# Slip length for a viscous flow over spiky surfaces

Alexei T. Skvortsov<EMAIL_ADDRESS>Defence Science and Technology Group,
Melbourne, VIC 3207, Australia  Denis S. Grebenkov
<EMAIL_ADDRESS>Laboratoire de Physique de la Matière
Condensée,
CNRS – Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau,
France

###### Abstract

For a model of a 3D coating composed of a bi-periodic system of parallel
riblets with gaps we analytically derive an approximate formula for the
effective slip length (an offset from the flat surface at which the flow
velocity would extrapolate to zero) as a function of the geometry of the
system (riblet period, riblet height, and relative gap size). This formula is
valid for an arbitrary fraction of gaps (i.e from narrow riblets to narrow
gaps) and agrees with the known analytical results for the 2D periodic coating
of riblets without gaps.

##  I Introduction

The viscous flow over surfaces covered by sharp elements (riblets, grooves,
spikes, or pillars) has been the key component in many problems of
microfluidics (lab-on-a-chip Stone_2004 ; Rothstein_2010 ), geophysics (canopy
flows Monti_2022 ), and biomechanics (the so-called shark skin phenomenon
Bechert_1989 ; Luchini_1991 ; Dean_2010 ; Martin_2014 ). A spiky coating has a
remarkable (and, perhaps, counterintuitive) property of drag (shear stress)
reduction of the viscous flow compared to a flat surface, although in the
former case, the contact area between the fluid and solid is much higher
Deyn_2022 ; Ran_2021 ; Lee_2016 . This property has made spiky coatings an
attractive candidate for many practical applications (e.g., drag reduction of
ships and drones Domel_2018 , improvement of propeller performance Chen_2023 ,
micro-pump design Rothstein_2010 ; Huang_2016 ) and stimulated many
experimental and theoretical studies Crowdy_2011a ; Prosperetti_2007 ;
Prosperetti_2007a ; Ling_2017 ; Bazant_2008 ; Golovin_2017 ; Quere_2008 ;
Crowdy_2011 ; Truesde_2006 ; Philip_1972 ; Lauga_2003 ; Asmolov_2012 . This is
an active area of research with extensive literature, see Deyn_2022 ; Ran_2021
; Zeng_2022 ; Li_2023 ; Crowdy_2022 ; Modesti_2021 , and references therein.

The effect of a coating of complex morphology on viscous flow has been
conventionally quantified by a parameter called effective slip length
Rothstein_2010 ; Bechert_1989 ; Crowdy_2011a ; Bazant_2008 ; Philip_1972 ;
Lauga_2003 ; Asmolov_2012 . This parameter can be introduced with the
following arguments. Near a flat surface the velocity of flow is directed
along the surface, it is zero at the surface (no-slip boundary condition) and
can be modeled by a linear profile

$v(y)=Jy,$ (1)

where $y$ is the distance from the surface, constant $J$ is related to the
friction drag at the surface $\tau=\mu dv/dz=\mu J$, and $\mu$ is fluid
viscosity. With a coating of complex morphology, the flow just above and
inside the coating can be very complex. Nevertheless, far from the surface the
linear relation $v(y)$ is restored but with an additional parameter $\lambda$

$v(y)=J(y-\lambda),$ (2)

where $\lambda$ (can have either sign) has a dimension of length and is called
the effective slip length Rothstein_2010 ; Bechert_1989 ; Crowdy_2011a ;
Bazant_2008 ; Philip_1972 ; Lauga_2003 ; Asmolov_2012 . This parameter is the
aggregated measure of the effect of coating morphology on the hydrodynamic
properties of the surface. Condition $y=\lambda$ corresponds to the fictitious
coordinate at which the flow velocity would extrapolate to zero (relative to
the surface $y=0$). Likewise, the parameter $\lambda$ can be introduced by
postulating a radiation boundary condition at the surface Prosperetti_2007a :

$v+\lambda\frac{\partial v}{\partial y}=0.$ (3)

The effective slip length may also incorporate the effect of changing boundary
conditions at some parts of the surface (from no-slip condition $v=0$ to no-
stress condition $\tau=0$) due to the air bubbles trapped between the spikes,
see Fig. 1. Evidently, the patches of no-stress areas of the surface (e.g.,
due to trapped air) may lead to a significant reduction of viscous drag and
that is often referred to as hydrophobic properties of the coating.
Alternatively, roughness can block access of the flow to some parts of the
surface so that the intrinsic hydrophobicity of the surface (as its
physicochemical property) can be significantly amplified Quere_2008 . This
necessitates investigation of the interplay of the effects of hydrophobicity
and roughness. To incorporate the effect of hydrophobicity of the spikes due
to the no-stress and no-slip patches of the spike surface the spikes can be
modeled with the radiation boundary conditions (see below). All these cases
are depicted in Fig. 1.

Figure 1: Examples of 2D riblet coating: a \- coating with a periodic system
of riblets; b \- a system of riblets with trapped air pockets (in grey); c \-
coating with a periodic system of partially hydrophobic riblets modeled with
partially absorbing (radiation) boundary condition (depicted in grey). The
flow velocity v is normal to the image plane.

