ABSTRACT
I. Introduction ................................................................ 351 II. Homopolymers in Solution......................................... 354
A. Dilute solutions ................................................... 355 B. Semidilute Solutions ........................................... 358 C. Mean-Field Picture of Semidilute
Solutions .............................................................. 360 III. Homopolymers at an Isolated Surface....................... 361
A. General Aspects .................................................. 361 B. Proximal, Central, and Distal Parts in a
Scaling Picture .................................................... 364 C. SCF modeling ...................................................... 366
1. The Scheutjens-Fleer Discretization Scheme .......................................................... 367
2. From the Potential Fields to the Concentrations .............................................. 367
3. From the Concentrations to the Potential Field............................................... 372
4. Fixed Point of SCF Equations ...................... 374 5. Critical Adsorption Energy........................... 375 6. Thermodynamics........................................... 377
D. The Equivalence of the Propagator and the Edwards Equation............................................... 380
E. Analytical Descriptions for Homopolymers Near a Single Surface ......................................... 381 1. General .......................................................... 381 2. Exact Solutions for u¼ 0............................... 384 3. Ground-State Approximation (GSA) ............ 388 4. Depletion ....................................................... 391 5. Adsorption from semidilute
good solvents ................................................. 395 6. Adsorption from Dilute Solutions ................ 399
F. Numerical Results ............................................... 408 IV. Colloidal Stability ....................................................... 417
A. General Aspects................................................... 417 B. Concentration Profiles in the Gap...................... 419
1. Semidilute Solutions..................................... 420 2. Dilute Solutions ............................................ 421
C. Grand Potential and Gibbs Energy of Interaction ........................................................... 422
D. Depletion.............................................................. 423 1. The Adsorption Route to Calculate V .......... 423 2. The Gibbs Energy of Interaction.................. 425
E. Adsorbing polymer .............................................. 429 1. The Disjoining Pressure ............................... 429 2. Attraction Due to Bridge Formation............ 432 3. Repulsion Due to Tails.................................. 434 4. Total Gibbs Energy of Interaction................ 437 5. Numerical examples of Ga(h) ....................... 438 6. The Gibbs Energy of Interaction
at Contact ...................................................... 445
V. Concluding Remarks .................................................... 447 References............................................................................ 450
I. INTRODUCTION
Theoretical insights into the behavior of polymers at interfaces are to a large extent based on a self-consistent field (SCF) analysis of the problem. The basic equation that describes the interfacial layer is the Edwards equation [1-4], which considers the chains as walks where each step is weighted in a self-consistent potential field, wherein the excluded volume of the chains and nonideal interactions between the segments are (to first order) accounted for. The presence of the surface is incorporated essentially through the boundary condition. We have to mention, however, that in this approach the chains in the bulk are ideal (Gaussian) and do not change their conformation due to the presence of excluded-volume interactions; the swelling in good solvents is disregarded. Consequently, SCF-results for polymers at interfaces are only approximate. For example, the adsorbed amount is systematically underestimated slightly. Apart from this inherent drawback, mean-field results are internally consistent and thermodynamically sound. They allow for interpretations in terms of simple scaling behavior, which can be supplemented with the numerical prefactors. The predictions can readily be compared to the experimental results. Therefore it is worthwhile to investigate the problem in this simple scheme and consider the various intricacies. The results may, in the end, be extrapolated to mimic real systems by exchanging the mean-field power laws by scaling laws consistent with chain swelling. By doing so, we lose information on the numerical prefactors, however.
