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Effect of waviness filtering on surface area ratio measurements in
microscopy: a numerical study
M A Rodríguez-Valverde*1, P J Ramón-Torregrosa1, A Amirfazli2 and M A Cabrerizo-Vílchez1
1
Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, Campus de Fuentenueva; E18071, Granada, Spain
2
Department of Mechanical Engineering, York University, Toronto, Canada, M3J 1P3
Roughness of a surface is typically measured as a deviation from waviness in the topography. To consider exclusively the
role of surface roughness, it is necessary to filter those topography features associated to surface motifs different of
roughness ones. Waviness filtering requires knowing a posteriori the directionality of the addition of roughness over the
original wavy surface, before the roughening process. However, morphology of rough surfaces is controlled by their own
dynamics (chemical etching, ion sputtering ,...). In other respects, 3D microscopy misses surface features such as voids,
overhangs. Surface area ratio (waviness-weighted roughness factor) is a hybrid roughness parameter which directly
quantifies the impact of surface roughness on interfacial phenomena such as wetting and tribology. In this work we
analysed numerically the influence of the directional character (vertical or normal) of the addition between waviness and
roughness on the surface area ratio of rough wavy surfaces. Different double-scale surfaces were simulated as
perturbations, in both directions, of an analytically-defined wavy surface using a fractional Brownian noise.
Keywords: roughness; waviness filtering; surface area
1. Introduction
The behavior of engineered and natural surfaces can be understood from their three– dimensional roughness [1, 2]. In
the field of micro-/nanostructured surfaces, e.g. superhydrophobic surfaces, where the systems contain dual scale
structure roughness, it is very important to be able to characterize accurately the surface topography. Development of
advanced measurement techniques in surface microscopy relies on the proper definition of surface topography and their
reconstruction methods [3]. Physically real surfaces contain multicomponent topographic data, which should be
properly processed to isolate the information of interest.
Surface roughness is governed by man–made or natural processes, which modifies the initial surface at arbitrary
directions. These processes can be roughly classified into ballistic [4], if the rough profile is created on the previous
non-planar (horizontal) surface following the vertical direction; and growth/erosion [5], where the perturbation of the
wavy surface is performed following its normal direction. Many surfaces evolve by curvature flow [6, 7]. For instance,
in reactive systems, the asperities and grooves may act as preferable sites for reaction, diffusion, adsorption, nucleation,
etc... This affects the evolution of the interface leading to its roughening.
Surface features at different scales may affect differently the function of the surface [8]. Surface area is especially
relevant in interfacial phenomena such as wetting, tribology or cell adhesion, where the interactions depend on the
contact area between bodies/phases [1, 9, 10]. The surface area ratio, r, also referred to as relative surface area, enables
a global evaluation of the surface roughness. The surface area ratio is a hybrid roughness descriptor [11] that quantifies
the difference between the actual surface area, A, and the apparent or nominal surface area, Aapp, as follows:
A
(1)
r=
Aapp
where Aapp is the area of the smooth reference surface. This reference surface, known as waviness, is the surface with
the same shape and dimensions than the actual surface. If the surface were rough without waviness, Aapp would be the
area of the horizontal plane resulting from the vertical projection of the rough surface. In microscopy, the area Aapp is
known as geometrical area, Ag, and it is numerically equal to the scan–area (i.e. the product of the transversal and
longitudinal scan–lengths). Some authors prefer to define the surface area ratio as A/Ag when the surface deviations are
referred to the smoothness (no roughness) and the plainness (no waviness) indistinctly.
Unfortunately, surface area evaluated by microscopy is very sensitive to the measuring conditions [1, 7, 12].
Actually, the true value of surface area is experimentally inaccessible because it would require the acquisition of infinite
points (i.e. a continuous topography) and this is technically impossible. Instead, the calculation of surface area ratio
from contact angle measurements is widely reported [7, 13, 14, 15, 16].
The current microscopes (AFM or confocal) provide surface textures grouping waviness and roughness indistinctly,
i.e. a compounded account of intermediate and short length scales. To assess the frictional or wetting behavior of a
surface, since both properties depend on roughness, one needs to filter the acquired morphology of the surface. But any
filtering process of raw topography results in certain loss of information. A wrong choice of filtering tool (using some
kind of high-pass and low-pass filters) can lead to misleading values of the statistical parameters [17].
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In general, a surface morphology can be considered as the mathematical superposition of the spatial elevations
associated with waviness and roughness. From this hypothesis, regardless of its physical nature, the generating process
of a rough surface can be grouped, at least, into vertical (ballistic deposition) and normal (growth/erosion), according to
the direction of the superposition of heights. This simplistic classification can be used to understand the generic