The main focus of many theoretical studies of spiky coatings was the
analytical derivation of the value of parameter $\lambda$ as a function of
coating morphology. For the steady unidirectional flow of viscous fluid
without pressure gradient, in which the velocity vector is directed along the
$z$ axis and depends only on the other two coordinates (i.e., $v\equiv
v(x,y)$), the equations of motion reduce to the 2D Laplace equation
Bechert_1989 ; Bazant_2016

$\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}=0.$
(4)

A wealth of analytical results for 2D coatings have been derived by employing
the property of conformal invariance of Eq. (4) Bechert_1989 ; Crowdy_2022 ;
Luchini_1991 . For instance, for the 2D comb-like boundary or riblets shown on
Fig. 1, the results are as follows. For the no-slip boundary condition at the
spikes and on the base, Fig. 1a, Bechert_1989 ; Skvortsov_2014

$\lambda=\frac{W}{\pi}\ln\left[{\cosh(\pi H/W)}\right]\quad(\textrm{no-slip on
the base}),$ (5)

and for the no-slip boundary condition at the spikes and the no-stress
(‘hydrophobic’) boundary condition on the base, Fig. 1b, Skvortsov_2018

$\lambda=\frac{W}{\pi}\ln\left[{\sinh(\pi H/W)}\right]\quad(\textrm{no-stress
on the base}),$ (6)

where $W$ is the period of the comb-like structure (distance between spikes)
and $H$ is the height of the spikes. As $H\rightarrow\infty$, both equations
(5) and (6) behave similarly as

$\lambda\approx H-\frac{\ln 2}{\pi}W,$ (7)

so $\lambda$ tends to $H$ minus a universal offset proportional to the period
of the structure. For the riblets of other shapes (e.g., semicircle,
triangular, rectangular cross-sections) the results are similar and can be
found in Bechert_1989 ; Luchini_1991 .

For a periodic configuration of alternating (no-slip and no-stress) stripes on
a flat surface oriented perpendicular to the flow velocity Philip_1972 ;
Lauga_2003 ; Asmolov_2012 , one has

$\lambda=\frac{W}{2\pi}\ln\left[{1/\sin\left(\frac{\pi}{2}\sigma\right)}\right],$
(8)

where $\sigma$ is the surface fraction of the no-stress stripes, $W$ is the
period of the stripes.

For the 3D morphological structures of the coating (e.g., spikes, pillars, or
hemispheres) conformal transformation cannot be applied and there are only a
limited number of papers in which the parameter $\lambda$ has been derived
analytically, see Davis_2009 ; Davis_2010 ; Ng_2911 ; Ybert_2007 ;
Lindsay_2017  and references therein. In particular, the authors of Refs.
Davis_2009 ; Davis_2010  considered the model (that we refer to as the disk
model) in which a viscous flow exists only above ‘nanoforest’ composed of a
lattice of identical cylindrical pillars. In this model, the effect of the
coating on the viscous flow is reduced to the friction forces acting at the
top disks of each pillar (the flow satisfies the no-slip boundary condition on
the top of the circular pillar, $y=0$) whilst the flow inside the ‘nanoforest’
(e.g., between the pillars) is disregarded (at $y=0$ the no-stress boundary
condition was assumed everywhere except the top disks). It was found that at
the limit of a small areal density of the ‘nanoforest’ pillars (or surface
fraction of the top disks), $\sigma\ll 1$, the effective slip length obeys the
scaling law Davis_2009

$\lambda=H-\left(\frac{A}{\sqrt{\sigma}}-B\right)W,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma\rightarrow 0,$ (9)

where $W$ is the period of the pillar lattice (for simplicity assumed to be
the square lattice), $A=(3/16)\sqrt{\pi}$ and $B=(3/2\pi)\ln(1+\sqrt{2})$
Davis_2010 . It was also found that this scaling law is geometry specific,
viz., for the elongated (quasi-one-dimensional) cross-section of the
structural elements (wall-like in the current context) it changes from the
power-law (9) to the logarithmic form Davis_2010 :

$\lambda=H-\frac{1}{3\pi}\ln\left(\frac{4}{\pi\sigma}\right)W,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma\rightarrow 0.$ (10)

Formulas (9) and (10) are valid for $\sigma\ll 1$ Davis_2009 , Davis_2010 .
The limit $\sigma\to 0$ (no disks) corresponds to $\lambda\to-\infty$ or no
friction (drag) on the surface so that the approximate relation (2) does not
make sense anymore. The letter condition follows from Eq. (3) when the first
term, which is proportional to $1/\lambda$, becomes insignificant. The second
terms in Eqs. (9) and (10) become zero at $\sigma=(A/B)^{2}\approx 0.18$ and
$\sigma=4/\pi\approx 1.27$, respectively, instead of $\sigma=1$ (uniform
surface with the no-slip boundary condition), which is due to the
inapplicability of Eqs. (9) and (10) at high $\sigma$.

The aim of the present paper is to derive the self-consistent expression for
the effective slip length of the spiky coating as a function of the height of
its 3D structural element, similar to Eqs. (5), (6) for 2D.

##  II Riblets with periodic gaps

To appreciate the effect of the pillar height, we need to incorporate the flow
between pillars. We begin with the 2D solutions (5, 6) which we modify to
capture the 3D effects. More specifically, we assume that the riblets have the
periodic identical gaps with the relative size of the gaps being identical as
shown in Fig. 2. In some regards this model is complementary to the disk model
discussed above because all drag is generated by the viscous flow acting on
the side of the solid parts of the riblets whilst the contribution of the flux
from the top surface area of the riblets is neglected.

Figure 2: Models of coatings: a \- periodic system of riblets; b \- periodic
system of riblets with periodic gaps.

