The name is absent

surface S' (Figure 2.1, Top). Thus, the total force applied on the tip is given as the surface
integral

F = ∫T ∙ ndS          (2.9)

s

where n is a unit normal vector pointing into the surface S,

_ r'(z')z-r           (2.10)

∏ = ~~) ~~~^........-

ʌ/l + r^,2

in which r is the tip curvature given as

r = y∣radius² - z²             (2∙ 11 )

T is the total stress tensor [50, 53]:

T = (∏ + ɪ ¾E ∙ E)I - ¾EE     (2.12)

in which ∏ is the osmotic pressure term

∏ = 2n₀k_BT(cosh(ey//k_BT)-ï) (2.13)

I is the unit dyadic, E is local electric field vector. The tip-sample force measured by

the AFM can be described as the z component of Equation 2.9.

⅞ л г 1                      -∣             ʌ

F. = [ r' TI +-εε_nE² - εε_nEE +εε_aEE_r 2πrdz (2.14)
ZJ    ɔ U   UZ  Ur    Xz

Λ L ²             J ;

Equation 2.14 was numerically calculated based on the electrostatic field values and
potentials exported from the simulation. Zi and Z2 are the z-axis limits of the sphere. By
changing the tip-sample separation, force curves were simulated. These curves were
compared to the silicon nitride reference data, and σ_tip and σ_sampι_e were adjusted to
achieve a good match (Figure 2.3). Once the tip charge density was known, the same
procedure was carried out on the lipid data to determine the long-range fit.

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