erimentalown that
urfactantsre water.ore size isions are in
Journal of Colloid and Interface Science 273 (2004) 589–595www.elsevier.com/locate/jcis
Capillary imbibition of surfactant solutions in porous mediaand thin capillaries: partial wetting case
V.M. Starov,a,∗ S.A. Zhdanov,a and M.G. Velardeb,c
a Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UKb Instituto Pluridisciplinar, Universidad Complutense, 1, Paseo Juan XXIII, 28040 Madrid, Spain
c International Center for Mechanical Sciences, CISM, Palazzo del Torso, Piazza Garibaldi, 33100 Udine, Italy
Received 30 July 2003; accepted 6 February 2004
Abstract
The capillary imbibition of aqueous surfactant solutions into dry porous substrates is investigated from both theoretical and exppoints of view in the case of partial wetting. Cylindrical capillaries are used as a model of porous media to study the problem. It is shif the mean pore size is below a critical value, then the permeability of the porous medium is not influenced by the presence of swhatever the value of the concentration: the imbibition front moves exactly in the same way as in the case of the imbibition of puThe critical radius is determined by the adsorption of the surfactant molecules onto the inner surface of the pores. If the mean plarger than the critical value, then the permeability increases with increasing surfactant concentration. These theoretical conclusagreement with the experimental observations. 2004 Elsevier Inc. All rights reserved.
Keywords: Partial wetting; Capillary imbibition; Porous media
ofap--
olu-hasthe-bes,
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1. Introduction
Surfactants are widely used to control the kineticsliquid–liquid displacement, capillary penetration, and evoration. The adsorption of surfactants accelerates the impregnation of hydrophobic porous bodies by aqueous stions [1–10]. The process of immiscible displacementbeen intensively investigated both experimentally andoretically using model porous bodies and networks, tuslits, and periodically shaped capillaries [11–17].
In particular, because world oil resources are limited,exploitation of oil fields to the highest degree is desirabNew methods are required to improve the recovery rateoil fields and to get the oil trapped in the pores of rocIn secondary oil recovery water is injected to move outoil (water flooding). The injection of surfactants (surfactflooding) to reduce the interfacial tension between theand water phases, allowing the recovery of oil trappedsmall pores, has also been suggested [18].
* Corresponding author.E-mail addresses: [email protected] (V.M. Starov),
[email protected] (M.G. Velarde).
0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.02.033
Surfactant-enhanced remediation has become a promiing techniques to remove nonaqueous phase liquidssoils and aquifers [19,20]. It is a promising innovative tenology for the restoration of such contaminated sites.
In most of the above-mentioned processes it is assuthat surfactant molecules reach the moving water–oil inface. However, if they do not reach the moving frontresults is a waste of surfactants and no enhancement at
Generally, it is difficult to separate the effects of the cillary shape and of surface tension on the capillary presof the moving meniscus. Hence the interest of using mocylindrical capillaries.
Earlier studies [1,2] of the kinetics of the capillary imbbition of aqueous surfactant solutions into hydrophobic cillaries have shown that the rate of imbibition is controllby the adsorption of the surfactant molecules ahead omoving meniscus on the dry hydrophobic surface of the cillary. This process results inthe partial hydrophylization othe surface of the capillary in front of the moving meniscand gives the possibility of penetration into the initially hdrophobic capillary. The same idea has been used in [2describe the spreading of aqueous surfactant solutionshydrophobic substrates.
590 V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595
ntanesentac-f the
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d theithe
pil-yetlta-ts ohel tonalap-inre
verm-wi-llaryft forv-fines isolu-ater
ion
ac-
e–andes
-
s sobareted.-al to
qua-
lue,mn-
nceted
In this paper we consider the imbibition of surfactasolutions into porous substrates, nitrocellulose membrpartially wetted by water. This case is considerably differfrom that of a hydrophobic porous medium in that surftants do not reach the moving three-phase contact line imean pore size is below some critical value.