Let $x,y$ denote the horizontal and vertical axis, respectively, and the $z$
axis is directed along the riblets (and flow velocity) as shown in Fig. 2. The
boundary condition at the solid part of the riblets is $v=0$ (no-slip) and at
the gaps, the boundary condition is $\partial_{x}v=0$ (no tangential stress).
Applying the aforementioned arguments, the alternating boundary conditions at
the riblet surface ($y<H,x=\pm W/2$) imply that this surface (grey spikes in
Fig. 1c) can be treated with boundary homogenisation framework
Berezhkovskii_2004 ; Skvortsov_2015a ; Hewett_2016 ; Lindsay_2018 , or, more
specifically, it becomes partly hydrophobic and can be modeled with a
radiation boundary condition:

$\frac{\partial v}{\partial x}+\frac{v}{\lambda_{s}}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x=\pm W/2,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0<y<H,$ (11)

where $\lambda_{s}$ is given by Eq. (12)

$\lambda_{s}=\frac{L}{\pi}\ln\left[1/{\sin\left(\frac{\pi}{2}\sigma_{s}\right)}\right],$
(12)

where $L=s+g$ is the period of the solid-gap structure of an individual
riblet, $s$ is the width of the solid part of the riblet (per period), $g=L-s$
is the width of the gap, $\sigma_{s}=s/L$. For the case $g=0$ (no gaps) we
return to the solution given by Eqs. (5, 6).

With the homogenised boundary condition the original 3D problem reduces to a
2D problem that can be tackled analytically (although due to the radiation
boundary condition at the riblets conformal mapping is not helpful). Assume
that $v=v_{0}=const$ for some $y=\delta\gg H$ (Couette flow) and for a given
$\lambda_{s}$ we can derive offset $\lambda$ as $\delta\rightarrow\infty$. The
parameter $\lambda$, being a function only of the geometry of the coating, is
independent of $\delta,v_{0}$, and $\mu$.

Formally, due to the periodicity of the system, we need to find a solution of
the Eq. (4) for $0\leq y\leq\delta$ and $|x|<W/2$ with the following boundary
conditions (see Fig. 1c):

$\frac{\partial v}{\partial x}+\frac{v}{\lambda_{s}}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x=\pm W/2,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0<y<H,$ (13) $\frac{\partial
v}{\partial x}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x=\pm W/2,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ H<y<\delta,$ (14) $v=v_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ y=\delta,$ (15)
$v=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
y=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{no-
slip base,}$ (16)

or

$\frac{\partial v}{\partial y}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ y=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{no-stress base}.$ (17)

The parameter $\lambda$ for this setting can be derived from the solution of
Eqs. (13 – 17) by assuming that far above the coating ($H\ll y\ll\delta$) the
solution takes the form (2) and then by matching this solution with the one
inside the coating ($0\leq y\leq H$); this solution will be presented
elsewhere (see also Grebenkov22 ). For the purpose of this study, we derive a
simpler (approximate) solution for $\lambda$ that can straightforwardly be
deduced from the fact that the Robin boundary conditions (13) can be replaced
with the homogeneous boundary conditions $v=0$ but imposed on the equivalent
boundaries at $y=W/2+\lambda_{s}$ and $y=-W/2-\lambda_{s}$. From here the
approximate solution is immediately given by Eqs. (5, 6) with substitution
$W\rightarrow W+2\lambda_{s}$:

$\displaystyle\lambda$
$\displaystyle=\frac{W+2\lambda_{s}}{\pi}\ln\left[{\cosh(\pi
H/(W+2\lambda_{s}))}\right]\quad(\textrm{no-slip on the base}),$ (18)
$\displaystyle\lambda$
$\displaystyle=\frac{W+2\lambda_{s}}{\pi}\ln\left[{\sinh(\pi
H/(W+2\lambda_{s}))}\right]\quad(\textrm{no-stress on the base}),$ (19)

where $\lambda_{s}$ is given by Eq. (12).

This is the main result of the present Letter. It provides insights into the
dependence of the effective slip length on two-dimensional arrangements of the
pillars and their height. At $\lambda_{s}=0$ we return to the previous results
for the 2D case, Eqs. (5), (6). As $H\rightarrow\infty$, we recover the
asymptotic relation similar to Eq. (7):

$\lambda\approx H-\frac{\ln 2}{\pi}(W+2\lambda_{s}).$ (20)

In view of Eq. (12) the limit $\sigma_{s}\to 0$ in this formula recovers the
logarithmic dependency similar to Eq. (10). Finally, for
$\lambda_{s}\rightarrow\infty$ and $H$ fixed (sparse configuration of the
needle-like pillars), one finds

$\lambda\approx\frac{\pi H^{2}}{2(W+2\lambda_{s})},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{no-slip base},$ (21)
$\lambda\approx\frac{2\lambda_{s}+W}{\pi}\ln\left(\frac{\pi
H}{W+2\lambda_{s}}\right)-\frac{W}{\pi},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{no-stress base}.$ (22)

