COMPUTATIONAL SIMULATION OF CAPILLARY FORCES
ACTING BETWEEN PARTICLES CONNECTED WITH LIQUID
“BRIDGE”
Y. Tselischev, V. Valtsifer, V. Strelnikov
Institute of Technical Chemistry of UB RAS, ac. Koroleva, 3, Perm,
Russia
tselishch@yandex.ru
Abstract
A method for computational simulation of capillary forces acting
between dispersive components’ particles connected with liquid
“bridge” has been developed. The method considers dimensions and
shape of particles, distance between them, a type of their “contact”,
volume and shape of liquid “bridge”, surface tension of liquid, and
wetting angle. The offered method is based on good correlation between
computational and experimental values of capillary forces.
Investigations in capillary interaction between variously shaped
particles–sphere, cone and plane – have been conducted. Computational
simulation has resulted in ascertainment of basic parameters of particles
and of liquid “bridge” exerting appreciable influence on capillary force
and capillary pressure values. A conclusion has been drawn that the
developed model enables to qualitatively appraise how humidity of air
influences equilibrium moisture load in powders.
Introduction
Capillary forces are most significant forces acting between
particles of dispersive components during their storage, processing, and
use under natural conditions. These forces are generated in a liquid
“bridge” which is formed within a narrow gap between particles due to
absorption of moisture from both the environment and its capillary
condensation. Values of these forces are dependent on various
parameters characterizing particles of dispersive components, and on
properties of liquid interlayer and of the environment. Immediate
measurement of capillary forces and investigation in their variations
caused by variable parameters of particles and of liquid “bridge” are
difficult enough to perform due to small values of objects and forces
under investigation. Among pioneer works theoretically describing
capillary forces acting between model spherical particles of ideal soil
221
were works published by W.B. Haines and R.A. Fisher as early as in
1920s [1-3]. Later, results of more complete theoretical and
experimental investigations in capillary forces acting between spherical
particles were mentioned in a number of works by various authors [4-7].
Despite undisputed achievements, results obtained in this field of
investigations as from these years are often either different or
contradicting; this phenomenon may be explained by considerable
complexity of records, of descriptions and, in a number of cases, of
necessary simulation of various parameters in dispersive systems.
This work aims at design of a computational model describing
capillary forces between particles of dispersive components, and at
determination of basic parameters of particles and of liquid interlayer
which exert appreciable influence on these forces.
Various dependences and methods for determination of capillary
forces between particles regarded, as a rule, as uniform spheres, are
known [1-7]. Capillary interaction between variously shaped and sized
particles, as is usually the case with real powders, had been investigated
much more rarely [8, 9]. Such status of research works is one of reasons
why a method for calculation of capillary forces which would consider
combined influence of various parameters of powders and could be
experimentally confirmed is currently not available.
Calculation part
This work presents a computational method for determination of
capillary forces between variously sized particles. This method considers
various parameters of a powder and of liquid ‘bridge”.
Due to interaction of liquid’s molecules between both themselves
and dispersive particles, surface of “bridge” is bent.
Microscopic investigations have shown the shape of liquid
“bridge” between two model particles (Fig.1) to be adequately described
by circumferences (O1 and O2) with radii r1 and r2, respectively [10].
Hence, investigation results enable approximating meniscus of “bridge”
as a circular arc. This approximation was used to ascertain dependences
needed to determine a capillary force value and a quantity of capillary
liquid.
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(а)
(б)
Fig.1. Liquid “bridges” between model particles (a), and shapes of their menisci
(b).
In the first stage of investigations, two spherically-shaped particles
with liquid “bridge” between them were regarded as a computational
cell (Fig. 2). At this point, the liquid “bridge” is assumed to be formed
from (a) moisture from the environment and, (b) liquid components from
dispersive compositions.
Fig.2. Calculation cell for determination of capillary force acting between
spherically-shaped particles
The obtained dependences describing capillary interaction
between particles of computational cell are mentioned further [10].

