The Grahame equation

In many cases we have an idea about the number of charged groups on a surface. Then we might want to know the potential. The question is how are surface charge a and surface potential V o related This question is also important because if we know a fo) we can calculate da/dtpo- This is basically the capacitance of the double layer and can be measured. The measured capacitance can be compared to the theoretical result to verify the whole theory. 

Grahame derived an equation between a and based on the Gouy-Chapman theory. We can deduce the equation easily from the so-called electroneutrality condition. This condition demands that the total charge, i.e. the surface charge plus the charge of the ions in the whole double layer, must be zero. The total charge in the double layer is /0°° pe dx and we get [59] 

Using the one-dimensional Poisson equation and the fact that at large distances the potential, and thus its gradient, are zero (dtp/dx z=00 = 0) we get 

For low potentials we can expand sinh into a series (sinh x = x, t3/3 +. ..) and ignore all but the first term. That leads to the simple relationship 

With a Debye length of 3.04 nm at 25°C the surface charge density in SI units is a = 1.60 x 10 19As/16 x 10-18m2 = 0.01 Asm-2. With this we get 

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Equation (69), known as the Grahame equation, describes the variation of charge density with potential at the surface with no limitations as to the value of the potential. Example 11.2 considers an application of this relationship. 

In order to estimate the surface charge we use the Grahame equation a = a/8 c eeff T-smh ertl)o/2kBT). With... 

The Gouy-Chapman portion of the model is expressed in the Muller equation, which is commonly but erroneously referred to as the Grahame equation (Tatulian, 1999),... 

For the flat surfaces of a nanochannel, the surface charge density can be approximated using the Grahame equation. Especially, for the monovalent electrolyte solution, the surface charge density can be simply described by... 

The surface potential is related to the surface charge density through the charge balance, also known as the Grahame equation ... 

Once the zeta potential has been found, it can be used to calculate the surface charge density using the Grahame equation. 

Why do alkyltrimethylammonium surfactants form roughly spherical aggregates on silica but cylindrical aggregates on mica The answer probably lies in the far greater density of electrostatic adsorption sites available on mica. Zeta potential measurements [40] in 1 mM KCl have shown that at pH 6 (a typical imaging condition), the surface potentials % of silica and mica are —80 and —120 mV, respectively. Using the Grahame equation... 

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

It is also important to note here that the first integral of the Poisson-Boltzmann equation leads to an important relationship, known as the Grahame equation, and also as the Contact theorem [5]. This, fundamentally, is a relationship between the surface charge density, ctq (which is defined as cTo = / Pedy, with a SI unit of C/rc ) and the limiting value of the ionic density profile at the substrate-fluid interface. For a single flat surface with an infinite extent of... 

The above equation is a general result, known as the Grahame equation, and can be used to assess the accumulation of various ions at a charged interface essentially as a monolayer. 

In nonpolar media, less charge is needed to reach a certain surface potential. Quantitatively, this is deduced from the Grahame equation (4.32). At low potentials, the surface potential is given by Eq. (4.33) ... 

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