2. Theoretical analysis
Let us consider the imbibition of an aqueous surfacsolution into a cylindrical capillary, whose walls are partiawetted by water. A cylindrical capillary is used as a moof a porous medium. The situation in this case is differfrom the case of a hydrophobic capillary [1,2]. Indeed, wter can penetrate into the capillary even in the absencsurfactant molecules at the moving meniscus. Howeverpresence of surfactant molecules at the moving meniscusults in a lower contact angle relative to the pure water cand, hence, leads to a higher capillary pressure behinmeniscus. Accordingly, the imbibition rate increases wthe increased concentration of surfactant molecules at thmoving meniscus.
The moving meniscus covers fresh parts of the calary walls, where surfactant molecules had not adsorbedThus the imbibition process is accompanied by the simuneous adsorption of surfactant molecules onto fresh parthe capillary walls in a vicinity of the moving meniscus. Tstrength of adsorption in a porous medium is proportionathe surface per unit volume, which is inversely proportioto the radius of the capillary in the case of a cylindrical cillary. On the other hand, the rate of imbibition is lowerthinner capillaries due to higher friction. This gives motime for diffusion to bring new surfactant molecules to cothe fresh part of the capillary walls. Thus we have two copeting trends. Accordingly, if the capillary radius is belosome critical value, adsorption proceeds faster than imbibtion, all surfactant molecules are adsorbed onto the capiwalls, and, hence, there are no surfactant molecules lethe meniscus and the surfactant concentration at the moing meniscus vanishes. Thus for very thin capillaries (orporous media) the imbibition rate of surfactant solutionindependent of the surfactant concentration in the feed stion and takes a value equal to that in the case of pure wWe provide below further insight into this question.
Let us consider the imbibition of a surfactant solutfrom a reservoir with fixed surfactant concentration,C0(feed solution), into a thin capillary of radiusR � L (Fig. 1),whereL is the capillary length. Pure water (at zero surftant concentration) partially wets the capillary; hence,
(1)γ (0)cosθa(0) = γsv(0) − γsl(0) > 0,
whereγ , γsv, γsl, θa are the liquid–air, the solid substratvapor, and the solid substrate–liquid interfacial tensionsthe advancing contact angle, respectively. All these valu
-
.
f
.
Fig. 1. Imbibition of a surfactant solution in thin capillaries:C0, concentra-tion at the capillary entrance (in the feed solution);Cm, concentration onthe moving meniscus;�(t), position of the moving meniscus.
are concentration-dependent. LetCm be the bulk concentration of surfactant behind the moving meniscus; then
γ (Cm)cosθa(Cm) = γsv(0) − γsl(Cm) > γsv(0) − γsl(0)
(2)= γ (0)cosθa(0).
It is assumed in Eq. (2) that the imbibition process goefast that transfer of the surfactant molecules onto thesurface in front of the moving meniscus can be neglecHence, the solid–vapor interfacial tension,γsv, does not depend on the surfactant concentration and remains equits value at zero concentration. Furthermore,γsl(Cm) is adecreasing function of the surfactant concentration. Etion (2) shows thatΨ (Cm) = γ (Cm)cosθa(Cm) is an in-creasing function of the concentration, with maximum vaΨmax = Ψ (CCMC), reached at the CMC, and minimuvalue,Ψmin = γsv(0)−γsl(0), reached at zero surfactant cocentration.
2.1. Concentration below CMC
Let the surfactant concentration at the capillary entrabe below the CMC,C0 < CCMC. The transfer of surfactanmolecules in the filled portion of the capillary is describby the equation
∂C(t, x, r)
∂t= D
∂2C(t, x, r)
∂x2+ D
1
r
∂
∂r
(r∂C(t, x, r)
∂r
)
(3)− ∂
∂x
(v(r)C(t, x, r)
),
whereC(t, x, r) is the local concentration of surfactant;t , x,r are time, axial, and radial coordinates, respectively;D isthe diffusion coefficient; andv(r) is the axial velocity distri-bution.