 1 1 
Fc   R1 sin 1 2 sin 1     R1 sin 1    ,
 r l 

223
(1)
r
R1 1  cos 1   R2 1  cos  2   h
,
cos 1     cos  2   
l  R1 sin 1  r  1  sin 1    ,
(2)
(3)
here, Fc – capillary force;
r, l – curvature radii of liquid “bridge”;
R1, R2 – radii of particles;
1, 2 – angles of liquid’s half-infilling in “bridge” for a
“contact” of particles with radii R1, R2 , respectively
(particles “1” and “2”);
θ – wetting angle;
h – gap between surfaces of particles.
A capillary force value, as in the case with uniformly-sized
particles [4], is dependent on two constituents. The first constituent (F)
is determined by surface tension of liquid acting along perimeter of
wetting area. The second constituent (Fp) is due to availability of either
depression or pressure caused by curvature of “bridge” surface; this
phenomenon is described by the Laplace equation. During investigation
in interaction of uniformly-sized spherical particles mimicked by
spherical segments with large radii of curvature, computational values of
capillary forces were in good accord with experimental data. This
circumstance had enabled using an approach based on the above
mentioned dependence of forces on two constituents, also for
investigation in capillary forces between variously-sized spherical
particles. At this point, “contacted” particles are to be implied not only
as particles with immediately contiguous surfaces, but also as particles
spaced some distance apart, this distance, however, not leading to
disruption of liquid “bridge”.
To investigate in capillary forces as influenced upon by
particles’ parameters, it is necessary to determine volume of liquid
“bridge” exerting appreciable influence on values of these parameters.
Taking into account a possibility to describe meniscus of “bridge” as a
circular arc, the Vc (volume of capillary liquid) value is determined as
volume of a figure formed by a circular arc with radius r turning around
vertical Y axis (Fig.3) minus volumes of particles submerged into liquid
(Vs):
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Vс  
y2
 F  y 
2
dy  Vs ,
(4)
y1
Here, F(y) is the equation of the curve forming a revolution
surface around the Y axis; y1, y2 are integration limits determined as
ordinates of upper and lower points of perimeters along which particles
are wetted with liquid.
The equation F(y) looks as follows:
F ( y)  r  l  r 2  y 2 ,
(5)
The Vs value is to be determined from the following dependence:
Vs 
4

3
 3 4 1 
2 1 
3
4 2 
2  2 
 R1 sin 2  3  2 sin 2   R2 sin 2  3  2 sin 2 





(6)
The volume of liquid “bridge” can be obtained after calculation of
upper and lower integration limits and of the Vs value, after substitution
of these values into Eq. (4, 5), after its integration and transformation.
Results and discussion
The obtained dependence of capillary force vs. quantity of liquid
within “bridge” during interaction of variously-sized spherical particles
(Dp1, Dp2) is presented in Fig.3.
Water was considered as capillary liquid wetting particles.
Particles were spaced 0.01 μm apart; this distance did not exceed 0.1%
of particles’ dimension. Results of more complete investigation in how
the h value exerts influence on values of capillary forces and of capillary
pressure in liquid “bridge” are presented below. To investigate in
maximally possible values of forces, a status of complete wetting of
particles with liquid (θ=0) was considered. As is apparent, augmentation
of liquid’s quantity is in all cases accompanied by maximal capillary
forces. This regularity was observed by several authors [4, 9, 11] during
investigation in interaction of not contacting uniform spherically-shaped
particles (h≠0). Capillary force values increase also, as dimensions of
particles increase (Fig. 3).
225
Fig.3. Patterns of capillary force vs. volume of liquid “bridge” dependence
during interaction of variously-sized spherical particles. Radius R1 of one
particle equals 5 μm. The R2/R1 ratios of particles are: 1–100; 2–10; 3–5;
4–2.5; 5–1.
However, as the R2/R1 ratio augments, it influences on capillary
forces to a lesser extent, especially with large volumes of liquid
“bridge”, when values of forces differ slightly. The obtained findings
had permitted not only a qualitative conclusion, as for example in [8, 9],
but also a quantitative demonstration of how differences in particles’
dimensions influence on values of capillary forces acting between them.