For the case of a square configuration of pillars ($W=L$) the plots of Eqs.
(18, 19) are depicted in Fig. 3. For the no-slip boundary condition on the
base (left panel), the parameter $\lambda$ is positive and exhibits a
monotonic increase with $\sigma_{s}$ (i.e., the fraction of the solid part of
the riblet). When $H$ is large, the dependence on $\sigma_{s}$ is very weak,
while $\lambda$ remains close to $H$, except for very small values of
$\sigma_{s}$ (note that $\lambda$ is rescaled by $H$ in this panel). In
particular, one sees that the asymptotic relation (20) accurately reproduces
the exact solution. In turn, in the limit $\sigma_{s}\to 0$ (no riblet), the
effective slip length vanishes according to Eq. (21), as it should. However,
this limit is achieved extremely slowly for large $H$. In fact, one has
$\lambda_{s}\approx(L/\pi)\ln(1/\sigma_{s})$, and this parameter should be
much larger than $\pi H/2$ or, equivalently, $\sigma_{s}\ll
e^{-\pi^{2}H/(2L)}$, to be able to apply Eq. (21). For instance, if $H/L=10$
(the red curve), one has $e^{-\pi^{2}H/(2L)}\sim 10^{-22}$, i.e., the
asymptotic relation (21) is not applicable for any reasonable $\sigma_{s}$. In
contrast, if the riblet height $H$ is much smaller than $W$, the asymptotic
relation (21) provides an accurate approximation for the whole range of
$\sigma_{s}$. In the intermediate case when $H\sim W$, the relation (21) is
applicable only for small $\sigma_{s}$ (left panel, middle curve).

The situation is quite different for the no-stress boundary condition on the
base (right panel). While the effective slip length still grows monotonously
with $\sigma_{s}$, it takes negative values as $\sigma_{s}\to 0$. Moreover,
$\lambda$ diverges to $-\infty$ in this limit, in agreement with Eq. (22). As
previously, the approach to this limit is very slow when $H$ is large, so that
Eq. (22) is not applicable. In this setting, one can use the large-height
expression (20).

Figure 3: The effective slip length $\lambda$ as a function of $\sigma_{s}$,
shown by symbols and given by Eqs. (18, 19) for the no-slip (left) and no-
stress (right) condition on the base, respectively. Here $L/W=1$ and three
values of $H/W$ are considered. Solid and dashed lines present the asymptotic
behavior (21, 22) for moderate heights $H/W=0.1$ and $H/W=1$, while dash-
dotted line indicates the large-height asymptotic relation (20). Note that
$\lambda$ is rescaled by $H$ on the left panel and by $W$ on the right panel.

The parameter $\lambda$ completely defines the relative change of drag due to
coating

$\frac{\tau_{\rm coat}-\tau_{\rm flat}}{\tau_{\rm
flat}}=\frac{\lambda}{\delta-\lambda}\approx\frac{\lambda}{\delta}$ (23)

for $\lambda\ll\delta$ and $\delta$ is defined in Eq. (15). Eqs (18) and (19)
and the data presented in Fig. 3 provide a clear guidance for targeted coating
optimisation.

##  III Discussion and future work

In summary, for a model of 3D spiky coating we derived an approximate formula
for the effective slip length as a function of the pillar height, and the 2D
arrangements of the pillars. There are several extensions that can easily be
incorporated into the proposed model. For instance, there is no need to assume
that riblets should have only one gap per period, since there is the formula
for $\lambda_{s}$ for an arbitrary number of gaps per period Crowdy_2011 ;
Skvortsov_2020 . This enables the analytical treatment of coatings with much
more complex structures.

Our results can also be extended for the pillars of arbitrary cross-section
(i.e., different from an infinitely thin interval) provided the momentum flux
through the top surface of the pillar can still be neglected. To this end, we
can apply the following rationale. It is known that the Stokes force (which is
proportional to the momentum flux over the surface of the pillar) is
proportional to the capacitance of the object (or logcapacity in 2D)
Hubbard_1993 . As a consequence, to translate the results of the proposed
framework to arbitrary pillars, it is sufficient to find a solid interval of
an equivalent logcapacity for a given pillar cross-section (note that the
logcapacity of an interval is $4s$ Landkof_1972 ). Moreover, the parameter $s$
is in the argument of logarithm, so the final result is insensitive to minor
inaccuracies in estimation of the equivalent logcapacity. Indeed, this
approximation can only hold for sparse configurations: $s/W\ll 1$. This
approach also allows us to make informative conclusions regarding the
applicability of the disk model for a coating of tall pillars (a brush). For
instance, the effective slip length due to the momentum flux through the top
surface and the side of the pillars can be characterised by the second terms
(offsets) in Eqs. (9), (20), respectively. By comparing these terms we arrive
at the simple condition of validity of the disk model

$\frac{1}{\sqrt{\sigma}}\ll\frac{(1+2\lambda_{s}/W)\ln 2}{\pi A}+\frac{B}{A},$
(24)

where constants $A$ and $B$ are defined in Eq. (9) and $\lambda_{s}$ is given
by Eq. (12). We note that both terms on the right side of this inequality are
positive and $B/A\approx 5.4$ so this condition is quite restrictive for
$\sigma$.

We believe that the presented results can be useful for the targeted design of
engineered coatings with desirable hydrodynamic properties before proceeding
with extensive computational simulations and experimental evaluation.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon request.

## Acknowledgements

A.T.S. grateful to Ian R. MacGillivray and Paul A. Martin and many insightful
discussions. D.S.G. acknowledges the Alexander von Humboldt Foundation for
support within a Bessel Prize award.