Integration of Eq. (3) from 0 toR, whereR is the capil-lary radius, results in
∂
∂t
( R∫0
rC(t, x, r) dr
)
= D∂2
∂x2
( R∫rC(t, x, r) dr
)+ R
(D
∂C(t, x, r)
∂r
)r=R
0
V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595 591
onbars
al to
dn.
thery is
llaryis
sur-stantan
ntrara-des
t-for
ep-
n--
dlowcen-
di-
asse.nce,
n
de-ted,that
v-ries
Fig. 2. Typical time evolution of the imbibition front. SDS concentrati0.1% and nitrocellulose membrane with mean pore size 0.22 µm. Errorare not shown because they would fit inside the square symbols.
(4)− ∂
∂x
( R∫0
rv(r)C(t, x, r) dr
).
The second term on the right-hand side of Eq. (4) is equ
(5)−D∂C(t, x, r)
∂r
∣∣∣∣r=R
= ∂Γ
∂t− Dsl
∂2Γ
∂x2,
whereDsl is the surface diffusion coefficient over the filleportion of the capillary andΓ is the surface concentratioHence, Eqs. (4) and (5) yield
∂
∂t
( R∫0
rC(t, x, r) dr
)
= D∂2
∂x2
( R∫0
rC(t, x, r) dr
)− R
(∂Γ
∂t− Dsl
∂2Γ
∂x2
)
(6)− ∂
∂x
( R∫0
rv(r)C(t, x, r) dr
).
The characteristic time scale for the equilibration ofsurfactant concentration in a cross section of the capillaτ ∼ R2/D ≈ 10−3 s, for R ∼ 1 µm andD ∼ 10−5 cm2/s.The characteristic time scale of the spontaneous capirise in partially hydrophilic nitrocellulose membranesabout 10 s (Fig. 2). The latter shows thatτ is much smallerthan the time scale of the imbibition process. Hence, thefactant concentration has enough time to reach a convalue in each cross section of the capillary, and the surfacconcentration,C = C(t, x), depends only on the position,x.Figure 1 presents a sketch of the system and the concetion profile inside the capillary. Taking the latter considetion into account Eq. (6) can be rewritten, after both si
tt
-
are divided byR2/2, as
2
R
∂Γ
∂t+ ∂C
∂t= D
∂2C
∂x2 + Dsl2
R
∂2Γ
∂x2 − v∂C
∂x,
(7)0 < x < �(t),
wherev = (2/R2)∫ R
0 rv(r) dr is the mean velocity (expecing no confusion in the reader, the same symbol is usedboth the mean and the local velocity). The mean velocity,v,is equal to the meniscus velocity,
(8)v = d�
dt,
where�(t) defines the position of the moving meniscus.To further simplify the problem we use the simplest st
wise adsorption isotherm,
(9)Γ (C) ={
Γ∞, C > 0,
0, C = 0,
which together with Eqs. (7) and (8) yields
(10)∂C
∂t= D
∂2C
∂x2 − d�
dt
∂C
∂x, 0 < x < �(t).
As a result of adsorption onto the capillary wall the cocentration of the surfactant molecules on the moving meniscus,Cm, is lower than at the capillary entrance and shoulbe determined in a self-consistent way. The solution beshows that a solution exists in the case of constant contration on the moving meniscus,Cm.
The solution of Eq. (10) must satisfy the boundary contions
(11)C(t,0) = C0,
(12)C(t, �(t)
) = Cm,
together with the condition on the moving front
(13)2Γ (Cm)
R
d�
dt= −D
∂C
∂x
∣∣∣∣x=�(t)
.
The latter condition expresses the conservation of mat the moving meniscus andthe three-phase contact linThe simplest way to deduce it is to use the mass bala2πRvΓ (Cm) = πR2j , wherej = −D(∂C/∂x)x=� is thediffusion flux to the moving meniscus.
