0.9988 0.9994
b N = 20
0.01 0.03
0.097 0.095
0.464(9) 0.351(6)
N = 30 0.01 0.068 0.408(5) 0.0121(0) 0.9993 0.03 0.067 0.318(8) 0.0094(1) 0.9980 a The estimated standard error of the fit in the last decimal digit is shown in parentheses. The correlation coefficient R2 is reported in the last column.
concentrations of MadsSxy-1 = 0.01 and 0.03, and the fitting curves are shown in Figure 9 with continuous and dotted lines, respectively. (The data points for the latter cases are not shown explicitly for visual clarity, and only the fitting curves are reported.) Table 1 shows that for both polymer lengths an increase in the chain surface concentration is related both to an increase in the short-range width of the distribution (∼xb), or equivalently to a broader correlation hole at z = 0 through a decrease in b, and to a longer tail at larger distances (∼exp(-cx2)) through a decrease in c. However, at a fixed surface concentration the increase in the chain length is mostly reflected in a broader width of the distribution through a decrease in c. From Table 1 it may be noted that the short-range power law behavior of F(z) for the end-adsorbed linear chains has an exponent that is significantly smaller than predicted by de Gennes for brush polymers8 (b = 2/3) though scaling arguments under conditions of nonoverlapping coils. This difference can be explained by considering that even at a low surface concentration (MadsSxy-1 = 0.01) some interactions between the adsorbed coils can be present, somewhat stretching the chain conformation in the mushroom regime with respect to the isolated coils assumed by de Gennes in the infinitedilution limit.
Conclusions The physicochemical behavior of polymeric plasticizers in dispersed phases is an open question for practical reasons (related to their technological relevance, such as their essential role in high-performance concrete) and from an academic point of view. In concrete preparation and in general for the stabilization of dispersed powder systems with nanosized particles, the most useful plasticizers are linear or comblike44,45 polyelectrolytes because of their dispersing activity and water retention. Fundamental molecular aspects of the adsorption of these macromolecules on the surface of cement particles, thus acting as viscosity modifiers, are not fully understood. In this article, low-molecularweight PEGs with degrees of polymerization of about 22 and 90 monomers are used as model plasticizers. In fact, these molecules were functionalized at one end by a monoester of a dicarboxylic acid (succinic acid), leaving a free terminal carboxylic group to interact with Ca2þ ions of the CaCO3 surface. A CaCO3 powder of known surface area was used as a model substrate for cement particles in experimental measurements of the adsorption isotherms. These isotherms had a convex shape with two plateaus, (44) Yoshioka, K.; Tazawab, E.; Kawaib, K.; Enohatac, T. Cem. Concr. Res. 2002, 32, 1507–1513. (45) Li, C. Z.; Feng, N. Q.; Li, Y, D.; Chen., R. J. Cem. Concr. Res. 2005, 35, 867–873.
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suggesting possible multistep adsorption that is probably related to a conformational rearrangement of the chains as a consequence of the increase in the adsorbed amount on the available surface. The shape of the adsorption isotherms was roughly the same, but the polymer with the larger molecular weight had a lower adsorbed amount, indicating that the smaller molecules are preferentially adsorbed in keeping with previous experimental31,32,34 and theoretical17-21 results. To find a relationship between the adsorption isotherms and the conformational properties of the adsorbed chain, a coarse-grained model of end-functionalized PEG adsorbing on a flat and uniformly charged surface was built. A Debye-H€uckel screened electrostatic potential was used to mimic the screening effect of the ionic double layer on the surface charges, thus accounting implicitly for the ionic strength of the solution. Two different nonbonded interaction potentials between the monomers were used to describe either good solvent conditions or a solvent of slightly lower quality. The adsorption isotherms were calculated by a Monte Carlo simulation of a dense system of chains using appropriate boundary conditions with a classical Metropolis sampling scheme. By defining an appropriate length-scale factor, it was possible to plot on the same scale the experimental and the calculated isotherms, allowing a semiquantitative comparison, even though the full isotherm shape could not be reproduced in detail. With a choice of energy parameters in the model description that allows for the frequent exchange of chains between the surface and the bulk phase during the simulation time, the calculated adsorption isotherms under good solvent conditions (athermal limit) have a convex shape that is qualitatively comparable to the SCF model of Ligoure and Leibler.17 Upon increasing the degree of polymerization, a decrease in the number of adsorbed chains was obtained in the simulations because of the larger steric repulsions. Under this condition, at a very small surface concentration the isolated adsorbed coils characterize the so-called mushroom regime and the calculated adsorption isotherms have a linear dependence on the total concentration, whereas afterwards the slope of the isotherms decreases monotonically with increasing concentration, corresponding to the brush regime due to both steric and electrostatic repulsive effects among neighboring adsorbed molecules. The nonbonded Lennard-Jones potential with a lower cutoff was also used to model chains in a solvent of lower quality, thus producing a weakly attractive potential among the beads. As a consequence, the calculated adsorption isotherms had a greater adsorbed amount, in particular, at a large total polymer concentration but with an unchanged initial slope related only to the parameters of the electrostatic energy between the surface and the charged end beads. A somewhat poorer solvent quality also increases the linear Henry’s region of the adsorption isotherms because of the smaller chain size and the attractive interactions between the adsorbed chains. The effect of the electrostatic screening due to the ionic strength of the solution, as typically found in the cement paste environment, was also investigated by changing the Debye length of the Debye-H€uckel potential while keeping the other parameters fixed. A decrease in the Debye length, corresponding to an increase in the ionic strength, leads to a decrease in the adsorbed amount at all concentrations as a result of the decrease in the range of the electrostatic interaction between the surface and the chains in solution. Even though no detailed parameter fitting was actually carried out, the best agreement between the experimental and the calculated adsorption isotherms was qualitatively found with chains of 10 beads with De = 1.0 and an energy parameter of the Lennard-Jones potential of ξ = 1.0 in the region across the mushroom-brush transition. Apart from the semiquantitative Langmuir 2010, 26(20), 15814–15823
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fitting of the experimental data, the calculated adsorption isotherms bring out the relationship between the conformational properties of the adsorbed chains in the mushroom or brush regime and the shape of the adsorption isotherms, considering the influence of the bead-surface and bead-bead interactions. Further investigation along these lines could consider polyelectrolyte chains to unravel the possible cooperative effect on adsorption phenomena and polymers with more complicated architectures (combs, stars, irregularly branched polymers, and
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copolymers) to assess the influence of the chain topology on the adsorption isotherms. Acknowledgment. We gratefully thank Dr. E. G. Timoshenko and Yu. A. Kuznetsov for the use of their Monte Carlo code, which formed the basic simulations engine needed for this work, and the CILEA consortium for computer time and technical support. We also thank the CTG Italcementi group for financial support.
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