properties of very different roughening processes. Since an arbitrary space direction can be discomposed into the
vertical and normal directions, a generic process, which evolves the unprocessed surface in an arbitrary direction, may
be understood as a combination of ballistic–like and growth/erosion processes.
a
Fig 1. (a) Addition of a constant to a sinusoidal profile (solid
line) in the vertical (dashed line) and normal (short dashed
line) directions. Addition of a random function to the
sinusoidal profile (b) in the vertical (c) and normal (d)
directions. Instead of (d), a stylus profiler will register the
profile (e) without overhangs.
b
c
d
e
An arbitrary surface can be mathematically interpreted
as the result of the addition of roughness to the waviness
following two directions: the vertical one or the normal
one associated to the wavy surface. Graphically, the
vertical and normal additions of roughness to waviness
provide different surface profiles as shown in Figure 1.
However, instead of topography profiles like Figure 1(d),
the standard devices (e.g. stylus profiler, confocal optical
microscope, atomic force microscope) acquire single–
valued topographic elevations (without overhangs, voids
...) like Figure 1(e), arranged in regular grids.
In this study, the following terminology is used to distinguish the surfaces studied. A pure rough surface is defined as
a surface with roughness but without waviness, a pure wavy surface is a surface with waviness but without roughness,
and a rough wavy surface is a surface with both roughness and waviness (see Figure 2). The surface area ratio of pure
rough surfaces, rrough, is also referred to as roughness factor. It is obvious from Eq. (1) that the surface area ratio of a
pure wavy surface is one. For rough wavy surfaces, depending on the direction (vertical or normal) of the addition of
roughness to the waviness, the surface area ratio will be denoted as rv for the vertical addition, and as rn for the normal
addition.
Fig 2. (a) Pure rough surface (roughness), (b) pure wavy surface (waviness), and the resulting rough wavy surface (c) obtained by the
addition of surfaces (a) and (b). The addition can be either in the vertical or the normal directions with respect to the wavy surface.
By inspection of an experimental topography, generally it is not possible to know a posteriori the precursor
roughening process that caused the evolution of the surface. This uncertainty in the directionality of the superposition of
roughness to waviness can lead to meaningless values of roughness parameters after the waviness filtering. The
directional character of waviness filtering and its influence on the roughness descriptors are not clearly discussed in the
literature [18].
To characterize surface roughness, special care should be taken for the filtering of raw topography data. The aim of
this study is to analyze numerically the effect of waviness filtering on the values of surface area ratio of rough and wavy
surfaces evolved by ballistic–like or growth/erosion processes, accordingly.
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2. Methods
To measure the roughness factor, and other quantities, of a rough wavy surface, the waviness component should be first
removed from the topography information. Processing of surface data requires to establish a reference or nominal
surface to identify the waviness from the experimental data. Ideally, this reference surface should coincide with the
surface shape (pure wavy surface), without roughness (see Figure 2(b)). Typically, once the reference surface has been
selected a priori, the waviness is removed. This process is called flattening or levelling. Commonly, the removal of
waviness is performed by subtracting, from the surface morphology, the reference surface with different methods such
as: least-squares fittings [19], bicubic-Bezier surface patches or Gaussian filters [3, 12]. The least–squares fitting is
typically based on the minimization of vertical offsets between the surface morphology data and a proposed
mathematical model [19].
The waviness removal in normal direction is not commonly achieved because it would produce stretching/shrinking
of the topography data, providing an irregular grid and a non-rectangular surface projection on the horizontal plane (see
Figure 1(d)). Therefore, after filtering, the distance between two adjacent points will be different so the original
sampling range and thus the planar surface would be distorted. Due to these artifacts, which complicate the numerical
treatment, the conventional software of different topometric devices only apply filtering in the vertical direction.
In general, waviness filtering is usually performed in the vertical direction by least–squares fitting or gaussian
filtering [3, 20]. All these operations assume the vertical expansion of surface data. For any rough wavy surface z(x,y),
which is a single–valued function, the roughness can be modeled as a noise function n(x,y) added linearly in the vertical
direction at each point of the wavy surface f(x,y):
z ( x , y ) = f ( x, y ) + n ( x, y )
(2)
The vertical superposition is mainly valid for roughening governed by ballistic–like processes [21]. Figure 2(c)
shows a rough wavy surface, designed by vertical addition of waviness and roughness, whereas the isolated
representations of the waviness and the roughness are shown in Figure 2(a) and (b), respectively.
Most processes of surface erosion or growth [5] produce perturbations in the local normal direction (e.g. acid etching,
sand-blasting, curved-surface machining, etc). For these cases, the roughness should be considered as a noise function
n(x, y) added following the local normal direction to the waviness and in consequence, the rough wavy surface z(x, y)
would be parameterized in Cartesian coordinates as:
 x = x '+ n ( x ', y ' ) xˆ ⋅ Nˆ 0 ( x ', y ' )

