Accepted Manuscript Detailed Statistical Contact Angle Analyses; ”slow moving” drops on inclining silicon-oxide surfaces M. Schmitt, K. Groß, J. Grub, F. Heib PII: DOI: Reference:

S0021-9797(14)00812-1 http://dx.doi.org/10.1016/j.jcis.2014.10.047 YJCIS 19941

To appear in:

Journal of Colloid and Interface Science

Received Date: Accepted Date:

29 September 2014 15 October 2014

Please cite this article as: M. Schmitt, K. Groß, J. Grub, F. Heib, Detailed Statistical Contact Angle Analyses; ”slow moving” drops on inclining silicon-oxide surfaces, Journal of Colloid and Interface Science (2014), doi: http:// dx.doi.org/10.1016/j.jcis.2014.10.047

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Detailed Statistical Contact Angle Analyses; ”slow moving” drops on inclining silicon-oxide surfaces

M. Schmitt*a, K. Großb , J. Grubc, F. Heibd

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Physical Chemistry, Saarland University, 66123 Saarbrücken, Germany E-Mail: [email protected]

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Keywords: sessile drop, silicon wafer surface, statistical analysis, contact angle, inclining

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1

Abstract

Contact angle determination by sessile drop technique is essential to characterise surface properties in science and in industry. Different specific angles can be observed on every solid which are correlated with the advancing or the receding of the triple line. Different procedures and definitions for the determination of specific angles exist which are often not comprehensible or reproducible. Therefore one of most important things in this area is to build standard, reproducible and valid methods for determining advancing/receding contact angles. This contribution introduces novel techniques to analyse dynamic contact angle measurements (sessile drop) in detail which are applicable for axisymmetric and non-axisymmetric drops. Not only the recently presented fit solution by sigmoid function and the independent analysis of the different parameters (inclination, contact angle, velocity of the triple point) but also the dependent analysis will be firstly explained in detail. These approaches lead to contact angle data and different access on specific contact angles which are independent from “user-skills” and subjectivity of the operator. As example the motion behaviour of droplets on flat silicon-oxide surfaces after different surface treatments are dynamically measured by sessile drop technique when inclining the sample plate. The triple points, the inclination angles, the downhill (advancing motion) and the uphill angles (receding motion) obtained by highprecision drop shape analysis are independently and dependently statistically analysed. Due to the small covered distance for the dependent analysis (< 0.4 mm) and the dominance of counted events with small velocity the measurements are less influenced by motion dynamics and the procedure can be called “slow moving” analysis. The presented procedures as performed are especially sensitive to the range which reaches from the static to the “slow moving” dynamic contact angle determination. They are characterised by small deviations of the computed values. Additional to the detailed introduction of this novel analytical approaches plus fit solution special motion relations for the drop on inclined surfaces and detailed relations about the reactivity of the freshly cleaned silicon wafer surface resulting in acceleration behaviour (reactive de-wetting) are presented.

2

Introduction Contact angle measurements by sessile drop techniques are commonly used to characterize solid surfaces in terms of wetting behaviour[1, 2], adhesion[3] and so on. Theoretical and practical aspects of contact angle determination are known[4-9] and the manufacturing techniques for surfaces are advanced to the point that fabricating of specimens with well defined, symmetrical and reproducible surface patterns[10-15] at both microscopic and sub-microscopic levels is possible. But nearly no advantages took place to develop the measuring technique for surfaces in terms of data evaluation and local resolution. Even more a huge number of publications exist ignoring the existence of different contact angles whereas some authors call attention to the experimental problems

[9, 16, 17]

.

The advancing θa and receding contact angle θr are essentially static characteristics of the wetting situation occurring for solid/liquid pairs[18]. Different procedures/definitions are usual to obtain the advancing angle θa and the receding angle θr. The values of these specific angles on real surfaces distinguish from each other and result in the so-called contact angle hysteresis ∆θ[10] which has also an own hysteresis[19] (other formulations: needs activation energy, have to compensate a pinning, depends on initial conditions and so on). The problem of contact angle hysteresis is introduced in detail in recent publications.[20-22] Therefore, commonly used definitions to identify the one advancing and the one receding angle by optical observation with human eye are hardly comprehensible and are very subjective. Even the conditions (measuring and data processing) are unclear and unfortunately often not named, such as the wetting parameters (static or moving triple line)[22], which were established in their experimental studies. Also questions like defining the angle before (leading to a limit), during or after the motion from the same or different images of the drop are in nearly all cases not answered. After recognising these problems some authors even critically questioned the notions themselves (especially the receding angle[23]). If significant surface roughness and/or chemical modification/non-homogeneities are present such basic procedures of defining one/two angles by optical observation are influenced to the point that no valid results can be obtained. Additionally, for flat, non-reactive and homogeneous surfaces differences in the movement are hardly observable by human eye. For example the drops/triple lines within this contribution imminently start to move with minimal velocity but “the observed angle θ depends on the way the system was “prepared””[24]. Hence, the initial angle even being some advancing angle leads to insufficient result. All following angles while inclining the surface are per definition dynamic ones, which are more or (as preferred) less affected by the velocity. From our point of view the determination of specific angles, static or dynamic advancing and receding angles from a contact angle measurements depends on multiple experimental parameters but also needs to be statistically analysed by automatic data processing routine and meaningful analysis procedures leading to values with minimal variances. In Schmitt and Heib[9] we presented a technique basing on fitting a 3

