Surface Complexation Models Statistical Mechanics

The adsorption of soluble polymers at solid-liquid interfaces is a highly complex phenomenon with vast numbers of possible configurations of the molecules at the surface. Previous analyses of polymer adsorption have ranged in sophistication from very simple applications of "standard" models derived for small molecules, to detailed statistical mechanical treatments of the process. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

Statistical mechanics, the science that should yield parameters like A/x , is hampered by the multibody complexity of molecular interactions in condensed phases and by the failure of quantum mechanics to provide accurate interaction potentials between molecules. Because pure theory is impractical, progress in understanding and describing molecular equilibrium between phases requires a combination of careful experimental measurements and correlations by means of empirical equations and approximate theories. The most comprehensive approximate theory available for describing the distribution of solute between phases—including liquids, gases, supercritical fluids, surfaces, and bonded surface phases—is based on a lattice model developed by Martire and co-workers [12, 13]. 

Previous works dealing with disordered surfaces have been dedicated mainly to random, or correlated topographies. In the latter case, the combination of heterogeneity and ad-ad interactions effects produce complex behaviors on the equilibrium properties. An exact statistical mechanical treatment is unfortunately not yet available and, therefore, the theoretical description of adsorption has relied on simplified models. One way of circumventing this complication is the Monte Cado (MC) method, which has demonstrated to be a valuable tool to study surface processes [3,4],... 

The Transition State Model was an attempt to overcome the several shortcomings of the Thermodynamics, the Kinetics and the Statistical Mechanics Models, by forging a strong and precise link between thermodynamics and kinetic variables. By considering a chemical reaction as a process in which a system passed over the top of the energy barrier between the initial and the final states, this model proposed that rate could be calculated by focusing attention on the molecular complexes at the col of the surface. Moreover, the application of statistical methods made possible the development of equations that related the concentration of the species involved to the rate of a reaction. 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The alternative to analytical theories are simulations computer experiments that generate representative statistical samples of the system from which macroscopic properties can be derived. The theory used to generate simulations is much simpler than the statistical mechanical theories of liquids. Nevertheless the results can be far more accurate and less assumptions go into the derivation of the results. Computer simulations are thus often used to test the reliability of analytical theories, using model potentials. Using realistic potentials, computer simulations can be used to understand and predict properties of large systems of complex molecules. This is not restricted to simple molecular fluids but may extend to complicated macromolecules, liquid crystals, micels, surfaces and chemical reactions. 

The other model of the adsorption system, the so-caUed three-dimensional model, seems to be more realistic and promising. In this model, we do not assume the existence of a distinct surface phase but consider the problem of a fluid in an external force field. From a theoretical point of view a solution of this task requires only knowledge of the adsorbate-adsorbate and adsorbate-adsorbent interactions. However, in practice, we encounter difficulties connected with the mathematical complexity of the derived equations. The majority of the statistical-mechanical theories of nonuniform fluids have been formulated for adsorption on homogeneous siufaces. Nevertheless, the recent results obtained for heterogeneous sohds are really interesting and valuable [16]. 

A new opportunity, which creates good prospects for avoiding many problems eoimected with the theoretical description of adsorption on heterogeneous surfaces, has appeared as a result of the introduction of computer simulation methods [17,18]. Over the last three deeades, computer simulations have grown into a third fundamental discipline of research in addition to experiment and theory. The study of adsorption on heterogeneous solid surfaces has especially benefited fi om the molecular simulation method, first of aU, because of the complexity of interactions of adsorbate molecules with differently distributed active centers that are not easily described by the methods of statistical mechanics. Computer simulation can, in principle at least, provide an exact solution of the assumed model. 

This book reviews the statistical mechanics concepts and tools necessary for the study of structure formation processes in macromolecular systems that are essentially influenced by finite-size and surface effects. Readers are introduced to molecular modeling approaches, advanced Monte Carlo simulation techniques, and systematic statistical analyses of numerical data. Apphcations to folding, aggregation, and substrate adsorption processes of polymers and proteins are discussed in great detail. Particular emphasis is placed on the reduction of complexity by coarse-grained modeling, which allows for the efficient, systematic investigation of stractural phases and transitions. 

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