Patterns reflecting capillary force vs. volume of liquid “bridge”
φl (φl – ratio of volume of liquid in “bridge” to volume of particles)
dependence at variable wetting angle were plotted for interaction
between uniform spherically-shaped particles (Fig. 4).
As is apparent from Fig. 4, either worsening of wetting, or an
increase in wetting angle from 0 (pattern 1) to 80 (pattern 3) is
accompanied by a lesser value of Fc.
Variations of F and of Fp constituents as influenced upon by
volume of liquid “bridge” and by surface tension were appraised (Fig.
5). The obtained dependences show that, as volume of liquid “bridge”
augments, values of capillary force (Fc) and of its constituent (Fp)
decrease, whereas value of the Fσ constituent increases. It is quite
noticeable that, subject to small- and mid-sized “bridges”, total value of
226
capillary force (Fc) (pattern 1) is determined by either the Fp constituent,
or capillary pressure in liquid “bridge”.
1
2
Fc∙106, N
2
1
3
0
-7
-6
-5
-4
-3
-2
-1
lg jl
Fig.4. Patterns of capillary force vs. volume of liquid “bridge” dependence at
variable wetting angle (deg.): 1 – 0; 2 – 50; 3 – 80; h= 2 nm, D1=D2=10
μm
1
Fc, Fσ, Fp,[·106], N
2
2
1
3
4
6
0
5
-1
-7
-6
-5
-4
lg jl
-3
-2
-1
Fig. 5. Patterns of Fc, F, Fp vs. volume of liquid “bridge” dependence: 1, 4 –
Fc, 2, 5 – Fp, 3, 6 – F; 1-3 – σ=72.8 mN/m; 4-6 – σ=22.8 mN/m; D1[2]
=10 μm; h=2 nm; θ=0
As volume of “bridge” augments, proportion of the Fσ constituent
increases as well (pattern 3). A decrease in coefficient of surface tension
from 72.8 mN/m (patterns 1-3) to 22.8 mN/m (patterns 4-6), which is
227
typical for “water-ethyl alcohol” mixtures, leads to diminution of all
constituents of capillary force. Variable coefficient of surface tension of
a liquid mixture exerting influence on structural formation of porous
materials (incl. variations caused by variable capillary forces between
particles) was considered by us during synthesis of silica-based
mesoporous materials [12].
Investigation in capillary pressure within liquid “bridge” has led to
ascertainment of the fact that volumes of “bridge” within which
capillary pressure equals zero may exist. The P=0 condition has been
ascertained to be determined by quantity of liquid in “bridge”, by
wetting angle value, and by a gap between particles. The obtained
dependence of “bridge” volume vs. wetting angle at variable distance (h)
between uniform spherically-shaped particles is presented in Fig. 6. The
P=0 condition has been ascertained to be attained at two values of
liquid’s volume, namely at minimal (Pφ.g.min=0, dotted curves), and at
maximal (Pφ.g.max=0, firm curves). As is apparent, the obtained patterns
at P=0 form an area with capillary depression (P<0, Fc=f (Fσ , Fp)=”+”),
characterized by combined action of both constituents of capillary force
and, accordingly, by its maximal value.
0
P > 1; Fp = "-"; Fc=f(Fσ)="+"
-2
4
Pφ l. max=0
3
-4
lg jl
P < 1; Fp = "+"; Fc=f(Fσ, Fp)="+"
2
1
-6
Pφ l. min=0
-8
Fc=f(Fp)="-"
-10
0
15
30
45
60
75
90
, o
Fig. 6. Patterns of liquid “bridge” volume vs. wetting angle dependence at P=0
and at variable gap h (% D): 1 – 5 nm (0.05); 2 – 10 nm (0.1); 3 – 0.1 μm
(1): 4 – 1 μm (10); σ=22.8 mN/m; D1[2] =10 μm
Area above the pattern of Pφ.g.max=0 is determined by Fσ
constituent, whereas area below the pattern Pφ.g.min=0 is determined by
228
Fp=”-“constituent. Should volume of liquid within “bridge” either
exceed maximal value, or be below minimal value, capillary pressure P
and the Fp constituent of capillary force promote repulsion of particles
[13]. It has been ascertained that either separation of particles, or
increased gap (h) between them (from 0.05% in pattern I to 10% in
pattern 4) lead to appreciable diminution of this area. It is noticeable that
worse wetting of particles’ surface is also accompanied by diminution of
this area, albeit to a lesser extent.