Gompertzian function[25] onto the courses of contact angles relative to the inclination angle which is suggested at this time as useful to describe the trend of the data with a minimum number of parameters and results in a characterisation of the average properties of the surfaces. In this publication the intention is to present and to explain in detail two/three techniques which allow the determination of reproducible angles while the triple line advances or recedes in a statistical manner basing on the Gompertzian function and on counting of events followed by a dependent and an independent analysis. Neither the statistical technique nor the fitting procedure is restricted on inclining surfaces as shown in an additional publication. The detailed statistical analysis of the data is possible even for a very flat and chemical homogenous surface when a “minimal” movement of the drop imminently starts while inclining the surface. Differences between two types of silicon wafers will be highlighted in the following. For sake of clarity the results of the detailed statistical analyses of the second surface can be found in the supporting information. By using high-precision drop shape analysis (HPDSA)[9] also non-axisymmetric droplets and contact angles of super-hydrophobic surfaces are evaluable, compare to subsection “Data processing - HPDSA”. The supporting information presents for example the measurements of hydrophobic surface. For this introducing study the measured 0.05 mL drops are the non-axisymmetric ones obtained by inclining the sample surface. The independent statistical analysis (≡ global) is still used within a recent contribution about the analysis of a hydrophobic functionalized silicon wafer surface[26]. Additionally to the differences because of the motion behaviour, the authors defined the advancing as downhill angle θd (front edge of the drop) and the receding as uphill angle θu (back edge of the drop) as performed by several researchers. This procedure takes into account the difference in the force distribution affecting the triple line (by variation of parameters like the effective mass, centre of gravity if inclining a surface). For the authors this difference is mandatory for example theoretical study is published by Krasovitski and Marmur[27], whereas experimental studies are published by ourselves[16,

28]

. Within this

publication the contact angles and the velocities of the triple points are obtained by HPDSA[9]1, but methods from purchasable software like the one of contact angle equipment’s also result in useful data (time, contact angles, triple points, inclination angle) but in general with reduced sensitivity and precession.

1

C-programs and control files with implemented HPDSA-routine are obtainable without charge for scientific use from M. Schmitt.

4

Experimental Measurements All dynamic sessile drop experiments of the same surface took place with 0.05 mL ultrapure water (diameter > 7.8 mm) on defined positions of the sample surface, with controlled temperature, in a closed measuring chamber and after a sufficient delay time (> 2h) to ensure that a constant and saturated vapour atmosphere (moist air) is archived. The measurements were monitored by an OCA20, dataphysics, Filderstadt, Germany with 768 x 574 pixels. Due to the optical magnification one pixel corresponds to a length of 18.2 µm. Two flat surfaces were investigated to introduce the special analytical approaches, Table 1. Table 1: Investigated silicon wafer samples and conditions for the dynamic measurements of contact angles. sample cleaned wafera) rinsed wafer

crystal lattice 100

111

storing time prompt after receiving from trader aged at least one year in atmosphere

surface treatment boiling in RCA solution

No. of positions 10

frame rate Temperature 25 fps 30.4 °C

inclination rate 0.621 °/s

rinsing pure solvents

15

25 fps 30.4 °C

0.469 °/s

by

note Influenced/ accelerated motion no acceleration/ b) retardation

a) Detailed results within supporting information; b) the word “pinning” is avoided on purpose, roughness is in the range of 10-1nm to 100nm. The freshly received p-type (Boron) silicon wafer, native oxide, MicroChemicals, Ulm, Germany was cleaned by RCA solution consisting of 1 part of aqueous H2O2 (30 wt%), 1 part of aqueous NH4OH (29 wt%) and 5 parts of deionized water and by rinsing with ultrapure water. After this procedure the sample was placed in the closed measuring chamber. The second wafer a silicon wafer, aged native oxide, Siltronic AG, München, Germany, Table 1, was rinsed by pure solvents followed by an evaporation of the solvents. This rinsed surface results in a more ideal behaviour of the drop motion, see below, so that the detailed statistical analysis for this surface is presented within the manuscript whereas the results of the cleaned surfaces are presented within the supporting information. The rinsing with pure solvents removes only the weakly physically bonded adsorbates and will not result in reactive surface areas. The video-files are transferred to loss-free image files. The implemented fitting routine of the OCA equipment ellipse and tangent fitting are not able to independently analyse the uphill and the downhill angle and do not lead to valid contact angles[29] so that the self-developed HPDSA-routine[9] is used.

5

Data processing - HPDSA The image files were analysed by HPDSA routine implemented in software version 14.42 resulting in two contact angles per image which are independent from each other. The major objective of this analysis procedure is the transformation of the drop contour in mean radii of curvature by semi-circle fitting (keyword: La-Place equation). Hence this procedure is very suitable for strongly curved droplet (hydrophilic (compare to supporting information) and super-hydrophilic ones) but as presented within this study also suitable for dynamic evaluations of contact angles at least down to 20°. If the resolution and the quality of the image is well enough this procedure leads to the possibility to investigate the dependence of the two-phase interface on the distance of the three phase interface. The additional benefit is a tangent free computation of contact angle θm, ∆


 = 90° + arcsin    ± 

(eq.1)

The mean radius R, the inclination angle αBL of the baseline (= arctangent from the slope) and the difference of height coordinates Δy between the centre of the circle (xMP:yMP) and the triple point (xTP:yTP) are calculated for both sides of the drop for every image, Figure 1. The minimal length of the calculation arc (≈ length of the considered meniscus, fit range, Figure 1) can/have to be varied in dependence from the radius R to ensure convergence of the fitting procedure. For the data of RCA cleaned wafer this length was 1.0 mm (θm,min ≈ 20°) for the rinsed one 0.5 mm (θm,min ≈ 55°). Additional details of HPDSA procedure are published elsewhere[9]. The limits for the gradient in colour values were 10/pixel3. Diverging from published procedure an in-situ automatic baseline detection, Figure 1, took place to calculated the inclination angle of the baseline/camera. The baseline was determined by the intersection of two linear functions per side of the droplet, the one of the drop contour and the one of contour of the drop reflexion with the distance in-between 1 and 10 pixel distances (< 200 µm) from an given initial baseline. This procedure can dynamically control the baseline of the droplet but in this publication the average values for the triple points from around 30 images from inclination angles smaller than 40% of the maximal angle are computed and a baseline was determined. For sake of clarity the alternative determination of the contact angles from the performed linear regressions and the contact angle dependent shift of the intersection points relative to the real triple points will be shown in an additional publication. Different procedures are reasonable to define the coordinates of the triple points (xTP:yTP). The coordinates which were calculated during the image transformation are strongly influenced by the noise/resolution of the image[9] and the ones of the linear regression to calculate the baseline can be slightly shifted from 2

C-programs and control files with implemented HPDSA-routine are obtainable without charge for scientific use from M. Schmitt. 3 Note that for bmp-format with red, green and blue bit masks the maximal gradient corresponding to a black to white transition is ± 229.5/Pixel.