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11institutetext:  Institute of Mathematical and Computer Sciences, University
of São Paulo, São Carlos, São Paulo, Brazil
Lyles School of Civil Engineering, Purdue University, West Lafayette, Indiana,
USA
London Mathematical Laboratory, London, England, UK
Mathematics Institute, University of Warwick, Coventry, England, UK
São Carlos Institute of Physics, University of São Paulo, São Carlos, São
Paulo, Brazil

Complex systems Dynamics of social systems Structures and organization in
complex systems

# Cultural heterogeneity constrains diffusion of innovations

Aruane M. Pineda 11 Sandro M. Reia 22 Colm Connaughton 3344 José F. Fontanari
55 Francisco A. Rodrigues 111122334455

###### Abstract

Rogers’ diffusion of innovations theory asserts that the cultural similarity
among individuals plays a crucial role on the acceptance of an innovation in a
community. However, most studies on the diffusion of innovations have relied
on epidemic-like models where the individuals have no preference on whom they
interact with. Here, we use an agent based model to study the diffusion of
innovations in a community of synthetic heterogeneous agents whose interaction
preferences depend on their cultural similarity. The community heterogeneity
and the agents’ interaction preferences are described by Axelrod’s model,
whereas the diffusion of innovations is described by a variant of the Daley
and Kendall model of rumour propagation. The interplay between the social
dynamics and the spreading of the innovation is controlled by the parameter
$p\in[0,1]$, which yields the probability that the agent engages in social
interaction or attempts to spread the innovation. Our findings support Roger’s
empirical observations that cultural heterogeneity curbs the diffusion of
innovations.

###### pacs:

89.75.-k

###### pacs:

87.23.Ge

###### pacs:

89.75.Fb

##  1 Introduction

Everett Rogers proposed in 1962 a theory for the diffusion of innovations that
explained how novel ideas spread and are perceived by individuals in a
community [1]. Among the goals of Rogers’ ambitious enterprise was the
identification of the elements that determine how innovations spread among
individuals, communities, societies, and nations [2, 3]. The social fabric, or
the relationships and connections that individuals make with each other, is
one of these elements: Rogers found that the degree of cultural heterogeneity
of a community influences how far information spreads among its members,
concluding that cultural similarities facilitate the diffusion of innovations.

Individual heterogeneity refers to the variety of traits and behaviors that
characterize the individuals within a community. Individuals that exhibit a
wide range of traits and behaviors may also have varying requirements and
preferences that can affect both their interests and the rate of adoption of
novelties. Individual heterogeneity can result in a more varied and effective
information flow, which can speed up the diffusion of innovations. The degree
of shared values, beliefs, and norms among members of a community is referred
to as cultural similarity [1]. People of similar cultural backgrounds may have
similar requirements and preferences. According to Rogers, cultural similarity
can speed up the spreading of innovations by lowering uncertainty and
fostering interpersonal trust [4, 1, 5].

To facilitate mathematical analysis, individual heterogeneity is usually not
taken into account in the models for the propagation of rumours, innovations,
ideas, or even infectious diseases [6, 7]. However, individual heterogeneity
and cultural similarity can easily be incorporated in those models by
borrowing the agents’ representation and the interaction rules introduced by
Axelrod in his celebrated model of culture dissemination [8]. Axerold’s model
offers an explanation for the fact that although social interactions increase
the similarity between individuals – a phenomenon known as social influence –
there is still cultural diversity in human societies. In a two-dimensional
lattice where the agents interact with their nearest neighbors only, the model
exhibits two classes of absorbing configurations: ordered (monocultural)
configurations in which all agents (or the vast majority of them) exhibit the
same culture, and disordered (multicultural) configurations in which multiple
cultures coexist in the lattice. The absorbing configuration is the result of
the interplay between the disorder of the initial configuration, tuned by the
two parameters of the model, viz., $F$ (number of cultural features) and $Q$
(number of values each cultural feature takes on), and social influence [9,
10]. Interestingly, the model shows that multicultural configurations are
possible outcomes of a dynamics where agents become more alike after each
interaction.

The diffusion of new ideas in the academic world has been modeled using SIR-
like epidemic models [11, 12], but the standard epidemic model for studying
the spreading of rumours is the DK model proposed by Daley and Kendall in the
1960s [13]. In the DK model, the agents can assume three possible states,
viz., $S$ (ignorant), $I$ (spreader) and $R$ (stifler). Ignorants are not
aware of the rumour, spreaders are aware of the rumour and spread it, whereas
stiflers are aware of the rumour but do not spread it.

Figure 1: Mechanisms of cultural assimilation and rumour propagation. Agent
$i$ takes part in the cultural dynamics with probability $p$ and in the
spreading of the rumour with probability $1-p$. In the cultural dynamics,
agent $i$ selects a random neighbor $j$ and assimilates a divergent feature
with probability $P_{ij}$. In the rumour dynamics, partially represented here,
a spreader agent $i$ transmits the rumour to an ignorant neighbor $j$ with
probability $\lambda P_{ij}$. Figure 2: Absorbing configurations of the
coupled culture-rumour model. For a fixed number of cultural features $F$, the
diversity of the community is captured by the number of cultural traits $Q$.
In a scenario of low cultural diversity, interactions increase even further
the similarity between agents, which in turn facilitates the spreading of the
rumour. For highly diverse communities, the steady state is characterized by
the presence of many distinct cultural domains, which hamper the spreading of
the rumour.  The top row shows the initial (a) and final (b) cultural states
of the agents for $F=3$ and $Q=4$. Each color corresponds to a cultural
vector, so we can observe the ordering effect of social influence on the
disordered initial configuration. Panel (c) shows that all agents became
stiflers (blue) at the final state when there is a single spreader in the
initial configuration. The bottom row shows the initial (d) and final (e)
cultural states of the agents for $F=3$ and $Q=14$. In this case, social
influence is not strong enough to homogenize the cultural states of the
agents. Panel (f) shows that the rumour spreads across different cultural
domains, but some ignorants (light blue) remain in the community. The other
parameters are $L=10$, $\lambda=1$, $\alpha=0.01$ and $p=0.1$.