As R � L (Fig. 1) the liquid flow inside the capillary cabe approximated by a Poiseuille flow; hence
(14)
d�
dt= R2
8η
2γ (Cm)cosθa(Cm)
R
1
�= Rγ (Cm)cosθa(Cm)
4η�,
whereη is the dynamic (shear) viscosity, taken here inpendent of surfactant concentration. As earlier indicawe continue assuming, in a self-consistent procedure,the surfactant concentration remains constant at the moing meniscus (similar to the case of hydrophobic capilla[1,2]).
592 V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595
ndsndi-
-y,em
erothen
ons(9)ins
inlemLet
tem
for
osi-
l-
tantthisbe-en-
twozeroeen
Solution of Eq. (14) using the initial condition�(0) = 0gives
(15)�(t) = K√
t, K =√
Rγ (Cm)cosθa(Cm)
2η.
Substitution of Eq. (15) into Eq. (10) results in
(16)∂C
∂t= D
∂2C
∂x2 − K
2√
t
∂C
∂x,
with boundary conditions (11), (12), and
(17)Γ (Cm)K
R√
t= −D
∂C
∂x
∣∣∣∣x=K
√t
,
instead of (13).Let us introduce the following similarity coordinateξ =
x/(K√
t) and assume that the solution of Eq. (16) depeon this coordinate only. Then Eq. (16) and boundary cotions (11), (12), (17) take the form
(18)λ2C′′ = C′′[1− ξ ],(19)C(0) = C0,
(20)C(1) = Cm,
(21)C′(1) = −Γ (Cm)Ψ (Cm)
2Dη,
whereλ2 = (2D)/K2 = (4Dη)/(RΨ (Cm)) is a dimension-less small parameter. For aqueous surfactant solutionsD ∼10−5 cm2/s, η ∼ 10−2 P, andγ ∼ 102 dyn/cm; hence,λ2 ∼ (10−9 cm)/R. In what follows we consider only capillaries with R > 0.1 µm = 10−5 cm and, consequentlλ2 < 10−4 � 1. Equations (18)–(21) show that the problunder consideration has self-similar solutions.
If the concentration on the moving meniscus is above z(the case of zero concentration is considered below),boundary condition (21) can be rewritten as
(22)C′(1) = −Γ∞Ψmax
2Dη,
and Eq. (18) should be solved with boundary conditi(19), (20), and (22). Note that it follows from Eqs. (2) andthat hereΨ is independent of the concentration and attaits maximal value,Ψmax.
The small parameterλ2 affects the highest derivativeEq. (18). Hence, we have a singular perturbation proband matching of asymptotic solutions ought to be used.us introduce the following local variable,z:
(23)z = 1− ξ
λ.
Using (23) the inner solution of Eq. (18) satisfies the sysof equation and boundary conditions
C′′ = −zC′, 0 < z < ∞,
C′(0) = Γ∞Ψmaxλ
2Dη,
(24)C(0) = Cm,
together with
(25)C(∞) = C0.
The solution of the problem (24), (25) is
(26)C(z) = Cm + Γ∞Ψmaxλ
2Dη
z∫0
exp
(−z2
2
)dz.
Using the boundary condition (25) and the solution (26)the unknown concentration on the moving meniscus,Cm, wehave
C0 = Cm + Γ∞Ψmaxλ√
π
23/2Dη
or
(27)Cm = C0 − Γ∞
√πΨmax
2DηR.
The concentration on the moving meniscus should be ptive, Cm > 0; hence
(28)C0 > Γ∞
√πΨmax
2DηR
or, in other words,
(29)R >Γ 2∞πΨmax
2DηC20
.
Let us define
(30)Rcr = Γ 2∞πΨmax
2DηC2CMC
.