6

the real triple point (depending on the contact angle). The intersection points of the computed drop

contour (semicircle) with the baseline (m = slope; b = axis intercept), Figure 1, are less influenced by the optical observation, 1 =

2 =

  ! " # !  $%  ∙ ! ∙# ! $%

()* = ±+1 , 20 " , 2 -)* = . ∙ ()* + /

(eq.2), (eq.3), (eq.4), (eq.5).

The physically practical signs in Equation 2 and Equation 4 depend on the considered drop side. In total more than 6.000 (RCA wafer) / 9.000 (rinsed wafer) images were transferred in coordinates and were analysed.

Figure 1: Example for the baseline detection, the fitting, the determination of the intersection point (baseline (length 7.8 mm) with semicircle) and the calculation of the contact angle (right-angled triangle). Only for the purpose of visualization the camera is slightly tiled by 0.57° and every 20th data point is individually marked (the fit range of around 0.5 mm contains 37 data points).

Data analysis by Gompertzian function For analysis with a Gompertzian[25, 30] function only the knowledge about the inclination angle φ and the calculated contact angle θm is necessary,
 1" =
23456 + 7 × exp , exp ,

(eq.6).

This procedure allows describing the whole measurement with only four parameters. As shown in our recent publications the fit solution provides a well analytic representation of the trend of the data[9, 26]. Additional analyses by this procedure are explained within the “Results and Discussion”.

Data analysis by statistical approaches The velocity or its changing is the condition which is normally used to define specific angles like advancing and receding angles by the optical observation of triple line movement. To free this procedure from the subjectivity of the experimenter the statistical data analysis starts with the velocity determination and classification of the contact angle events. Therefore the rates of coordinates of the triple points (x:y) relative to the inclination angel φ are obtained by three point linear regression of three temporally neighbouring points (No. = image number), ? ?

BC.%$ BC.%$ ∑BC.%$ @( BC.$(4 14 " × 3 , ∑BC.$(4 " ∑BC.$14 " I. H J eq. 7", A = BC%$ 0 0 @1 BC. ° ∑BC%$ BC.$14 " × 3 , ∑BC.$14 ""

BC%$ BC.%$ ∑BC.%$ @BC.$-4 14 " × 3 , ∑BC.$-4 " ∑BC.$14 " I. A = H J eq. 8". BC.%$ 0 0 @1 BC. ° ∑BC.%$ BC.$14 " × 3 , ∑BC.$14 ""

The total velocity of the triple point, vel(φ)4 is given by (eq.3), T# 0  TU BC.

OPQ1"BC. = f × S

T 0 V  H ° J eq. 9", TU BC.

+

where f is +1 for movement in downhill (front edge) and -1 for movement in uphill direction (back edge). Additionally, the covered distance relative to the chronological first triple points (dis) is calculated[9]. To define the spectrum of contact angles that was being considered for the statistical analysis the contact angles were classified by two conditions: Contact angles during a constant speed/velocity (zero or limited) and contact angles during an acceleration/deceleration. In the case of acceleration definition of an angle before, during or after the acceleration/deceleration took place to catch commonly used definitions of correlations with a jump in the drop motion (or limit before

4

8

The velocity in unit µm/s is obtained by multiplying with the inclination rate in deg/s

motion) to define θa and θr. These conditions have to be transferred in logic operations for automatic processing. Therefore the change in velocity (∆vel), ∆OPQBC./BC.±X = ± OPQ=1YZ. ±Q"> , OPQ=1YZ. ">

(eq.10),

between different images leads to values which can be compared with a limit value (lv). For the different classified conditions the logic operations in Table 2 are defined resulting in four spectra of contact angle events5, Figure 2, that were being considered for the statistical analysis.

Figure 2 A,B: Example for the identified downhill contact angle events (marked as vertical lines; lv = 40 µm°-1) and velocity relative to the inclination of the rinsed wafer (A) and of the cleaned wafer (B). The graph (A) contains the markings of the contact angle events “before acceleration” and “during acceleration” whereas the graph (B) contains the markings of the contact angle events “after acceleration” and “constant speed”. The difference in motion behaviour (strong acceleration for (B)) is found within all experiments and indicates a self-induced non-homogeneity on the cleaned wafer. Similar but smaller differences exist for the motion on the uphill side of the droplets. A slow movement of the triple line is nearly imminently recognisable for both surfaces and sides of the drops.

5

Contact angle events are a set of four parameters (φ:θ:vel:dis) which are obtained if the logical condition for the change in velocity ∆vel are true, Table 2. For horizontal set-up exempli gratia the volume V can be the first parameter.

9

Table 2: Important definitions and logical conditions for the automatic determination of specific contact angle events relative to the limit value (lv) = 40 µm/°. Contact angle before acceleration during acceleration

conditions ∆velNo./No.+1 > lv |∆velNo./No.-1| < lv |∆velNo./No.-1| > lv

after acceleration

|∆velNo./No.+1| < lv |∆velNo./No.+2| < lv ∆velNo./No.-1 < - lv |∆velNo./No.-2| < lv |∆velNo./No.-1| < lv |∆velNo./No.+1| < lv |∆velNo./No.+2| < lv

constant speed

note sensitive to acceleration sensitive to deand acceleration deceleration followed by constant speed

A counting and calculation algorithm of the former obtained contact angle events is the core of the statistical analyses, Table 2. The four investigated parameters p for every event are the inclination (φ), the contact angle (θ), the velocity of the triple point (vel) and the covered distance relative to the chronological first triple point (dis). The expectation values E(p) and the standard deviations of a sample σ(p) were calculated by eq. (11) and eq. (12) with n as number of considered events leading to the so-called “global values” for every event(≡ independent analysis), Table 2, [" =

∑ \] ^

_[" = +∑` , 1"$[4 , [""0 "

(eq.11), (eq.12).