Results of computational investigations in the Fc vs. h dependence
are presented in Fig. 7, 8. As is apparent, maximal values of capillary
force are attained under condition of “contact” between particles (pattern
I – h/D=410-5). With small- and mid-sized volumes of “bridges”
(patterns 1-4 in Fig. 7), a slightly increased distance between particles
leads to a considerable decrease in capillary force. As volume of
interlayer augments, a horizontal section of patterns appears, where Fc is
either not, or slightly dependent on separation of particles. As follows
from the obtained dependence, a powdered material even with moderate
content of liquid will ruin while slightly strained; augmentation of
liquid’s volume will increase plasticity of powder. Hence, larger-sized
“bridges” will exert determining influence on strength properties of
powder, since they exist in a larger diapason of variable distances
between particles.
Fc·107, N
9
6
1
3
2
3
4
5
6
7
8
0
-10
-9
-8
-7
-6
lg (h, m)
Fig. 7. Patterns of Fc vs. h dependence at variable φg: 1 – 110-8; 2 – 110-7; 3 –
110-6; 4 – 110-5; 5 – 110-4; 6 – 110-3; 7 – 110-2; 8 – 110-1; D=5 μm;
σ=72.8 mN/m
229
As is apparent from Fig. 8, worse wetting of particles with liquid,
alongside with the θ angle varying from 0 (patterns 1, 8) to 50
(patterns 1’, 8’), weakens their capillary cohesion forces; at this point,
qualitative character of this dependence remains.
8
Fc·107, N
6
8'
4
1
1'
2
0
-10
-9
-8
-7
-6
lg (h, m)
Fig. 8. Patterns of Fc vs. h dependence at variable wetting angle θ (deg.): 1, 8 –
0; 1’, 8’ – 50; and volume φg: 1, 1’ – 110-8; 8, 8’ – 110-1; D=5 μm;
σ=72.8 mN/m
Numerous types of capillary interaction between particles of
dispersive components may be described as “contacts” of particles
shaped as sphere, plane, and cone. While applying the approach used in
description of capillary interaction between spherically-shaped particles
and equations (1-6) derived for this case to the offered types of
“contacts”, we can use computational dependences to determine
capillary forces and volumes of “bridge” [10].
The obtained dependences of capillary forces vs. volumes of
“bridge” with various types of “contact” are presented in Fig. 9.
230
Fig. 9. Patterns of capillary forces vs. volumes of “bridge” dependence with
various types of “contact” between particles: 1 – ‘plane-plane”; 2 –
“sphere-plane”; 3 – “sphere-sphere”; 4 – “cone-cone” (=150); 5 –
“cone-plane” (=90); 6 – “cone-cone” (=90); h=0.01 μm; 2, 3 – D=1
μm; θ=0.
As is apparent from these patterns, a type of “contact” exerts
appreciable influence on character of dependence and on value of
capillary forces. With small quantity of liquid, value of capillary forces
is slightly dependent on a type of “contact”. Its influence starts growing,
as volume of “bridge” augments. For “plane-plane” contact, maximal
value of capillary force is attained (pattern 1); at this point, dimension of
flat particles appraised from diameter of their surface wetted with
capillary liquid (d0.3 μm) is less than dimension of spherically-shaped
particles (with wetting diameter up to 1 μm). Minimal value of force is
observable for “cone-cone” contact (pattern 6) with lesser angles at their
vertices. For “sphere-sphere” (pattern 3) and “sphere-plane” (pattern 2)
contacts, dependences have extreme character with availability of
maximums.