Then two latter inequalities (28), (29) become
(31)C0 > CCMC
√Rcr
R
and
(32)R > RcrC2
CMC
C20
,
respectively.Let us consider two cases: (i)R < Rcr and (ii)R > Rcr.In the first case, (i), condition (32) is violated for all va
ues of the concentration in the feed solution,C0, betweenzero and the CMC. Thus the concentration of surfacmolecules on the moving meniscus is equal to zero inconcentration range. Accordingly, there are two regionshind the moving meniscus: one, close to the capillarytrance, where the concentration changes fromC0 in the feedsolution to zero on the moving boundary between theregions; the second, where the concentration remainsduring the whole process. Let the moving boundary betwthese two regions be located at
(33)�1(t) = Kβ√
t,
whereβ < 1 is yet to be determined.
V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595 593
ains5),
ionle
e
-in-the
evi-
en-e
on-erooes
ue,cange,ne
oves
n-ov-ves
ortpil-oven,-esetheo-ndid-
nlycen-
n-ionn
theof
hecon-firstntly,is
s inby
be-
n-
In this case the concentration on the meniscus remzero and the meniscus moves “slowly” according to Eq. (1
(34)�(t) =√
RΨmin
2η
√t .
The concentration profile in the first region is the solutof the following problem (using the same similarity variabas before)
(35)λ2C′′ = C′(1− ξ), 0 < ξ < β,
(36)C′(β) = −Γ∞Ψminβ
2Dη,
(37)C(0) = C0,
(38)C(β) = 0.
As λ � 1 we have
(39)β = 1− λχ,
whereχ is a new unknown quantity. The solution of thproblem (35)–(39) gives the equation forχ
C0 = Γ∞
√πΨmin
2DηRexp
(−χ2),and hence
(40)χ =(
Γ∞C0
√2πΨmin
DηR
)1/2
.
Thus, the concentration changes fromC0 at the capillary en-trance,x = 0, to zero atx = K(1−λχ)
√t and remains zero
in the neighborhood of the moving meniscusK(1 − λχ) ×√t < x < K
√t , wherex = K
√t is the position of the mov
ing meniscus. It appears that the adsorption process in thenough capillaries consumes all surfactants and the imbibition is not influenced by the presence of surfactants infeed solution whatever the value of its concentration.
Let us consider the second caseR > Rcr. Note that thecapillary radius is assumed to be bigger than in the prous case. If the concentration in the feed solution,C0, is lowenough, that is, condition (31) is violated, then the conctration on the moving meniscus,Cm, is equal to zero and thmeniscus moves “slowly” according to Eq. (34).
If, however, the concentration in the feed solution,C0, ishigh enough, that is, condition (31) is satisfied, then the ccentration of the surfactant molecules is different from zon the moving meniscus and the imbibition process g“faster” according to Eq. (15):
(41)�(t) =√
RΨmax
2η
√t .
Hence, if the capillary radius is bigger than a critical valthen the whole concentration range in the feed solutionbe subdivided into two parts: a low concentration ranC0 < Ccr = Γ∞
√(πΨmax)/(2DηR), where the adsorptio
consumes all surfactant molecules, the concentration on th
moving meniscus is equal to zero, and the meniscus m“slowly” (34); and a high concentration range,C0 > Ccr =Γ∞
√(πΨmax)/(2DηR), where the adsorption cannot co
sume all surfactant molecules, the concentration on the ming meniscus is different from zero, and the meniscus mo“faster” according to (41).
2.2. Concentration above CMC
If the concentration in the feed solution,C0, is above theCMC, then two zones form inside the capillary after a shinitial transient period. In the first region, close to the calary entrance, the concentration inside the capillary is abthe CMC. This region is followed by the second regiowhere the concentration is below the CMC. The concentration is equal to the CMC at the boundary between thtwo regions. Hence, two regions can be identified insidecapillary: the first region, from the capillary inlet to some psition,�M(t), where the concentration is above the CMC athe surfactant solution includes both micelles and indivual surfactant molecules. The second region, from�M(t) to�(t), is where the concentration is below the CMC and oindividual surfactant molecules are transferred. The contration is equal to the CMC atx = �M(t). In the secondregion,�M(t) < x < �(t), things occur as in the case of cocentrations below the CMC. Accordingly, the concentraton the moving meniscus,Cm, is below the CMC even whethe concentration at the entrance,C0, is above the CMC. Weconsider below only the transport in the first region.