The allocation of the events in classes (class size = σ(p) × 0.5 for the figures and class size = σ(p) × 0.125 for Table 6) result in the raw data for following statistical analyses. Counting of every value within the classes leads to density distribution relative to the independent parameter (≡ dependent parameter/analysis). Hence, within every class the expectation values and the standard deviations are additionally computed by eq. (11), eq. (12). It is also reasonable to use the expectation value of the independent parameter instead of the centre of the class for the presentation in histograms[29]. The expectation value (together with the standard deviation as wide) also contains information about the distribution of the independent parameter within the class size, compare to all following figures of the statistical analyses.

10

Results and Discussion Gompertzian/average analysis The fitting of the data (φ:θ) of every position by Equation (6) is the first step of the procedure. Different initial conditions on the motion which is a known aspect of the tilted plate[21, 22] results in possible outlines which can be identified by performing this first step. Within this contribution

neither the Gompertzian nor the statistical analyses remove these outliners to keep external nonautomatic intervention as low as possible. The second step of the procedure consist of an averaging

of the Gompertzian fittings for every position leading to the averages data slopes presented in Figure 3 A and Figure 3 C.

Figure 3 A,B: Gompertzian analysis and selected residuals for the rinsed wafer. Specific angles with lowest standard deviation are marked; C,D: Gomperzian analysis and selected residuals for the RCA cleaned surface. For the RCA cleaned surface an acceleration of the drop motion is recognisable by the large variances from the fit curve. In this case additionally to the specific angles with the lowest standard deviation the ones at the limit before the discontinuity is marked.

Information about the motion of the drop are not directly observable from these slopes, the effect of motion is hidden in the specific slopes / standard deviations. The Gompertzian procedure resulted in only small variances from the data for slow average velocities which are for example for the rinsed wafer < 100 µm/°. All drops starts to move with minimal but by HPDSA measurable velocity if inclining the surfaces. The averaged data within these figures can again also be described by a Gompertzian function, eq. (3), resulting in Equation (13) and Equation (14) abc 1" = 69.01° + 6.94° × exp , exp ,0.2632f°1 , 2.97°"

abg 1" = 71.45° , 18.93° × exp , exp ,0.1211f°1 , 7.34°"

(eq.13), (eq.14),

which describe the average behaviour of a 50 µL drop on this rinsed surface with an inclination rate of 0.469 °/s. The RCA cleaned surface leads to Equation (15) and Equation (16), abc 1" = 27.92° + 19.98° × exp , exp ,0,106f°1 , 3.07°" abg 1" = 38.85° , 26.29° × exp , exp ,0.082f°1 , 4.10°"

(eq.15), (eq.16),

(inclination rate of 0.621°/s). The reader should not be confused by the differences in the first term of eq. (13) to eq. (16), the start of the Gompertzian function is at minus infinity[25] so that physically reasonable results can only be obtained within the fitting range. Within this range the trend of the data is described with the desired resolution and a minimum number of parameters. If analysing the Gompertzian average data in Figure 3 A and Figure 3 C more in detail a range with smaller standard deviations are recognisable. It can be assumed that in an ideal case an intersection point for the functions of all positions exists. For the rinsed wafer the points with the smallest standard deviation of the contact angle θd/u, which can be named the downhill / uphill respective the advancing / receding angle, Figure 3 A,
T 8.1°" = 74.4°
i 8.7°" = 63.3°

(eq.17), (eq.18),

are located within the fitting range of the inclination angle φ and therefore called “real”. In this range the velocity of the droplet is below or in the range of the onset of the macroscopic movement (= sliding drop, high velocity range). For the RCA cleaned wafer the points with the smallest standard deviations of the contact angle θd/u, Figure 3 C,
T 14.1°" = 42.7°

12


i 10.3°" = 24.4°

(eq.19), (eq.20),

are beyond the fitting range of the inclination angle φ and therefore physically less-meaningful and called “imaginary”. In this case it is more reasonable to present the angles at the limit of the fitting range, the average largest inclination angles φ before the unsteady motion of the droplets, Figure 2,
T 8.5°" = 39.5°
i 8.8°" = 25.5°

(eq.21), (eq.22).

These are also the last contact angles which are not influenced by the spontaneous motion dynamic of the drop which were identified for the RCA-cleaned wafer, see below. This influence/disturbance is clearly recognisable by comparing the residuals of the individual measurements of their Gompertzian functions, Figure 3 D, with the residuals of the non-effected measurements of the rinsed wafer, Figure 3 B. As the statistical analyses confirms, see supporting information, a pulling of the drop at the front edge to non-wetted surface area causes this motion dynamic. These aspects will be highlighted within the subsection “Properties of the test surfaces”. It can be summarized that the Gompertzian fitting procedure together with the residual analysis is an interesting technique which for example allows observing the accumulation of energy before starting the movement of the drop or a transition from surfaces with different liquid-solid interfacial tensions.

Data analysis by statistical analysis The statistical procedures allow investigating multiple analyses. For this introducing contribution with two flat and homogenous surfaces the discussion is restricted (not strongly) to the relations of the contact with the inclination angle for the rinsed surface. The upper-limit of the inclination angle is chosen individually for every measurement and is in the range of the fast movement of the droplet, Figure 2 (for the rinsed surface < 19°) to visualize effects of the moving droplet. Even more detailed analysis is possible if the upper limit is below the macroscopic movement of the drop. Due to the logical conditions to count an event, Table 2, the acceleration events are complementary to the ones with the constant speed. Independent analysis, global values If the parameters p are independently investigated so-defined “global values” for the expectation values E(p), eq. (11), and the standard deviation of a sample σ(p), eq. (12), are accessible, Table 3, Table SV1 to Table SV8. For this surface the individual count numbers for the events (for every measurements on its own) which can be regarded as similar to a jump in the motion of the droplet (“before” and “after acceleration”), Table 2, are small. In Table 4 and Table 5 selected results from the specific angle event “before acceleration” for different positions are presented.

