If we assume that the potential barrier determining ka does not change with the accumulation of material, then ka merely acts as a linear scaling factor on the adsorption rate (Eq. 35), and does not affect the form of the rate law. In contrast, the function 0 affects its shape, and the main goal of many quantitative adsorption studies is to identify the function 0 and correlate it with the molecular properties of the system under study

It is obvious that 0 = 1 for a totally empty surface with Q = 0. Q is the fraction of the surface covered by adsorbed particles, and is related to v and total adsorbed mass M by

where a is the occupied area per particle and m its mass. The analysis of high-resolution kinetics (such as those obtainable with OWLS) allows a to be determined with an accuracy equal to or better than with atomic force microscopy. A totally full surface (Q = 1) will have 0 = 0. What is less obvious is that a surface may be less than fully occupied (Q < 1) but nevertheless may not be able to offer any free place to an incoming particle. Even polygons that can tile the plane will only reach Q = 1 if they are perfectly aligned. If they are placed randomly, 0 will already approach zero for Q as little as 0.5.

Given that 0 is a function of Q, we can quite generally expand it in powers of Q:

The so-called random sequential adsorption (RSA) has been analysed extensively in two dimensions (e. g. Evans 1993) from purely geometrical considerations. The coefficients bi, b2, and b3 have been determined for

Mode |
bi |
b2 |
b3 |
Application |

Langmuir |
1 |
0 |
0 |
Clustering, large receptors |

RSA (spheres) |
4 |
3.808 |
1.407 |
Irreversible adsorption |

GBDa |
4(1 |
- j) 3.808 - |
1.407+ 4.679j - |
Nucleation and growth |

0.180/ - 3.128j2 |
25.58j2 + 8.550j3 |

a j is the ratio of probabilities p/p', p' being the probability of a particle adsorbing on a hitherto unoccupied patch of the absorbent at which it arrives, and p the probability that it adsorbs afterhaving reached a space large enough to accommodate it byapath of correlated lateral diffusion in the immediate vicinity of previously adsorbed particles. In the former case a protein arrives at an empty patch of surface, whereas in the latter case it arrives at a region where previously arrived particles are already adsorbed, and migrates to the edge of the cluster before becoming attached to the surface. j = 0, the lowest possible value, would correspond to pure random sequential adsorption (i. e. binding takes place solely independently of preadsorbed particles at empty patches of surface), and higher values correspond to increasingly favoured particle clustering.

a j is the ratio of probabilities p/p', p' being the probability of a particle adsorbing on a hitherto unoccupied patch of the absorbent at which it arrives, and p the probability that it adsorbs afterhaving reached a space large enough to accommodate it byapath of correlated lateral diffusion in the immediate vicinity of previously adsorbed particles. In the former case a protein arrives at an empty patch of surface, whereas in the latter case it arrives at a region where previously arrived particles are already adsorbed, and migrates to the edge of the cluster before becoming attached to the surface. j = 0, the lowest possible value, would correspond to pure random sequential adsorption (i. e. binding takes place solely independently of preadsorbed particles at empty patches of surface), and higher values correspond to increasingly favoured particle clustering.

spheres and ellipsoids (ellipsoids with an aspect ratio of 1:4 have virtually identical coefficients to those of spheres (Viot et al. 1992), for irreversible and reversible deposition, and deposition of laterally mobile particles (reversibility and mobility only affect b3), etc. Experimental limitations make it unnecessary to expand 0 beyond the third power of Q. Table 5 gives some values of the coefficients.

Practically, by plotting the rate of adsorption against the amount adsorbed, i. e. in direct accordance with Eq. 35, it is often possible to identify the adsorption mechanism merely by visual inspection. Four principle types of behaviour are observed:

1. dv/dt ~ constant implies that 0 = 1. This is characteristic of the formation of isolated aligned chains of adsorbate (e.g. lysozyme dissolved at low ionic strength (Ball and Ramsden (1997), with images given in Ramsden (1998));

2. dv/dt concave (i. e. progressively slower) implies that 0(Q) is a characteristic polynomial function of Q, implying that the proteins interact via excluded volume only, i. e. "pure" random sequential adsorption (e. g. common blood proteins such as transferrin and serum albumin (Rams-den 1993c, Kurrat et al. 1994));

3. dv/dt ~ linear, implying that 0 = 1 - Q (Langmuir adsorption): for adsorption onto a continuum, however, the annihilation of exclusion zones implied by this linear relation between 0 and Q can only occur upon 2D clustering or crystallization (e. g. cytochrome P450 added to a membrane (Ramsden et al. 1994));

4. dv/dt concave/convex implies generalized ballistic deposition (GBD) (e.g. phospholipase A2 (CsĂșcs and Ramsden 1998b));

5. dv/dt bell-shaped implies nucleation and growth of deposits (e. g. lyso-zyme on silica in the presence of thiocyanate ions (SCN-) (Ball et al. 1999).

Equation 35 with the appropriate expression for 0 can be fitted to numerically differentiated data, or the numerically integrated version of Eq. 35 fitted to the (M, t) data, by a least-squares optimization procedure, with free parameters a, ka (and j for a GBD process, see Table 5), and kd for reversible adsorption. If there is a desorption phase, it is more robust to globally fit both adsorption and desorption to Eq. 35 with an appropriate change of boundary conditions at the moment of flooding.6

Experimental values of the jamming limit Qj (i. e. the value of Q when 0 = 1) can be compared with theoretical values for spheres (Schaaf and Talbot 1989), and spherocylinders and ellipsoids (Viot et al. 1992) in order to deduce the shape of the molecule. When Q is very close to Qj, the final approach to jamming follows a power law whose exponent depends on the number of degrees of freedom of the adsorbing particle. Ellipsoids are specified by both position and orientation, i. e. they have one more degree of freedom than spheres, but the experimental data is usually too noisy to permit determination of the exponent sufficiently precisely to enable the number of degrees of freedom to be unambiguously established.

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