Inside the first region, 0< x < �M(t), the concentrationof free surfactant molecules is constant and equal toCMC. Hence, the transfer is determined by the diffusionmicelles. The total concentration isC = Cmol + CM, andCmol remains constant and equal to the CMC; hence,
(42)∂C
∂t= DM
∂2C
∂x2− d�
dt
∂C
∂x,
whereDM denotes the diffusion coefficient of micelles. Tadsorption on the membrane pores is determined by thecentration of the free molecules, which is constant in theregion and so the adsorption is also constant. Consequethe diffusion of adsorbed molecules in the first regionomitted in Eq. (42). The transfer of surfactant moleculethe second region (region free of micelles) is describedEq. (10).
The boundary conditions on the moving boundarytween the first and second regions,�M(t), are
(43)
DM
(∂C
∂x
)x=�M−
= D
(∂C
∂x
)x=�M+
, C(�M, t) = Cc.
As in Refs. [1,2], if follows that
(44)�(t) = K√
t, �M(t) = KM√
t ,
whereK is given by Eq. (15) in terms of the unknown cocentration on the moving meniscus,Cm, andKM, which isalso an unknown value yet to be determined.
594 V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595
ayC,
d-daryraictwo
s-les.tlythe
tlyf a
sionalidh
ofe-, 1,
ed inhereart
am-veraingper-lledthe, 4,
ws,ntforim-age
mov-tion,) PC,
wasthental
ce:
n an
cm)hee
con-t.
eseµm)mliq-as
erete.
e-
hethe
m ofter-sur-
bi-m-data
ncesed
anescase
t-m.the
me-
Let a similarity variable be introduced in the same was in the previous case of concentration below the CMi.e., ξ = x/K
√t in Eqs. (10) and (42). Using the boun
ary conditions (43), the expressions (44), and the bounconditions (19)–(21) a system of two nonlinear algebequations can be deduced for the determination of theunknowns: the concentration on the moving meniscus,Cm,and the position of the boundary,KM. This system includea parameterε = DM/D < 1, which is the ratio of the diffusion coefficients of micelles and free surfactant molecuAs in Refs. [1,2], the solution of the new system only slighdeviates from the solution in the previous case whereconcentration is below the CMC. Hence, the constantK andthe expression for the critical radius (30) are only slighdifferent from their corresponding values in the case oconcentration below the CMC. Consequently, the concluconcerning the existence of a critical radius remains veven at concentrations above the CMC, in agreement witexperiment as we show below.
3. Experimental setup
Figure 3 shows the setup to control the permeabilityinitially dry porous layers. The time evolution of the permability front was monitored. The nitrocellulose membranewas attached to an up/down moving support, 2, and placa thermostatted and hermetically closed chamber, 3, w100% humidity (to prevent evaporation from the wetted pof the membrane) and fixed temperature (20± 0.5 ◦C) weremaintained. To prevent temperature fluctuations the chber was made from brass and in the chamber walls sechannels were drilled to allow circulation of a thermostattliquid. The chamber was equipped with a fan. The temature was obtained using a thermocouple. A container fiwith water helps maintaining a constant humidity insidechamber. At the bottom of the chamber a small Petri dishwas placed filled with different SDS water solutions.
The chamber was equipped with optical glass windo5, for observation of the imbibition front of the surfactasolution. A CCD camera, 6, and a VCR, 7, were useddata acquisition and storage. Automatic processing ofages was carried out on a PC, 8, using the Scion Im
Fig. 3. Sketch of the experimental set-up: (1) membrane, (2) up/downing device, (3) thermostatted chamber, (4) Petri dish with SDS solu(5) optical glass windows, (6) CCD camera, (7) video tape recorder, (8(9) light source.
l
image processor. The duration of each experimental runin the range from 2.5 to 30 s. A discretization of time inprocessing ranged from 0.04 to 2 s in different experimeruns; the pixel size in an image was 0.01 mm.