13

Table 3: Example statistical overview of the independently computed contact angle events (≡ “global values”) for all measurements for the specific contact angle events of the rinsed wafer, Table 1. Contact angle uphill (receding)

σ(θu)

before acceleration

E(θu) [°] 60.2

[°] 5.3

Count number 475

during acceleration

59.8

5.3

1505

after acceleration

61.6

5.0

259

constant speed

63.8

4.2

4186

Contact angle downhill (advancing)

σ(θd)

before acceleration

E(θd) [°] 75.5

[°] 3.7

Count number 479

during acceleration

75.3

3.7

1459

after acceleration

74.4

3.2

250

constant speed

73.6

2.7

4225

Table 4: Global expected values E(p) and standard deviation of a sample σ(p) for all parameters p for selected measurements, compare to Table SV1 to Table SV8, and in total for the uphill direction (receding) of the specific angle event “before acceleration” for the rinsed wafer. measurement

M01 (uphill)

M04 (uphill)

M08 (uphill)

M12 (uphill)

M15 (uphill)

total

14

E(ϕ) [°] E(θ) [°] -6 -1 E(vel) [10 m° ] -6 E(dis) [10 m] 12.0 59.0 122.0 406.3 13.4 61.9 60.2 281.1 8.4 62.6 24 104 8.8 61.9 1 77 14.9 54.4 173 685 12.5 60.2 100 403

σ(ϕ) [°] σ(θ) [°] σ(vel) [10-6m°-1] -6 σ(dis) [10 m] 5.1 6.4 146 459 5.9 3.7 105 338 5.0 4.5 96 166 5.3 4.1 40.3 96.4 5.0 5.6 179 663 6.0 5.3 162.1 510.1

count number

29

44

33

49

35

475

Table 5: Global expected values E(p) and standard deviation of a sample σ(p) for all parameters p for selected measurements, compare to Table SV1 to Table SV8, and in total for the downhill direction (advancing) of the specific angle event “before acceleration” for the rinsed wafer. measurement

M01 (downhill)

M04 (downhill)

M08 (downhill)

M12 (downhill)

M15 (downhill)

total

E(ϕ) [°] E(θ) [°] E(vel) [10-6m°-1] -6 E(dis) [10 m] 12.6 75.6 115 655 12.0 76.8 74 399 11.0 72.4 145 436 15.2 75.6 127 610 14.7 76.4 199 1003 12.8 75.5 127 571

σ(ϕ) [°] σ(θ) [°] σ(vel) [10-6m°-1] σ(dis) [10-6m] 4.9 3.2 98 468 6.7 1.6 118 461 5.3 2.2 213.8 498.1 5.9 2.4 130 508 5.4 2.2 179 837 6.0 3.7 172 599

count number

23

37

29

28

38

479

This contact angle event has the advantages to be less affected by a change in the limit of the inclination angle, Figure 2, and have a relative large count number ≈ 475, Table 3. By analysing the individual measurements it can be concluded that they are independent random experiments, Table 4 and Table 5. Statistical outliers like M05 in Table 4 are identifiable. But in this case the count number is too low for an unequivocal explanation for the different motion behaviour. The expectation values (all positions at once) for the covered distance for the downhill angle is around 600 µm (≈ 33 pixels) and for the uphill angle around 400 µm (≈ 22 pixels). Similar differences are recognisable for the velocities and the distances, Table 4, Table 5. These differences in the movement/distance for the advancing and receding (downhill and uphill) side are likely present due to the difference in force distribution in relation with the measuring setup.[16, 28] But the low extent leads to the assumption that no pinning of the droplet is present and the movement of the drop on the rinsed surface is nearly ideal which is also confirmed by the steady increase of the velocity, Figure 2 A. Altogether it can be concluded that the so-defined global values of the contact angles which are correlated with an acceleration of the droplet, Table 2, can be used to build a standard method of measuring the uphill/downhill (advancing/receding) contact angles on a surface, Table 3. It is to expect that the specific angles for the event “constant speed” are significantly influenced by 15

an initialisation period, so that in this case the simple procedure of calculating “global values” is not sufficient. Note that the inclining plate method have the advantage to measure both angles simultaneously (uphill and downhill) but for small droplets an effect due to a correlation between both sides of the droplet will not be negligible, compare to the next section. This effect and possible

line tension correlations[31, 32] were the reason to choose the drop volume of 0.05 mL for the presented investigation. The drawing of the drop (back edge) is even observable for this quite large drop volume if the acceleration is as large as for the RCA cleaned surface, Figure 2 B and Figure 3 D.

Dependent analysis, contact angle correlations

Figure 4 A,B: φd / Velocity dependent analysis of θd for the event “before acceleration” of the rinsed wafer. E(p) and σ(p) are marked by lines, whereas the σ(p)’s within the classes are marked by error bars respective by the broadness of the columns for fp(p). Clearly to observe are a bimodal distribution for E(φd) and a domination of events with low veld.