Here, we combine the Axelrod model and the DK model to study the diffusion of
innovations (i.e., rumours) in a more realistic scenario where the agents are
culturally heterogeneous at the outset, but their cultural dissimilarities
change dynamically as they interact. This approach allows us to further
investigate Rogers’ empirical observations regarding the relationship between
cultural heterogeneity and the success of innovations. The coupling between
these models is that the spreader’s attempt to pass on the rumour to an
ignorant or to a stifler is conditioned on the cultural similarity between
those agents. In that sense, our approach differs starkly from previous
studies of diffusion of innovations in Axelrod’s model in which the innovation
was a novel value of a cultural feature [14, 15]. The cultural or Axelrod’s
dynamics takes place with probability $p$, whereas the rumour or DK dynamics
takes place with probability $1-p$, so $p$ can be viewed as a proxy for the
openness of agents to social influence. A similar idea was used to verify the
impact of rumour propagation on disease spreading [16, 17] and to study
interacting diseases [18]. Henceforth we will use the words innovation and
rumour interchangeably.

Since we keep the number of cultural features fixed to $F=3$ throughout this
paper, the cultural diversity of the community is controlled solely by the
parameter $Q$: the greater the number of distinct values that each cultural
feature can take on, the more cultural diverse the community is. The main
focus of this study is the fraction of agents that are aware of the rumour
after the dynamics reaches an absorbing configuration, which is given by the
ratio between the number of stiflers and the community size. We note that in
an absorbing configuration there can be only stiflers and ignorants.
Henceforth, we will denote the average of this ratio over many independent
runs by $\langle R\rangle$. Clearly, $\langle R\rangle$ is a proxy for the
success of the rumour: the greater it is, the more agents are aware of the
rumour. The cultural diversity $Q$ and the openness to social influence $p$
will be the leading independent variables in our study. In particular, we find
that $\langle R\rangle$ decreases with increasing $Q$ and increases with
increasing $p$. Therefore, heterogeneity makes it difficult for a rumour to
spread in a community and so different narratives may emerge without need of
topological changes [19, 20, 21].

##  2 Methods

The agent based model we study here is illustrated in Fig. 1. Agents are
placed in a square lattice of side $L$ with periodic boundary conditions and
neighborhood given by the four nearest sites. As mentioned before, the Axelrod
and the rumour dynamics take place with probability $p$ and $1-p$,
respectively. The cultural and the rumour dynamics are, in principle,
independent. In the following, we present the details of the implementation of
both dynamics.

###  2.1 The Axelrod Model

Each agent $i$ is assigned a cultural vector
$\Phi_{i}=(\phi_{i1},\phi_{i2},...,\phi_{iF})$ of length $F$. Each component
of this vector represents a cultural feature prone to social influence, and
each feature can take on $Q$ values (traits), i.e.,
$\phi_{ik}=\\{1,\ldots,Q\\}$ [8]. The number of features ($F$) and traits
($Q$) determine the degree of disorder of the initial configuration since
there are $Q^{F}$ possible cultural states (or cultures) for each agent. The
cultural dynamics proceeds as follows:

  * •

An agent $i$ is selected at random.

  * •

An agent $j$ $\in$ $V_{i}$, where $V_{i}$ is the set of neighbors of agent
$i$, is selected at random.

  * •

With probability given by the cultural similarity of agents $i$ and $j$, viz.,

$P_{ij}=\frac{1}{F}\sum_{k=1}^{F}\delta_{\phi_{ik},\phi_{jk}},$ (1)

a feature $k$ such that $\phi_{ik}\neq\phi_{jk}$ is selected at random and
$\phi_{jk}\rightarrow\phi_{ik}$. Here $\delta_{\phi_{ik},\phi_{jk}}=1$ if
$\phi_{ik}=\phi_{jk}$ and $0$ otherwise

These steps are repeated until an absorbing configuration is reached. In an
absorbing configuration, any pair of neighbouring agents must share no
cultural features (i.e., $P_{ij}=0$) or have identical cultures (i.e.,
$P_{ij}=1$).

Figure 3:  Competition of cultural diversity and openness to social influence
on the spreading of rumours. Increase of cultural diversity $Q$ decreases the
mean fraction of stiflers $\langle R\rangle$, whereas increase of openness to
social influence $p$ decreases $\langle R\rangle$.  Panel A: $p=0$, i.e., the
culture of the agents is frozen in the initial random configuration. Panel B:
$p=0.5$. Panel C: $p=1$, i.e., the rumour propagates only after the cultural
dynamics reaches an absorbing configuration. Panels D, E and F show the
variances of the fraction of stiflers for $p=0,0.5$ and $1$, respectively. The
peak of the variance for large $L$ yields the approximate location of the
transition point between the regimes $\langle R\rangle\approx 1$ and $\langle
R\rangle\approx 0$. The lattice sizes $L$ are indicated in Panel F. The other
parameters are $F=3$, $\lambda=1$ and $\alpha=0.01$.

