Elementos Primarios de Temperatura

10-6 10-4 10-2 100 10-8 2 3 κ ∗ 0 ξ∗_η∗ 0 Ha (a) 10-6 10-4 10-2 100 10-6 10-4 10-2 100 10-8 2 3 κ ∗ 0 ξ∗_η∗ 0 Bo (b) Figure 2.6: Peak curvature κ∗

0as a function of ξ∗η∗0obtained (a) for different modified Hamaker numbers

in the range Ha ∈ 10−8,10−1 and a fixed Bond number of Bo = 10−10, and (b) for different Bond

numbers in the range Bo ∈ 10−11,10−2 and a fixed Ha = 10−3. Numerical solution of eq.(2.1) for

[•] the stable and [⋄] the unstable branches, and the analytical solution for [ ] the stable and [ ] the unstable branches given by eqs.(2.7) and (2.13). [ • ] κ∗

0,crit is obtained from the evaluation

in eq.(2.13) with ξ∗

critη0,max∗ , in turn obtained when solving eq.(2.14), and the values of Clk shown in

Fig.2.7. Arrows indicate the growth of the corresponding parameter. of D∗

min, whereas enlarging Bomoves D∗minslightly toward smaller values, as illustrated in Figs.2.5a and

2.5b, respectively. In order to find the bifurcation point, we have implemented a dichotomy method, which allows to control the trueness of the predicted value of D∗

min. In this study, D∗minis forecast with

an error smaller than 10−3.

As shown in Fig.2.5c, the value of η∗

0,max decreases slowly when increasing the magnitude of Ha,

from an average value of η∗

0,min ≈ 0.315 in the range Ha ∈10−8,10−4 toward a value slightly below

η∗

0,min ≈ 0.3 for Ha = 10−1. In contrast, indicated in Fig.2.5d, η0,max∗ remains constant for the tested

values of Bo. It is important to mention that the employed methodology is based on a D∗min accuracy

control, giving, as a consequence, a bigger error in the calculation of η∗

0,max. Herein, the prediction of

η∗

0,max can present an error smaller than 7.3 ×10−2, apart form the value obtained for Ha= 10−8, which

is of the order of 10−1_.

2.2 Surface apex behavior

When analyzing the apex behavior for a given combination of Haand Bo, it is found that κ∗0grows along

with ξ∗_η∗

0, which in turn becomes larger when D∗ decreases, as observed in Fig.2.6. The peak curvature

κ∗

0, specially when it approaches its critical value κ∗0,crit, shows a remarkable simple dependency on the

reduced deformation ξ∗_η∗

0, which is given by:

κ∗₀∝ (ξ∗_η∗

0)3/2. (2.5)

From fig.2.6, a horizontal logarithmic displacement is produced when modifying the dimensionless num- bers: significantly toward larger values of ξ∗_η∗

0 when Ha increases; slightly to smaller values of η0∗ when

Bo grows. Whereas ξ∗η∗0 approaches ξ∗critη∗0,max, κ∗0 reaches its critical value κ∗0,crit≈ 3.5 × 10−2, which

stays nearly constant for small values of Ha. When increasing Ha, the critical curvature slowly decreases.

For Ha = 10−1, its value is slightly below κ∗_0,crit≈ 2.9 × 10−2. When increasing Bo, κ∗_0,crit increases

significantly, but barely reaching κ∗

0,crit≈ 1.0 × 10−1.

For the case of local probes, R is always much smaller than λC, and as a consequence Bo≪ 1. As

a consequence, the hydrostatic term is negligible compared to that of the interaction terms near the symmetry axis, which implies that the surface deformation, and more precisely the apex deformation

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 0.1 0.2 0.3 0.4 0.5 0.6 Clk Ha (a) 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 0.1 0.2 0.3 0.4 0.5 0.6 Clk Bo (b)

Figure 2.7: Bulk proportionality constant Clk as a function (a) of the modified Hamaker number Ha for

a fixed Bond number of Bo= 10−10, and (b) of the Bond number Bofor a fixed Ha = 10−3. The values

of Clk were obtained by fitting the data shown in Fig.2.6 in (2.13). [◦] Numerical solution and [ ] the

average value Clk ≃ 1.54 × 10−1± 1.58 × 10−2.

η∗

0, is mostly controlled by the balance between the probe/liquid interaction potential and the capillary

force. Thence, at r∗= 0, eq.(2.1) can be reduced to:

Π∗ pl    ₀= −2κ ∗ 0 Ha , (2.6)

Besides, the position of the sphere center can be written as a function of the apex product ξ∗_η∗ 0 and

the peak curvature κ∗

0. Solving eq.(1.17b) for the dimensionless probe position D∗ at r∗= 0, and using

eq.(2.6), yields the following relationship:

D∗= ξ∗η∗₀+ s 1 + 2κ∗0 Ha −1/3 . (2.7)

Also, from the geometry shown in Fig.1.4, an expression for D∗ _{is inferred in terms of the the apex}

position η∗

0, the initial gap ξ∗and the reduced equilibrium gap ε∗0:

D∗= 1 + ξ∗η₀∗+ ξ∗ε∗₀, (2.8)

which also leads to the relationship:

ε∗₀+ η₀∗= 1,

which results from the dimensionless definition of the initial gap, ξ∗_{= D}∗_{−1. As a consequence, because}

η∗_{≥ 0 and ε}∗_{≥ 0, the two variables are confined in the range [0, 1].}

A comparison between eq.(2.8) and the first-order approximation of eq.(2.7) (see Appendix D), gives the following expression of the peak curvature in terms of the initial and equilibrium reduced gaps:

κ∗₀= Ha

16 (ξ∗_ε∗ 0)3

. (2.9)

Following an equivalent approach, if we consider that the force due to the surface deformation can be taken as a linear function of the apex position:

Fγ ∼ πγ (Rξ∗η∗₀) , (2.10)

2.2. SURFACE APEX BEHAVIOR 35 force between a sphere and a flat surface (Israelachvili, 2011):

Fpl∼ HplR

6 (Rξ∗_ǫ∗ 0)2

, (2.11)

the equilibrium between these two forces, including a proportionality constant Clk, allows to obtain a

relationship between the apex position and the reduced gaps:

η∗₀= 1 Clk Ha 8ξ∗(ξ∗_ε∗ 0)2 , (2.12)

which, in combination with eq.(2.9) gives a power law relationship between the apex variables:

κ∗₀= (Clk)3/2r 2

Ha(ξ

∗_η∗

0)3/2. (2.13)

Eq.(2.13) is in agreement with the behavior observed in Fig.2.6, specifically near the critical conditions, and has been previously mentioned in eq.(2.5).

The proportionality constant Clk is directly obtained by fitting the data shown in Fig.2.6 with

eq.(2.13). As shown in Fig.2.7, the dependence of Clk on Ha is negligible in comparison with its submis-

sion to the value of Bo. For any Ha, the proportionality constant shows an average value with a sparse

deviation, Clk ≃ 1.54 × 10−1± 1.58 × 10−2, whereas, it is undeniably a function that grows along with

the magnitude of Bo.

Combining eqs.(2.7) and (2.13), gives the dependency of D∗ _{on ξ}∗_η∗

0, leading to the bifurcation

diagrams construction. The resulting relationship supplies the two physically possible solutions of eq.2.1, stable and unstable branches. Bifurcation curves obtained with this analytical approach are also shown in Fig.2.2a, for a fixed Bo = 10−10 and a range Ha ∈ 10−8,10−1, and in Fig.2.2b, reckoned with

Ha = 10−3and Bo∈10−11,10−2. Furthermore, the corresponding derivative of η∗₀ with respect to D∗,

as well as the peak curvature κ∗

0, are compared to the numerical results in Fig.2.3 and Fig.2.4, respectively.

In all cases, a very good accordance with the numerical solution is observed at the bifurcation points and the nearby region.

As it has already been mentioned, the value of D∗

minindicates the minimum axial position, at which

the sphere can be located, to prevent the liquid from jumping to contact the sphere. This particular situation corresponds to an unbound magnitude of the probe/liquid interaction potential. In addition, this minimum distance D∗

min provokes the critical initial gap ξcrit∗ , the maximum apex deformation

η∗

0,max and the critical peak curvature κ∗0,crit, which represent the critical stable conditions of the surface

deformation. As well, as it is observed from the bifurcation diagrams, at D∗

minthe apex position diverges

dη∗

0/dD∗→ ∞ and, as a consequence, one can make the derivative dD∗/dη0∗= 0.

It is well known that the system stability is deeply associated with the stationary points of the interaction potentials (Ruckenstein and Jain, 1973; Forcada, 1993; Christenson, 1994). Thus, consistent with the aforementioned hypothesis of the apex position divergence, differentiation of eq.(2.7), once again combined with eq.(2.13), gives rise to the polynomial equation:

ξcrit∗ η∗0,max
3_{+ Θ ξ∗} critη∗0,max
5/2 − Θ₄ 2 = 0, (2.14)

where Θ = pHa/2Clk. The roots of this polynomial are meant to be found in order to estimate the

critical apex product ξ∗

minη0,max∗ of the maximum apex deformation and the critical initial gap. The

acquired value is used in eq.(2.13) to determine the peak curvature. Afterwards, the minimum separation distance D∗

min is obtained when eq.(2.7) is evaluated. Then, the critical initial gap is given by:

ξcrit∗ = D∗min− 1, (2.15)

and the maximum apex position can be finally deduced. Both D∗

min and η∗0,max were analytically

acquired for the range Ha ∈10−8,10−1 and shown in Fig.2.5. Since eq.(2.14) does not depends on Bo,

the reference value for Ha = 10−3 and Bo = 10−10 is shown in the figure. Nevertheless, this constant

value is close to the numerically calculated values of D∗

min and η∗0,max, because their dependence on Bo

10-2 100 102 104 106 10-20 10-15 10-10 10-5 100 |X p | r∗ Boξ∗η∗ 2κ∗ HaΠ∗pl λ∗ H λ∗T λ∗C N T F (a) 10-2 100 102 104 106 10-20 10-15 10-10 10-5 100 |Yp | r∗ κ∗a κ∗ m λ∗ H λ∗T λ∗C N T F (b)

Figure 2.8: Characteristic length-scales determination. Different terms (a) Xp from eq.(2.1), for which

P_X

p = 0, and (b) Yp from eq.(1.23), for which P Yp = 2κ∗, as functions of r∗, for Ha = 10−3,

Bo = 10−10 and D∗ = Dmin∗ = 1.168. The upper-case letters N, T and F designate the near-field,

transition and far-field zones, respectively, which extents are bounded by the characteristic length-scales

λ∗

H, λ∗T and λ∗C.

depict the correct trend of D∗

min and η0,max∗ , in agreement with the numerical results. Further details

on this analysis are given in Appendix B.