Figure 5 A,B: φd / Velocity dependent analysis of θd for the event “constant speed” of the rinsed wafer. E(p) and σ(p) are marked by lines, whereas the σ(p)’s within the classes are marked by error bars respective by the broadness of the columns for fp(p). Clearly to observe by the high count number and the “small” contact angles θd are a initialising period relative to the E(φd) and to E(veld). In the previous section the parameters p are defined as independent from each other which is an approximation. It is to expect that the analyses of these correlations lead to a reduced variance in the calculated values, to additional information about the motion of the droplet, and about the surface (especially in terms of roughness and homogeneity). During this procedure, one parameter is defined as independent and the three others are defined as dependent from the first one, compare to Sub-

Sec. “Data analysis by statistical analysis”. The independent random experiments are combined to increase the basic population of the contact angle events. The basic population of the experiments are restricted due to the frame rate of the OCA equipment (in total ≈ 9.000 images are analysed). The presentation starts with the correlations of the downhill (advancing) angles. For all analysed contact angle events, Table 2, two maxima in the distributions of the inclination angle and two “steps” in the course of contact angle relative to φ, Figure 4, Figure 5, Figure S1, Figure S3, are to identify. The distributions of the velocity veld for the events “after”, “during” and “before acceleration” result in a clear plateau with significant count numbers and similar contact angles, Figure 4, Figure S1 and Figure S3. For the contact angle with the event “constant speed” a large (by number) initialization period is recognizable, Figure 5. Table 6: Overview about the detailed, dependent statistical analysis of contact angle relative to the ranges in inclination angle φa) for the rinsed wafer. (call size = ) a) E(E(vel)) E(E(dis)) range E(E(φ)) Count σ(E(vel))b) E(E(θ)) σ(E(θ))b) c) [°] [°] number [°] [°] [µ µm/°] [µ µm] [µ µm/°] 0.04:3.01 1.80 71.75 0.86 -18.86 14.91 7.2 50 before 3.76:11.93 8.05 74.49 0.91 10.87 20.37 123.1 137 acceleration 12.68:21.6 16.87 76.54 2.31 206.96 97.87 878.3 292 0.08:3.11 1.85 71.87 0.81 5.20 5.8 8.6 166 during 3.86:12.19 8.65 74.05 0.65 46.09 28.47 151.6 325 acceleration 12.95:22.0 17.46 76.24 2.00 309.38 117.73 1065 968 0.11:2.90 1.63 70.31 1.00 -9.17 4.5 9.2 21 after 3.60:14.07 9.98 73.43 0.76 40.79 34.17 222.1 117 acceleration 14.77:21.8 17.63 76.20 1.68 208.53 66.24 904.8 112 0.04:5.09 2.76 71.29 1.02 6.25 2.64 14.1 1482 constant 5.72:15.20 9.97 74.49 0.64 40.18 23.88 162 2373 speed 15.83:21.5 17.69 77.24 0.78 177.6 76.06 742.2 370 a) b) b) E(E(vel)) E(E(dis)) Contact angel range E(E(φ)) Count σ(E(vel)) E(E(θ)) σ(E(θ)) uphill [°] [°] number [°] [°] [µ µm/°] [µ µm] [µ µm/°] 0.04:3.06 1.86 68.52 0.77 -19.09 2.85 5.2 53 before 3.81:12.10 8.11 63.58 2.18 -8.76 13.54 47.1 151 acceleration 12.85:21.9 16.95 56.59 1.33 184.05 101.46 679.2 271 0.08:6.24 3.39 67.71 1.45 4.21 3.23 12.4 301 during 7.01:12.40 10.07 62.23 1.41 28.34 23.28 74.7 265 acceleration 13.17:21.6 17.62 56.56 1.15 272.54 102.83 825.7 939 0.15:6.01 3.22 68.19 1.34 -14.61 2.94 13 60 after 6.74:11.86 9.59 62.92 1.45 1.77 13.21 56.8 66 acceleration 12.59:21.4 16.45 57.94 1.64 162.44 76.39 575 133 0.04:5.18 2.7 67.92 1.13 3.54 0.57 10.1 1449 constant 5.82:10.32 8.08 63.79 1.19 7.50 3.12 37.8 1365 speed 10.96:21.9 14.26 59.33 1.36 80.7 66.08 257.8 1372 a) The ranges are given relative to the centre of classes (class size = σglobal(φ) × 0.125). b) Additional equations are explained

Contact angel downhill

in the supporting information; ∆E(p) are in the same range as in the figures, Table 3 and Table 4. c) The velocity in µm/s can be calculated by multiplying with 0.469 °/s

In this context the measurements are analysed by defining ranges in inclination/contact angle distributions, the results are summarised in Table 6. So, the identification of the initialisation periods is possible and leads to the conclusion that not only the event “constant speed” but all events are clearly influenced at the beginning of the measurement.

17

It is reasonable to obtain reproducible and valid results for the contact angle events if the expectation values of the ranges with large count number after the initialization (dis ≈ 170 µm; vel ≈

16 µm/s) and before the continuous motion are considered, for definition compare to Table 2: •

Downhill contact angle for the event “before acceleration”

74.5°

•

Downhill contact angle for the event “during acceleration”

74.1°

•

Downhill contact angle for the event “after acceleration”

73.4°

•

Downhill contact angle for the event “constant speed”

74.5°

We are convinced that by the aid of the other parameters the understanding of contact angle motion behaviour can be advanced. The presented and possible further assumptions have to be verified by additional measurements with varied parameters and on different surfaces. Differences in the force

distribution on the triple line[16, 28] leads to differences between the correlations of the uphill (receding) angles and the ones of the downhill angles. The variation in the contact angles, the velocities and in the distributions of the inclination angle proves these differences if comparing the “downhill” with the “uphill” sides of the drops, Figure 4 to Figure 6 and Figure 5 to Figure 7.

Figure 6 A,B: φu / Velocity dependent analysis of θu for the event “before acceleration” of the rinsed wafer. E(p) and σ(p) are marked by lines, whereas the σ(p)’s within the classes are marked by error bars respective by the broadness of the columns for fp(p). Clearly to observe are two maxima in the distribution of φu. Similar to the relations for the downhill side bi- to trimodal distributions for the inclination angle for the uphill side are recognisable, Figure 6, Figure 7, Figure S2 and Figure S4. Hence the definition of different ranges resulting in the values in Table 6 is meaningful. Three ranges with similar contact angles are identified for all events of the uphill angle. These are the first one, non-moving equal to initialisation, the second one, also nearly non-moving, and the third one with significant velocity. The third one is strongly influenced by the downhill motion.