###  2.2 The Rumour Model

Here we use the variant of the DK model introduced by Maki and Thompson [22],
which we will refer to as MT model. Whereas in the DK model both agents can
have their states updated in a single interaction, in the MT model only one
agent is updated per interaction. Despite this difference, the outcomes of
both models are similar and well described by the same mean-field approach,
with the DK model exhibiting a higher dispersion of mean values than the MT
model [23]. In addition, in both rumour models the agents are homogeneous, in
the sense that they have (i) the same probability of receiving and re-
transmitting the rumour when they become spreaders, and (ii) the same
probability of becoming stiflers.

The spreading of the rumour is a contact process driven by the following
rules:

  * •

An agent $i$ is selected at random.

  * •

If agent $i$ is a spreader, then:

    * –

An agent $j\in V_{i}$ is selected at random.

    * –

If agent $j$ is ignorant, then agent $j$ becomes a spreader with probability
$\lambda P_{i,j}$.

    * –

If agent $j$ is a spreader or a stifler, then agent $i$ becomes a stifler with
probability $\alpha P_{i,j}$.

These steps are repeated until the dynamics reaches an absorbing configuration
where there are no spreaders in the lattice. To ensure convergence, we
introduce the recovery probability $\delta=10^{-6}$ for a spreader to turn
into a stifler spontaneously [24].

We note that the propagation of the rumour is modulated by the cultural
similarity of the interacting agents. In other words, the propagation of the
rumour from agent $i$ to agent $j$ is possible if and only if these agents
have at least one cultural feature in common to trigger a social interaction
(i.e., $P_{ij}>0$). The cultural dynamics, however, is not influenced by the
rumour dynamics.

##  3 Results

The mean fraction of stiflers $\langle R\rangle$ in the absorbing
configurations is frequently used as an order parameter in epidemic models
[25] and so here we use this measure to characterize the asymptotic regime of
the coupled culture-rumour dynamics. As already mentioned, the cultural
dynamics is controlled by the parameters $F$ and $Q$, whereas the (pure)
rumour dynamics is controlled by $\lambda$ and $\alpha$. For fixed $F=3$,
increase of $Q$ leads to a sharp transition from a monocultural to a
multicultural regime [9, 10], as illustrated in Fig. 2.

Figure 4: Effect of the basic reproduction number on the spreading of rumours.
Mean fraction of stiflers $\langle R\rangle$ as function of the basic
reproduction number $\mathcal{R}_{0}=\lambda/\alpha$ for $Q=1,5$ and $14$ as
indicated.  In low diversity communities ($Q=1$ and $Q=5$), the rumour spreads
over a sizeable portion of the community provided that $\mathcal{R}_{0}>1$,
but for high diversity communities ($Q=14$) it is limited to a tiny region
around its creator, regardless of the value of $\mathcal{R}_{0}$.  The other
parameters are $F=3$, $L=50$, $p=0.5$ and $\alpha=0.01$. We keep $\alpha$
fixed and vary $\lambda$ to change the ratio $\lambda/\alpha$.

Figure 3 shows that $\langle R\rangle$ is strongly influenced by the cultural
heterogeneity of the community. We observe two asymptotic regimes: a regime
where the rumour spreads to a sizeable portion of the lattice (i.e., $\langle
R\rangle\approx 1$), and a regime where the rumour is restricted to a
microscopic portion of the lattice (i.e., $\langle R\rangle\approx 0$). The
transition point between these regimes depends on the cultural and
epidemiological parameters of the coupled culture-rumour dynamics. However,
since our primary interest is the effect of cultural heterogeneity on the
propagation of the rumour, we set $\lambda=1$ and $\alpha=0.01$, which
correspond to the basic reproduction number
$\mathcal{R}_{0}=\lambda/\alpha=100$. This guarantees that the rumour spreads
to the entire lattice ($\langle R\rangle=1$) in the case of homogeneous agents
(i.e, for $Q=1$) as can be seen in the top panels of Fig. 3.

These results show that $\langle R\rangle$ decreases as $Q$ increases,
regardless of value of the parameter $p$, which measures the openness of the
agents to social influence. Thus, cultural heterogeneity always impairs the
spreading of the rumour. As $p$ increases from $p=0$ to $p=1$, the curves
$\langle R\rangle$ vs. $Q$ shift to the right (top panels of Fig. 3),
revealing that the spreading of the rumour is facilitated by the openness to
social influence.

In fact, recalling that for $p=0$ the rumour dynamics takes place in the
frozen initial configuration of the agents’ cultures, which corresponds to the
most heterogeneous (or disordered) configuration for a given $F$ and $Q$.
Consequently, we observe a very small range of $Q$ where $\langle
R\rangle\approx 1$, indicating that the spreading of the rumour is heavily
impaired by cultural heterogeneity. On the other hand, for $p=1$ the cultural
dynamics proceeds until it reaches an absorbing configuration, and only then
the rumour dynamics sets in. In this case, we observe a wider range of $Q$
where $\langle R\rangle\approx 1$, which highlights that openness to social
influence facilitates the spread of rumours.

