MATERIALES Y MÉTODOS
ENSAYOS CONDUCTUALES
The interactions between surfaces in aqueous environments are complicated by the presence of water molecules and electrolytes. Suspended and surface-absorbed macromolecules in biological fluids add additional contributions to the effective tractions between adhering surfaces. A brief summary of the primary interactions involved in adhesion of cells and lipid membranes is given in Table 2.3. The long-range contributions are dominated by electrostatic interactions, van der Waals forces, and hydrophobic interactions. In some cases, thermally driven membrane undulations contribute an important repulsive contribution to the effect interactions. Entropic forces associated with confining the interfacial medium between the adhering surfaces usually dominate at close separations. That medium can either be surface-absorbed macromolecules (steric forces), suspended macromolecules (depletion forces), or the water itself (solvation forces). The effective tractions can be estimated by superposition of the relevant contributions. This section focuses on van der Waals interactions combined with electrostatic interactions in solution (DLVO theory), although other forces summarized in Table 2.3 are discussed throughout.
Electrolytes can induce or alter surface charge interactions when bodies adhere in an aqueous environment. Surface charging mechanisms include surface ionization, ion
absorption, and ion dissolution. The presence of surface charge alters the ion distribution in the surrounding solution; oppositely charged ions (counterions) collect at the surface, whereas ions with the same charge (coions) are dispelled from the surface region. The equilibrium distribution of ions is driven by minimization of the entropic mixing energy of coions and counterions and the potential energy associated with maintaining charge at a fixed distance z from the surface. The so-called “double-layer” interactions depend on
geometry, the electrolyte concentration, temperature, and separation. These interactions are largely independent of van der Waals interactions (Leckband and Israelachvili, 2001).
The repulsive double-layer interaction potential is generally expressed in the form:
dl dle z l
u =C − , (2.23)
where Cdl is a constant of proportionality that depends on geometry and l is the Debye
length (Leckband and Israelachvili, 2001). The later depends on the type and concentration of electrolyte in the environment and on temperature. For example, l =
0.8 nm at physiological conditions, i.e. NaCl concentration of 0.15 M
(
moles liter)
and a temperature of 310 K. The constant Cdl is determined from solution of a 2nd order,nonlinear differential equation known as the Poisson-Boltzmann equation (Israelachvilli, 1985; Leckband and Israelachvili, 2001). Analytic solutions are only possible for simple geometries with simplifying assumptions.
For two flat surfaces Cdl is given in units of energy per unit area by:
dl 2
C =
χ π
l . (2.24)The constant χ is defined in terms of the dielectric constant
ε
of the aqueousconstant k =1.38 10× −23J K, temperature T, the electron charge e= −1.60 10× −19C, the electrolyte valance
υ
, and the surface potential ψo:(
)
2 2(
)
o o b
64 kT e tanh e 4k T
χ
=π ε ε
υ ψ
. (2.25)The potential for flat, parallel surfaces given in (2.23) - (2.25) assumes a 1:1 electrolyte (e.g., NaCl), a weak electrostatic potential at the midpoint between the surfaces
(
z 2)
,and separations larger than about one Debye length l (Israelachvilli, 1985).
For a 1:1 electrolyte such as NaCl, with concentration denoted ρNaCl, the surface
charge
τ
of identical surfaces is related to the surface potential ψo by the Graham equation (Israelachvilli, 1985; Leckband and Israelachvili, 2001):(
)
o NaCl o
8 kT sinh e 2kT
τ = εε ρ ψ , (2.26)
which assumes the separation is far enough that the surface potential at the midpoint between the surfaces (z=0) is small. The later assumption is consistent with (2.25). As defined, ψo is called the Zeta surface potential to distinguish it from the total surface potential that would be measured in experiments. The other contribution to the potential is associated with a thin layer of ions bound to the material surface. The thickness of this thin layer, called the Stern layer, is often equated with the shear plane or “no-slip” plane in bulk flow experiments. Physically, the potential ψo measures the potential of “free” ions in the solution several molecular layers from the material surface.
The cumulative effect of van der Waals attraction and double-layer repulsion is described by superposition of the corresponding potentials, which results in the well- know Derjaguin,-Landau-Verwey-Overbeck (DLVO) theory. For flat, parallel plates the effective potential is given by:
(
)
2 o 2 o 2 32 tanh e 4 12 z l kT e e A u l kT z ε ε υ ψ π − = − + , (2.27)where the first term is taken from the van der Waals contribution in (2.11) and the second term is given from the double-layer interactions (2.23) - (2.25). The surface tractions are calculated in the usual way,
σ
=d du z. This potential is strictly only valid for moderate to large separations because of assumptions in (2.23) - (2.25). Additionally, repulsive solvation, depletion, and steric forces dominates the interactions at small separations (see Table 2.3), but these forces are not accounted for in (2.27) (Israelachvilli, 1985; Leckband and Israelachvili, 2001).The potential energy density (2.27) and corresponding tractions are plotted in Fig.
2.4 for T = 310K, A = 10−20J,
ε
=74, l= 0.8nm, υ= 1 (1:1 electrolyte), and severalvalues of ψo. The surface charge is related to the surface potential via (2.26). These
parameters are representative of adhesion between lipid surfaces (A ≈10−20J for hydrocarbons) under physiological conditions. The large repulsive barrier that occurs for moderate separations and moderate surface potentials is depicted in Fig. 2.4a,b. As mentioned above, the attractive van der Waals forces are balanced by other repulsive interactions at close separations that are not accounted for in the figure. The effect of including these repulsive short-range interactions is the development of a primary energy minimum at close separations. A relatively low-energy, stable equilibrium occurs outside of the repulsive barrier (see Figs. 2.4c,d). The energy density and tractions are 1000∼ ×
smaller than occurs for the repulsive barrier, whereas the equilibrium separation is more than 10× larger than occurs for molecular contact. The repulsive stresses are on the order MPa for physiologically relevant properties, which is extremely large considering
the Young’s modulus of a lipid bilayer is ∼8MPa (as estimated in Springman and Bassani, 2008). Equilibrium of adhered vesicles interacting only through van der Waals and double-layer interactions is, therefore, expected to occur at forces and separations characteristic of these secondary minima.
Attractive hydrophobic interactions between hydrocarbon surfaces can alter the adhesive interactions discussed above (see Table 2.3). The origin of these forces is not electrostatic, but is due to the high surface energies associated with hydrocarbon-water interfaces. Hydrophobic attraction is long-range and typically stronger than van der Waals attraction. Including this additional force in (2.27) increases the magnitude of the secondary energy minima and tractions relative to Fig. 2.4c, and also decreases the equilibrium separation. Additionally, attractive hydrophobic interactions can weaken the repulsive barrier, shifting equilibrium to the primary energy minima that occur at close separations.
The forces involved in the DLVO interaction potential described in this section are almost always important in adhesion of surfaces in aqueous environments. This potential generally has two equilibria separated by a repulsive barrier. The primary equilibrium occurs at relatively close separations, but is unlikely to be obtained in soft systems due to the large repulsive forces at moderate separations. However, the effective tractions can be altered by other physical forces discussed above and summarized in Table 2.3. Any effective potential can be characterized by their load maxima, their equilibrium separations, and the work required to separate the surfaces. These parameters depend, for example, on electrolyte concentrations, properties of absorbed and suspended proteins, temperature, and properties of the surfaces themselves. Relatively simple analytic
expressions used as approximate adhesion laws have been substantiated by experimental measurements (Israelachvilli, 1985; Leckband and Israelachvili, 2001).