Hence, the second range (dis ≈ 60 µm; vel ≈ 4 µm/s) leads to the best results in terms of independency, for definition compare to Table 2:

•

Uphill contact angle for the event “before acceleration”

63.6°

•

Uphill contact angle for the event “during acceleration”

62.2°

•

Uphill contact angle for the event “after acceleration”

62.9°

•

Uphill contact angle for the event “constant speed”

63.8°

Figure 7 A,B: φu / Velocity dependent analysis of θu for the event “constant speed” of the rinsed wafer. E(p) and σ(p) are marked by lines, whereas the σ(p)’s within the classes are marked by error bars respective by the broadness of the columns for fp(p). Clearly to observe are an dominating inducing period for θu relative to the E(φu) with nearly no motion (compare to E(velu)). Note that in this case the reduction of the range (e.g. < 100 µm°-1) and of the class size can further enhance the analysis. The influence of the motion at the downhill on the one at the uphill side, mentioned above, can also easily be shown by the statistical analysis procedure. Therefore, the relations between the distances and the velocities are compared, Figure 8. The non-linear behaviour (first two points have the highest count number) of the distanceu is the result of this correlation. But also the distanced leads to a deviation from the linear increase which means that the downhill side is affected by the motion of

the droplet. Due to the conditions to count an event, Table 2, the acceleration events are complementary to the ones with the constant speed.

Figure 8 A,B: Velocity dependent analysis of the covered distance for the event “after acceleration” of the rinsed wafer. In this case the difference between the linear relation for distanced and the nonlinear relation for distanceu are recognisable.

Properties of the test surfaces For the introduction of this novel technique to enhance the potential of optical observation of drops

like the sessile drop technique test surfaces, are chosen which are flat, rigid, pure (low number of compounds), defined and chemically non-reactive. The surface presented within the manuscript is cleaned by only rinsing with pure solvents whereas a second surface, the one whose statistical analysis can be found in the supporting information, is freshly received and cleaned by a RCAcleaning. The obtained differences in these obviously similar surfaces indicates that every kind of surface is analysable by the presented procedure most likely resulting in more complex specific distributions of the parameters. The authors are convinced that detailed analyses lead to an enhanced probe for local resolved surface properties. The apparently simple silicon wafer surface has an eminent importance for human life. Nevertheless only a limit number of contact angle studies

exist or is electronically accessible[33, 34]. For example Kissinger et al., 1991,[35] measure the apparent contact angles of silicon wafers with different treatments and storing times and concluded that “hydrophilic surfaces have angles of approximately 0° to 6° and hydrophobic ones of 50° to 70°”. In

Hermansson et al., 1991,[36] silicon wafer (out of the box) do not result in spreading and an apparent contact angle larger than 50° without surface treatment or after oxidising conditions is measured (inverted bobble technique). The detailed surface treatment studies within the the cited publication resulted in a plateau for the apparent contact angle of oxidised surfaces. Influences of the reactivity

or of lattice effects on the moving behaviour are not published. These publications lead us to the conclusion that comparing of an “out of the box” silicon wafer after removing not covalently bound contaminant by washing with pure solvents (ethanol and hexane) with a surface after cleaning by

RCA-solution is suitable to introduce the statistical analysis. For both surfaces front and back edge motion was imminently observable by HPDSA if inclining the surface. The small variances in averaged contact angles for the different events of the dependent analysis,


T̅ = 74.1° ± 0.4°
i̅ = 63.1° ± 0.6°

(eq.23), (eq.24),

Table 6, together with the smooth and homogenous change in velocity, clearly demonstrate that the rinsed surface is a practicable test surface with nearly ideal behaviour. As expected the dependent statistical analysis results for the freshly received and RCA cleaned wafer in smaller contact angles,
T̅ = 37.8° ± 0.2°
i̅ = 27.3° ± 0.4°

(eq.25), (eq.26).

As shown by these results the statistical analysis as performed within the supporting information is especially sensitive to “slow moving” before the sharp increase in velocity, Figure 2 B. It can be assumed that the RCA cleaned surface is also homogeneous but the ultra-pure water results in an increase of the surface tension of the solid-liquid interface (reactive (de-)wetting) of the wetted area as far as the bulk pressure foot[37] or the absorbed liquid layer[22] at the start of the measurements ranges. As long as the drop was so slowly that it did not catch up with influenced area the motion was smooth. Then, the drop was pulled to wet the non-aged area which resulted in a large acceleration, Figure 2 B, and in significant reduction of the contact angle. Retardation (pinning) of the frond edge movement can be excluded because of the non-significant difference in the covered distances which can be proven by the statistical analysis, supporting information, and by controlling the raw/crude data. Due to the velocity behaviour, Figure 1, and our expertise concerning the gravity effected acceleration of the drop motion, this unsteady behaviour can not only be an effect of the inclining. Further investigations with stable model surfaces are in preparation. Due to the chemically reactivity a reproducibility of contact angle of the freshly received and RCA cleaned surface is not to ensure, whereas aged, oxidised surface systems (after sufficient storing time in humid atmosphere), like the rinsed wafer are easily produced and are very stable. Nevertheless without a surface treatment (RCA etc.) the history of the silicon wafer surface until the stable surface condition is archived is also of importance. Alternative surfaces like a self-assembled monolayer of a perfluorinated mono-organyl siloxane produced through a vapour phase reaction[38] or poly dimethyl siloxane brush films are interesting systems[39]. First results using a perfluorinated mono-organyl siloxane which is interesting to produce well defined surfaces with larger contact angles are published using the global analysis of the “slow moving” procedure[26]. But especially for the siloxane formation with controlled or reduced content of water/acid the formation kinetic is hardly investigated[40], so that the parameters are hardly to control. Additionally, from aspects of interface relations/properties such complex surfaces require intensive investigations exempli gratia the influence/ thickness of, the correlations between the interlayers, the deformation of the brushes, 21

influence of bubbles (vapour pockets), effect of defect density (the coverage of the surface), effects of preparation[29] and the influence of the substrate itself are unknown or not fully demonstrated for dynamic measurements. The procedure presented in this study can help to investigate some of these parameters. To conclude we were able to characterize a clearly different behaviour of the nonreactive rinsed wafer and the less-reactive RCH cleaned wafer which seems to result in a reactive de/wetting.