(3)
 y = y '+ n ( x ', y ' ) yˆ ⋅ Nˆ 0 ( x ', y ' )

ˆ
 z ( x, y ) = f ( x ', y ') + n ( x ', y ' ) zˆ ⋅ N 0 ( x ', y ' )
where (x, y) are the coordinates of the final rough wavy surface; (x’, y’) are the coordinates associated to both pure
wavy and rough surfaces; N̂ 0 is the unit vector normal to the waviness (Eq. (A.5)); and { xˆ, yˆ , zˆ} is the Cartesian vector
basis. For relatively smooth topographies where the wavy surface slope is very small, Eq. (3) will be reduced to Eq. (2).
Most height equations in this context rely on a small-slope approximation (low height gradient) that allows one to
strongly simplify the analytical formulation. However, this simplification may be at the cost of inaccuracies in the
physical description. For instance, surface diffusion is severely overestimated in regions of high slopes.
We used synthetic (mathematically constructed) rough wavy surfaces with stochastic roughness [12]. These surfaces
were generated with a fix geometrical area (Ag = 4 a.u.) and a grid of 1024x1024 points. We selected this value of
resolution as it is the common resolution for many conventional instruments. The surface area ratio was computed using
a tessellation algorithm described in a previous work [1], to exclude the influence of the tessellation method on the
calculation of r-values [1, 22].
Roughness of the synthetic surfaces was produced with a fractional Brownian noise [23] with amplitude zr (peak-tovalley distance) and zero mean, <n> = 0. A zero-mean noise assures that the waviness should be parallel to the average
real surface. Different pure rough surface were produced by changing the fractal dimension from 2.1 to 2.9 in steps of
0.1. This variation in the surface texture did not significantly change the values of surface area ratio [12]. We used a 2D
sinusoidal surface as a simplified model of the waviness profile:
 2π 
f ( x, y ) = zw cos 
y
(4)
 λ 
where λ is the spatial period and zw the waviness semi-amplitude. The spatial period was fixed to 2 a.u. for all wavy
surfaces used in this study. We selected a symmetric waviness because in the limit n(x, y) = 0, rn = 1 as happens with rv
(see Appendix, i.e. Eqs. (A.9) and (A.17)). Finally, using the waviness described by Eq. (4) with to Eqs. (2) or (3), the
rough wavy surfaces were constructed with stochastic roughness for different values of zw/λ and zr/zw. To emulate the
3D microscopy, we also reproduced experimentally- accessible topographies, i.e. without overhangs, from the original
surfaces created by normal addition of roughness. The surface area ratio of these rough wavy surfaces with waviness
filtered in the vertical direction is denoted as rfn.
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All values of zw/λ and zr/zw are not allowed for the normal addition because surface points are overlapped when the
roughness added to the waviness holds the condition:
(
)
min κ 01−1 ( x, y ) , κ 02−1 ( x, y ) ≤ n ( x, y ) , ∀ ( x, y )
(5)
where κ01 and κ02 are the principal curvatures of the waviness. The surface overlapping is illustrated in Figure 3 when n
= 1/κ0i. For the sinusoidal waviness described by Eq. (4), the condition (5) is reduced to:
2
 λ 
zr