Experiments were carried out in the following sequen
• The membrane was placed in the chamber and left iatmosphere of 100% humidity for several minutes.
• The membrane was immersed vertically (0.1–0.2into a container with the SDS solution. After that tposition of the imbibition front was followed as timproceeded.
• Several runs for each type of membrane and eachcentration value of the SDS solution were carried ou
A 1.5 × 3 cm rectangular membrane was used. Thporous samples (mean pore sizes: 0.22, 0.45, and 3.0were cut from the nitrocellulose membranes purchased froMillipore. Each sample was immersed 0.1–0.2 cm into auid container and the position of the imbibition front wmonitored over time.
4. Experimental results and discussion
As in all our experimental runs we had� � ((kpc)/
(ηρg))1/2; hence, the action of gravity was neglected. Thwas unidirectional liquid flow inside the porous substraUsing Darcy’s law we can set
(45)�2(t) = kpct
η,
where�(t) defines the position of the imbibition front insidthe porous layer,k is the permeability of the porous membrane, andpc is an effective capillary pressure inside tporous sample. The permeability of the porous layer andcapillary pressure enter as a single parameter in the fora product in Eq. (45). Experiments were carried out to demine this parameter and its variation with the change infactant concentration. In all runs�2(t)/2 follows a straightline, whose slope gives the value ofkpc with kpc = K2η.
Figure 2 illustrates the typical time evolution of the imbition front (SDS concentration 0.1%; nitrocellulose mebrane with 0.22 µm mean pore size). The experimentalsatisfactorily follows Eq. (45).
Figure 4 shows experimentally determined dependeof kpc as a function of the SDS concentration in the fesolution for three different nitrocellulose membranes.kpc isindependent of the concentration in the case of membrwith 0.22 and 0.45 µm mean pore size. However, in theof membranes with 3.0 µm mean pore size, the value ofkpcincreases with increasing SDS concentration. Thus, the criical radius,Rcr, is somewhere between 0.45 and 3.0 µThe experimental data confirm our prediction concerningexistence of the critical pore radius below which the per
V.M. Starov et al. / Journal of Colloid and Interface Science 273 (2004) 589–595 595
on
ne 1)e in-
sizeental
en-
nsboth
ofverobro-ces
e ofer-nts,, thehened
fac-ean
ental
K–an-
3-
d
2.
0.
1
. 76
en-
ro-
253.
)
ffs,
ur.
ce
Fig. 4. Experimental values ofkpc versus concentration of an SDS solutifor nitrocellulose membranes with different mean pore sizes (inset).kpc re-mains constant in the case of membranes with both 0.22 µm (straight liand 0.45 µm (straight line 2) mean pore size, while it increases with thcrease in surfactant concentration for the membrane with mean pore3 µm (line 3). Each experimental point is an average of three experimruns.
ability is independent of the value of the surfactant conctration.
5. Conclusions
The capillary imbibition of aqueous surfactant solutiointo dry porous substrates has been investigated fromtheoretical and experimental points of view in the casepartial wetting. On one hand, cylindrical capillaries habeen used as a model of porous media to study the plem theoretically. On the other hand, partially wetted nitcellulose membranes were used to study the same proexperimentally. We have found that if the mean pore sizthe porous medium is below a critical value, then its pmeability is not influenced by the presence of surfactawhatever the value of the concentration. Consequentlyimbibition front moves in exactly the same way as in tcase of the imbibition of pure water. The above-mentio
-
s
critical radius is determined by the adsorption of the surtant molecules on the inner surface of the pores. If the mpore size is larger than the critical value, thenthe permeabil-ity increases with increasing surfactant concentration. Thestheoretical conclusions are in agreement with experimeobservations (Fig. 4).
Acknowledgments
This research was sponsored by the Royal Society (USpain Joint Project under Grant 15544) and by the Spish Ministerio de Ciencia y Tecnologia (Grant MAT20001517).
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