A word is in order about the phase transition between the asymptotic regimes
characterized by $\langle R\rangle>0$ and $\langle R\rangle\to 0$. Since a
phase transition occurs only in the thermodynamic limit $L\to\infty$, we can
only infer its onset for finite systems by considering lattices of different
sizes [26], as done in Fig. 3. A typical sign of the onset of a phase
transition is the increase of the size of the fluctuations, which in our case
is captured by the variance of the fraction of stiflers. Thus, the value
$Q=Q_{c}(L)$ at which the variance is maximum offers a rough estimate of the
transition point, as illustrated in the bottom panels of Fig. 3.

It is interesting that for $p=0$ and $\mathcal{R}_{0}\to\infty$ our problem
reduces to the bond percolation in the square lattice [27] where the
probability that a bond transmits the fluid is $1-\left(1-1/Q\right)^{F}$ and
so $Q_{c}=\left(1-2^{-1/F}\right)^{-1}$. For $F=3$, it yields $Q_{c}\approx
4.85$ that agrees very well with the results of panel A of Fig. 3. More
importantly, since in the bond percolation we have $0<\langle R\rangle<1$ at
the critical point $Q=Q_{c}$, the transition is discontinuous and only for
$Q=1$ (homogeneous agents) the rumour spreads over the entire lattice. We
conjecture that similar results hold for $p>0$ as well.

For the sake of completeness, Fig. 4 shows the influence of the basic
reproduction number $\mathcal{R}_{0}=\lambda/\alpha$ on $\langle R\rangle$. As
expected, provided that $\mathcal{R}_{0}>1$ and that $Q$ is small enough such
that the absorbing configurations of the cultural dynamics are monocultural,
the rumour spreads to a sizeable portion of the lattice, but never to the
entire lattice as pointed out before. The transition at $\mathcal{R}_{0}=1$ is
continuous [13, 22]. However, for large $Q$ such that the absorbing
configurations of the cultural dynamics are multicultural, the rumour reaches
only a few agents close to the location of the initial spreader, regardless of
the value of $\mathcal{R}_{0}$.

##  4 Conclusion

In his seminal book, Rogers has offered plentiful empirical evidence that
cultural similarity plays a major role in the fate of innovations [1]:
innovations are more likely to spread among individuals sharing similar
beliefs. The understanding of the role of heterogeneity in the spreading of
innovations has allowed companies to reduce risk and maximize the impact of
new products. In this paper, we add to Roger’s conjecture by exploring a
scenario in which the cultural heterogeneity of the community is described by
Axelrod’s model of culture dissemination and the diffusion of the innovation
is described by a variant of the Daley and Kendall model of rumour
propagation. Hence the interchangeability of the words innovation and rumour
throughout the paper.

Our results suggest that curbing the spread of rumours does not necessarily
require changes in the social fabric, i.e., changes in the topology of the
agents interaction networks [19, 20, 21]. In fact, this outcome appears
naturally in communities of heterogeneous agents who are allowed to hold
different opinions and beliefs, thus providing theoretical support for Roger’s
empirical observations. Specifically, we find that cultural heterogeneity
always impairs the propagation of the rumour and that, beyond a critical level
of heterogeneity, the rumour is stuck in the neighborhood of its creator.

Rogers’ theory of the diffusion of innovations is a sociological theory that
builds on many natural experiments (e.g., adoption of hybrid corn, Norplant
contraceptive, rap, cellular phones and Nintendo home video-game players) to
determine the characteristics of both the innovation and the target community
that best correlate with the success of the innovation [1]. The theory
contends that the number of adopters of an innovation increases in time
following an S-shaped growth curve (see [14] for a study of these growth
curves for the pure Axelrod dynamics). Since our entire analysis is based on
the characterization of the absorbing configurations, we have access only to
the ultimate number of adopters of the innovation, viz., $\langle R\rangle$.
In that sense, a direct comparison with real data is not possible, as those
data focus solely on the growth of the number of adopters. Nevertheless, our
results support Rogers’ conjectures and empirical observations in a
qualitative way. For instance, Rogers’ discussion of homophily as a barrier to
the diffusion of an innovation is in full agreement with the conclusions of
our study.

The study of the coupled culture-rumour dynamics adds to the rapidly growing
literature on social physics [28]. In fact, the availability of high quality
data on social patterns and human activity (see, e.g., [29]), as well as the
great appeal and relevance of social issues, has prompted physicists’
contributions to several interdisciplinary areas, including (but not limited
to) collective intelligence [30, 31], cultural polarization [32], criminal
behavior [33], human mobility [34], urban dynamics [35, 36, 37], traffic flow
[38, 39], social networks [40, 41], and moral behavior [42, 43]. These
advances have been made possible by the maturing of complex systems as a
research field, as attested by the 2021 Nobel Prize in Physics [44]. Perhaps
the key characteristic of a complex system is the emergent phenomena resulting
from its interacting components. In the coupled culture-rumour dynamics, the
emergent phenomenon is the discontinuous phase transition that separates the
regime where the rumour spreads over the community from the regime where it
dies out close to its creator. The existence of this transition could only be
revealed through a mathematical and computational modeling, hence the
importance of assessing Roger’s theory for the diffusion of innovations using
the tools of statistical mechanics.

###### Acknowledgements.

JFF was partially supported by Fundação de Amparo à Pesquisa do Estado de São
Paulo (FAPESP), grant 2020/03041-3 and Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), grant 305620/2021-5. AMP acknowledges the
support of São Paulo Research Foundation (FAPESP), grant 2019/22277-0. This
research was conducted using the computational resources of the Center for
Research in Mathematical Sciences Applied to Industry (CeMEAI) funded by
FAPESP, grant 2013/07375-0.

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