 ≤
π
2
z
z
w 
w

(6)
Fig 3. Limit regions for surface overlapping in the normal addition of
roughness to waviness. Note that the radius of both circles is 1/κ0.
3. Results and Discussion
Figures 4 and 5 show variation of the surface area ratios of rough wavy surfaces generated numerically by vertical and
normal additions, in terms of zw/λ for different values of zr/zw. The waviness filtered value of rn, i.e. rfn, is also shown in
Figures 4 and 5. Firstly, the surface area ratios rv and rn always underestimate the value of roughness factor rrough;
moreover, rn > rv for all cases and rn > rfn, except when zr/zw is 0.5. As the waviness is less fluctuating or with lower
amplitude (small values of zw/λ), both surface area ratios approach to the roughness factor because the waviness slope is
small. Likewise, only when the roughness amplitude is comparable to the waviness amplitude (large values of zr/zw),
there is a good agreement between rn and rfn regardless of the zw/λ value. Although the increase of surface area is
inherent to the addition of roughness as it is illustrated in Figures 1(c)-(d), the surface area ratio is indeed a wavinessweighted roughness factor (Eqs. (A.9)-(A.17)). However, the rough wavy surfaces generated in the normal direction
will always have greater surface area ratio than those generated following the vertical direction with identical roughness
due to the lateral growth. Therefore, rn is closer to rrough rather than rv. But, the vertical filtering of surfaces generated in
the normal direction produces greater underestimation of the rrough -values than without filtering.
(a)
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(b)
(c)
Fig 4. Relative variation of the surface area ratios of rough wavy surfaces generated numerically by vertical and normal additions in
terms of zw/λ for (a) zr/zw= 0.005, (b) zr/zw= 0.05 and (c) zr/zw= 0.5. The experimentally accessible values of the surface area ratio of
filtered rough wavy surfaces, rfn, are also plotted.
4. Conclusions
Algorithms based on the flattening of surface topography in the vertical direction should not be applied indiscriminately
to any surface. The waviness removal can be problematic because commonly it does not know the type of roughening
process that created the surface (ballistic or growth/erosion). How the roughness is added to the waviness in the surface
is unknown. This impedes the surface filtering. Besides, the waviness filtering algorithms for surfaces generated by
normal growth processes will be very complex and computationally expensive. The solution or alternative to avoid the
effects produced by the waviness removal, or to compare the overall behavior of general rough surfaces with different
waviness, is the waviness-weighted roughness factor. We found than the rough surfaces generated in the normal
direction will always have greater surface area ratio, without assumptions on small slopes or absence of overhangs, and
closer to the corresponding values of roughness factor.
Acknowledgements The supports by the Spanish MINECO (project MAT2014-60615-R) and the “Junta de Andalucía” (project
P12-FQM-1443) are gratefully acknowledged.
Appendix A. Exact derivation of surface area ratio for arbitrary rough wavy surfaces
Due to the different types of processes which can govern the evolution of a surface (i.e. the addition of roughness to the
unprocessed surface), the waviness filtering has a strong directional character. This will affect any hybrid estimator of
roughness [11]. Thus, an exact derivation of the surface area ratio for any general rough wavy surface, considering the
directional character of the mentioned addition, is required. Accordingly, in this appendix, roughness, waviness and
rough wavy surfaces will be modeled by univalued continuous functions.
In general, any curved surface S can be described by two generic curvilinear coordinates (q1,q2). This way, from the

local position vector r ( q1 , q2 ) , the unit normal vector to S is defined as:


1
∂r ∂r
(A.1)
Nˆ (q1 , q2 ) = 
×

∂r ∂r ∂q1 ∂q2
×
∂q1 ∂q2
whereas the area of the surface S will be:
A= 
S


∂r ∂r
×
dq1dq2
∂q1 ∂q2
Since the local position vector can be expressed in terms of the Cartesian unit vectors { xˆ, yˆ , zˆ} as:

r (q1 , q2 ) = x( q1 , q2 ) xˆ + y ( q1 , q2 ) yˆ + z (q1 , q2 ) zˆ
the cross product in (A.1) can be generally expressed as:
541
(A.2)
(A.3)
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Microscopy and imaging science: practical approaches to applied research and education (A. Méndez-Vilas, Ed.)


∂r ∂r
∂ ( y, z )
∂ ( x, z )
∂ ( x, y )
×
=
xˆ +
yˆ +
zˆ =
∂q1 ∂q2 ∂ (q1 , q2 )
∂ (q1 , q2 )
∂ (q1 , q2 )
=

∂ ( x, y )  ∂ ( y , z )
∂ ( x, z )
xˆ +
yˆ + zˆ  =

∂ (q1 , q2 )  ∂ (q1 , q2 )
∂ (q1 , q2 )

(A.4)
∂ ( x, y ) 
∂z
∂z 
yˆ 
 zˆ − xˆ −
∂ (q1 , q2 ) 
∂x
∂y 
where a coordinate transformation is achieved using Jacobians and their properties. Substituting the vector (A.4) into
(A.1), the unit normal vector can be rewritten in terms of the Cartesian map z(x,y) as:
1
(A.5)
Nˆ ( x, y ) =
( zˆ − ∇z )
2
1 + ∇z
=
being ∇ the surface gradient operator expressed in Cartesian coordinates. Further, if the magnitude of the vector (A.4)
is introduced into (A.2), the area can be expressed as:
A=

X
2
1 + ∇z dxdy
Y
(A.6)
where again the transformation property of Jacobians has been applied.
A.1 Addition of roughness and waviness: vertical direction
Roughness is considered as a noise function, n(x,y), added locally to a generically shaped surface S0, i.e. the waviness,
in the vertical direction ẑ . Let f(x,y) be the explicit function that defines the waviness. The overall surface S will be
described by:
z ( x , y ) = f ( x, y ) + n ( x, y )
(A.7)
Substituting (A.7) into (A.6), the surface area in terms of roughness and waviness is written as:
1 + ∇n ⋅ ∇ ( n + 2 f )
2
(A.8)
1+
A =   1 + ∇f
dxdy
2
X Y
1 + ∇f
Hence, the surface area ratio (1) is given by:
rv =
∇n ⋅ ∇(n + 2 f )
1+
1 + ∇f
(A.9)
2
S0
where the angled brackets stand for the surface averaging:
•
  • 1 + ∇f
≡
  1 + ∇f
X
S0
2
Y
X
2
Y
dxdy
(A.10)
dxdy
The expression (A.9) confirms the dependence of the surface area ratio on the first derivatives of both roughness and
waviness. Note that, as expected, for the case without waviness (i.e. f(x,y)= const.), rv becomes equal to:
rrough =
1 + ∇n
2
(A.11)
xy
where the brackets stand for the following surface average:
•
xy
≡
  •dxdy
  dxdy
X
Y
X
Y
(A.12)
Further, if there is no roughness (i.e. n(x,y)= const.), rv is the unit as expected.
A.2 Addition of roughness and waviness: vertical direction
Roughness is considered as a noise function added locally in the normal direction to a generically shaped surface S0.