While comparing patterns 3, 2 for “sphere-sphere” and “sphereplane” contacts, respectively, with patterns 4, 6 for “cone-cone”
contacts, a distinction in principle between them is visible, namely: for
the former, growth of capillary forces, as volume of liquid increases, is
observable only in a certain interval of variable volume of liquid;
231
whereas for “cone-cone” contacts, increase in volume of liquid leads to
growth of capillary forces in entire diapason of variable volume of liquid
within “bridge”. The presented dependences have permitted a conclusion
that variable type of “contact” may be accompanied by variable value of
capillary force in a wide diapason of several hundreds percent.
The offered method enables also determination of volume of
liquid within capillary “bridges” between particles of dispersive
components. As mentioned above, storage and processing of any powder
under real conditions is accompanied by formation of liquid “bridge”
within a narrow gap between “contacting” particles and by appearance
of capillary forces within it. A value of these forces is dependent on
volume of liquid which, in turn, is determined by moisture value of the
environment. Investigations in how moisture of the environment
influences volume of water within “bridge” and value of capillary forces
were conducted with use of the offered dependences (1-6) and while
taking into account: (a) the Kelvin equation (7) describing capillary
condensation:
  M  1 1 
p
    ,
(7)
  1  exp 
p2
  l RgT  r1 r2  
where  - relative pressure of saturated vapor over bent surface of liquid;
p1, p2 – pressure of saturated vapors of liquid over bent and flat
surfaces;
σ – surface energy (tension) of liquid on phase interface;
M – molecular mass of liquid;
ρ1 – density of liquid;
Rg – universal gas constant;
T – absolute temperature;
r1, r2 – curvature radii of main mutually perpendicular normal
sections of liquid’s surface;
and (b) the Laplace equation (8) describing capillary pressure in liquid:
1 1
P  pl  p g      ,
 r1 r2 
232
(8)
where pl, pg – pressure in liquid and gaseous phases, and under condition
of no mass exchange between medium and “bridge”.
0
3
lg jl
-2
-4
2
-6
-8
3
-10
-3
-5
-7
-9
lg (R, m)
Fig. 10. Patterns of relative volume of water in “bridge” vs. dimensions of
particles at variable relative humidity of air (j, %): 1 – 50; 2 – 90; 3 –
99.
Calculation results of relative volume of water in “bridge” vs.
dimensions of particles at variable relative humidity of air ( j, %) and at
zero wetting angle are presented in Fig. 10.
As is apparent from Fig. 10, volume of condensed water is
proportional to both humidity of air and dimensions of powder
material’s particles. Humidity of air influences most appreciably at the j
value exceeding 90%. With use of the obtained equations, volume of
water in “bridge” and capillary force values between aluminium powder
particles vs. relative humidity of air have been calculated. Basic data for
calculations: (a) the powder is composed of spherically-shaped particles
sized 20 μm; (b) ambient temperature 20C; (c) experimentally
determined wetting angle 65. The obtained dependences are presented
in Fig. 11. Comparison of computational values of water condensed in
“bridges” (in accord with dependences 1-10) with experimental data for
equilibrium moisture load of aluminium powder and of other
components differing in nature has manifested their qualitative
coincidence [14].
233
lg (Fc, N), lg φl
-4
-6
1
2
-8
-10
40
50
60
70
80
90
j, %ϕ, %
Fig. 11. Computational patterns of capillary force (1) and of relative volume (2)
of liquid “bridge” vs. relative humidity of air for particles with D=20
μm.
As is apparent from Fig. 11, volume of liquid in “bridge” starts
rapidly augmenting, as relative humidity of air exceeds 85% value. So,
variation of relative humidity of air from 10% to 85% results in twoorder augmentation of volume of liquid; 85%-100% variation – to fourorder augmentation. Such augmentation of quantity of water leads to
drastic growth of capillary forces acting between particles, as is reflected
by pattern 1. The noted tendencies are confirmed by authors in a number
of works [15, 16].
Conclusions
Thus, method for computational simulation of capillary forces
acting between dispersive components’ particles which considers
dimensions and shape of particles, distance between them, a type of their
“contact”, volume and shape of liquid “bridge”, surface tension of
liquid, and wetting angle has been offered.
Acknowledgements
This work was financially supported by Russian Foundation for
Basic Research (projects Nr. 14-03-00957_a and 14-0396009_p_ural_a).
234
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10.
11.
12.
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