Conclusions Three new methods are presented to analyse sessile drop experiments in a reproducible manner while inclining the sample surface. The necessary experimental data that means the positions of the triple points, the contact angles and the inclination angles are obtainable by using any user-friendly contact angle evaluation software. These approaches is also applicable for horizontal or other experiments using the optical observation of a droplet[41]. Thereby, they are independent from any subjectivity like analysing the video frames by human eye which is nearly impossible for these examples of very flat surfaces. Of course hardware aspects exist which have to keep in mind like pixel effects on the determined drop shape as critically highlighted in the first presentation of HPDSA[9]. Therefore, nearly all hard- and software aspects are monitorable and controllable by performing high-precision drop shape analysis, for example by the baseline determination firstly explained within this contribution. Within the presented study it was possible to define distributions in specific contact angles even for the droplet on flat and homogenous solids with an imminently starting motion leading to a covered distance of maximal 1 mm for both sides of the drop. The covered distance at the front side of the optimal range for the dependent analysis is with maximal 0.4 mm(RCA)/0.2 mm(rinsed) even smaller. Hence this procedure has a fine local resolution, so that even surface variations can be observed. Also because of the domination of events with low (≠ zero) velocity these approaches can be called an analysis of the “slow moving” drop (reduced flow effects, slip/stick contact angle behaviour). The behaviour of the deviations from the fitting procedure, Figure 3 B and Figure 3 D for large inclination angles (= high velocity, sliding range), clearly demonstrates, that the approaches can also not simply called dynamic contact angle determinations (capillarity numbers[19, 22] are smaller than 10-6). It will be shown in an additional publication that the analysis of the different limit conditions to count an event, Table 2, may be representative to gain access to the homogeneity of the surface and to analyse also the dynamic range. In dependence of the surface it is sufficient to analyse in detail the global values leading to reproducible results (independent analysis, every parameter on its own as shown in Table 3). The rinsed silicon wafer leads to global values for the acceleration events of 60.5° for θu and 75.1° for θd. The same events for the RCA cleaned surface results in of 26.3° for θu and 36.8° for θd even through the measurements 22

(not the performed analysis) is highly disturbed by the acceleration of the drop (most likely reactive de-wetting). The dependent analysis can result in more accurate contact angle values for the surfaces, because effects of inducing and high velocity range can be removed. Further analyses of the dependencies between the parameters are useful to additionally enhance the reliability for example of the contact angle distributions and to gain access to surface properties. For example depending on the performed experiment the dependent analysis relative to the velocity allows the investigation of static and dynamic effects. Additionally, it was also proven that the distribution for the uphill angle is not independent from the downhill angle (respectively motion) which was experimentally shown in Schmitt et al., 2014.[16, 28] Particularly the bimodal distribution in downhill contact angle (in relation with the inclination angle) is an effect which was unpublished before. The average drop behaviour relative to the inclination angle can be well described by the average Gompertzian functions, eq. (13) to eq. (16). The fitting procedure is also very suitable to identify non-ideal/unsteady behaviour of the drop motion and can also result in specific angles. The contact angles obtained by the dependent analysis as performed within this contribution, Table 6 (every 2th range), are nearly not moving (between -4 µm/s and 22 µm/s) and therefore less/non influenced by dynamic effects. Following to the presented and our recent results[9, 26, 29] three ranges can be defined, which are the non-moving (static), the “slow moving” and the dynamic (high velocity, “sliding”) range. It have to be confirmed by additional investigations, but it seems to be reasonable, that the presented fitting procedure leads to minimal variances for measurements until to dynamic range starts, Figure 3 A. Both statistical analyses can be varied to be sensitive for the “slow moving” (as presented in this publication), the non-moving and the dynamic range (“sliding” drop). All of the presented procedures as performed within this study are most likely less influenced by the hysteresis of the contact angle hysteresis[19] than a pure static contact angle determination and not influenced by the subjectivity of the operator. To summarize the problem of the contact angle hysteresis (advancing and receding angles) as studied with tilted (inclined) planes is an old[42-44] but still interesting field of research[21, 22, 45]. Many problems still remains unsolved, therefore the authors are convinced that for every contact angle measurement all parameters have to be controlled, stored and presented to be valuable in the future. Especially the Gompertzian fitting procedure describes the data for “slow movement” very well. But we are convinced that only in combination with the statistical analyses (dependent and independent) these approaches can lead to reliable contact angle determinations. The limit to count an event as exemplarily performed leads to four specific spectra of contact angle events that are being considered for the statistical analyses. Due to the chosen logical conditions, Table 2, the different spectra are complementing. Within the presented study the difference in contact angle and motion behaviour of a flat, rigid, pure, defined and chemically non-reactive and of flat, rigid, pure, defined and chemically reactive surface have been analysed and presented. 23

Acknowledgements Special thanks to R. Hempelmann for his confidence and support.

Notes and references a

Michael Schmitt, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected]

saarland.de. b

Katja Groß, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected].

c

Julia Grub, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected].

d

Florian Heib Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected].

Electronic Supplementary Information (ESI) available: The file contains additional figures, equations and tables containing the statistical data. See DOI

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27

M. Schmitt*a, K. Großb , J. Grubc, F. Heibd a

Michael Schmitt, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected]. b Katja Groß, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected]. c Julia Grub, Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected]. d Florian Heib Saarland University, Campus B 2 2, 66123 Saarbrücken, [email protected].

28

Graphical abstract