Let r0 (q1 , q2 ) be the vector that defines this initial topography (waviness) and n(q1,q2) the noise function (roughness)
[24]. If the unit normal vector to each point on the surface S0 is Nˆ (q , q ) , the position vector of the overall surface S
0
will be given by:
1
2
 
r = r0 (q1 , q2 ) + n(q1 , q2 ) Nˆ 0 (q1 , q2 )
(A.13)
If (q1,q2) are the curvilinear coordinates referred to the principal directions on each point over the surface S0, the
formulae of Rodrigues [24] reads:
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
∂Nˆ 0
∂r
= −κ 0i 0
(A.14)
∂qi
∂qi
with i=1, 2 and where κ0i represents the principal curvature on the i-principal direction [24]. Applying partial derivatives
to Eq. (A.13) and taking into account the result (A.14), each partial derivative of the position vector of the overall
surface can be expressed as:


∂r
∂r
∂n ˆ
(A.15)
= (1 − κ 0i n) 0 +
N0
∂qi
∂qi ∂qi

Further, if the curvilinear coordinates (q1,q2) are orthogonal, the cross product of the derivative vectors ∂r ∂qi is
reduced to:


∂r0
∂r0
∂n
∂n
−
1
κ
n
(
)
(1 − κ 01n )






02
∂
∂
∂
∂
q
q
q
q2 ∂r0
∂
∂
∂
r
r
r
∂r ∂r
1
0
×
= (1 − (κ 01 + κ 02 )n + κ 01κ 02 n 2 ) 0 × 0 − 2
− 1
(A.16)


∂q1 ∂q2
∂q1 ∂q2
∂q1
∂q2
∂r0
∂r0
∂q1
∂q2
From the magnitude of the vector (A.16) and the average surface (A.10), the surface area ratio (1) is
straightforwardly derived as:
2
rn = 1 − 2 H 0 n + K 0 n
2
 qˆ ⋅ ∇n   qˆ2 ⋅∇n 
1+  1
 +

 1 − κ 01n   1 − κ 02 n 
2
(A.17)
S0
where H0 and K0 stand for, respectively, the (local) mean and Gaussian curvatures of S0:
1
(A.18)
H 0 = (κ 01 + κ 02 ) ; K 0 = κ 01κ 02
2
and qˆi , the i−coordinate unit vector. It is straightforward to prove from Eq. (A.17) that the variation of local surface
area will be:
(A.19)
ΔA = ( 2 H 0 n − K 0 n 2 ) A0
in terms of the initial local surface area A0 (waviness).
It is worth to note that the surface area ratio (A.17) depends on the local waviness curvatures (first and second
derivatives of the surface). For non-zero constant noise, i.e. when the noisy wavy surface is mathematically parallel to
the wavy surface, the peaks are sharpened/flattened and the valleys are reciprocally flattened/sharpened (see Figure A1)
according to the sign of n (erosion or growth). Locally, from Eq. (A.18), the surface area of the peaks (H0 < 0)
increases/decreases and it decreases/increases for the valleys (H0 > 0). This leads to that, unlike rv (Eq. (A.9)), rn ≠ 1
when n = const.; rn becomes rrough (Eq. (A.11)), if there is no waviness (H0 = K0 = 0 or κ01 = κ02 = 0). Otherwise, it
becomes zero if n = 1/κ0i, i.e. when the non-overlapping condition (see Figure 3) is not fulfilled and the surface area is
ill-defined.
Fig A1. Initial profile (continuous line) and final profile (dashed line) produced by homogeneous (uniform) normal-directed erosion.
The roughening process depends on the local curvatures. The peaks are sharpened and the valleys are flattened.
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