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Spreading of surfactant solutions over thin aqueous layers at low concentrations: Influence of solubility
Journal of Colloid and Interface Science 329 (2009) 361–365
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Spreading of surfactant solutions over thin aqueous layers at low concentrations: Influence of solubility K.S. Lee, V.M. Starov ∗ Department of Chemical Engineering, Loughborough University, Loughborough, LE11 3TU, UK
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 7 July 2008 Accepted 4 October 2008 Available online 29 October 2008 Keywords: Surfactant solutions Marangoni effect
A small droplet of aqueous surfactant solution at concentration below CMC was deposited on a thin water layer. A moving circular wave in the centre was formed. The time evolution of the radius of the wave was monitored. Two surfactants of different solubility were used. It was shown that the time evolution of the moving front (i) proceeds in two stages: a fast first stage and slower second stage; (ii) the time evolution of the front motion substantially depends on the surfactant solubility. We suggest a qualitative explanation of the phenomenon, which reasonably agrees with our experimental observations. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Spreading behaviour in the presence of surface tension gradients frequently shows instabilities [1,2]: fingering instability of the liquid films driven by temperature gradients [1], stick-slip phenomena when a liquid containing surface-active molecules advances on a hydrophilic solid results in an instability of the moving front [2]. The presence of instabilities is an interesting phenomenon in its own rights. However, on the current stage it is difficult to extract information on properties of surfactants from experimental data on instabilities. Hence, it is important to select a method, which allows investigating the dynamic properties of surfactant solutions on the liquid–air interface without instabilities occurring. It has been shown earlier [3,4] that investigation of the motion of a front after deposition of a small droplet of aqueous surfactant solution on a thin aqueous layer allows investigating separately the influence of Marangoni phenomenon on the hydrodynamic flow. In this case the capillary and gravitational forces can be neglected [3]. The suggested method results in a stable motion of the wetting front. The latter allows extracting properties of surfactants on a liquid–air interface as described in [3,4]. The same method was used for an investigation of properties of trisiloxane solutions [5]. Let us consider a schematic of the experimental procedure (Fig. 1): a small droplet of an aqueous surfactant solution is deposited on the top of the initially uniform thin aqueous film (thickness h0 ) as in Fig. 1a. The latter results in a formation of a depression in the centre of the film and a moving front forms (Fig. 1b). When a droplet of a surfactant solution is deposited on a clean liquid–air interface tangential stress develops on the liquid surface.
*
Corresponding author. E-mail address: [email protected] (V.M. Starov).
0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.10.031
©
2008 Elsevier Inc. All rights reserved.
The latter is caused by the non-uniform distribution of the surfactant concentration, Γ , over a part of the liquid surface covered by the surfactant molecules. The latter leads to surface stresses and a flow (Marangoni effect) [6]:
η
∂ u (r , h) dγ ∂Γ = , ∂z dΓ ∂ r
(1)
where η and u are the dynamic shear viscosity of the liquid and tangential velocity on the liquid surface h, respectively; (r , z) are radial and vertical coordinates; γ (Γ ) is the dependency of the liquid–air interfacial tension on surface excess concentration of the surfactant. The surface tension gradient-driven flow induced by the Marangoni effect moves surfactant molecules along the surface and a dramatic spreading process takes place. The liquid–air interface deviates from an initially flat position to accommodate the normal stress occurring in the course of the motion. This process is referred below to as a spreading process. The consideration in [3] was restricted to the case of “insoluble” surfactants, that is, those solubility can be neglected over duration of the process (referred to below as “insoluble surfactants”). The current state of art in the area was presented in [3] with the proper references therein. Surface diffusion was neglected in [3] as compared to convective transfer. The latter assumption was proven to be valid [3,4]. Surfactants of different solubility at concentrations above CMC [4] were used to investigate the influence of surfactant solubility on the spreading process. It was shown that the time evolution of the moving front proceeds in two stages: a rapid first stage associated with disintegration of micelles, which is followed by a slower second stage, when all micelles were already disintegrated. The time evolution of the moving front substantially depended on the surfactant solubility. An exact solution for the evolution of the moving front was deduced for the case of insoluble surfactants at
362
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
(a)
(b) Fig. 1. (a) A small droplet of aqueous surfactant solution is deposited on a top of thin aqueous layer of thickness h0 ; (b) cross section of the system: R (t )—radius of a circular moving front, r (t )—radius of dry spot in the centre, H —the height of the moving front.
Fig. 2. The first stage. Small aqueous surfactant solution droplet deposited on the surface of pure water film. Concentration of surfactant inside the small droplet is c 0 . Radius of the droplet base is L (t ).
concentration above CMC. A qualitative theory was developed to account for the influence of the solubility on the front motion. Experimental observations [4] are in a good agreement with the theory predictions. The duration of that stage is fixed by the initial amount of micelles in the deposited droplet. The fast stage is followed by the second slower stage when the surfactant concentration decreases in the film centre but the total mass of surfactant remains constant over the whole film in the case of insoluble surfactants and decreases over time in the case of soluble surfactants. In both cases a solution provides a power law dependency predicting the position of the moving front, R (t ), as time proceeds [4]. 2. Theory In our experiments below a droplet of surfactant solution with concentration below CMC was placed in the centre of a thin aqueous film (Fig. 1) which the initial concentration of surfactant c 0 < CMC. Our experimental observations showed that as in the case of concentration above CMC two distinct stages of spreading take place: (i) a first fast stage, and (ii) a second slower stage. This is similar to the case of the concentration above CMC [4]. However, the physical reason behind those stages is substantially different from the case of high concentrations above CMC. During the first stage the deposited droplet serves as a reservoir of the surfactant molecules, hence, surface concentration of single molecules is kept constant during that stage in the centre. That is, the following boundary condition are satisfied during the first stage of the moving droplet contact line, L (t ) (Fig. 2):
Γ (t , 0) = Γeq (c 0 ),
(2)
where Γeq (c 0 ) is the equilibrium adsorption isotherm and c 0 is the initial concentration inside the droplet. The following assumptions have been adopted in the boundary condition (2): (i) there is an equilibrium on the edge of the surfactant droplet, L (t ), and the edge of the liquid film (Fig. 2); (ii) the radius of the small liquid droplet, L (t ), is much smaller than the radius of the moving front, R (t ), that is L (t ) R (t ); (iii) the dissolution of surfactant molecules from the small droplet into the underlying pure water can be neglected during the first stage. Assumption (i) is precisely the same, which was adopted earlier [3,4]. It is not in contradiction with our previous and current
experimental data. However, this can be violated in the case of surfactants, which undergo a reorientation on the liquid–air interface. Assumption (ii) allows using the boundary condition (2) at the centre of the liquid film. The spreading of liquid droplets on the surface of the film of the same liquid [7] obeys the same power law as the spreading over dry solid substrate in the case of the complete wetting (which is obviously the case under consideration) and proceeds according to the following power law dependency, L (t ) = A · t 0.1 , where r is the droplet base radius and the constant A was calculated in [7]. It will be shown below that R (t ) ≈ B · t 0.5 during the first stage, where B is a constant. That is, the development of the spreading front, R (t ), proceeds much faster than the spreading of the small droplet. The latter was confirmed by our direct experimental observations. Note, the surfactants solution from the droplet, penetrates into the underlying water film and, hence, dL /dt = v sp − v sh , where v sp is the velocity of spreading, which is equal to v sp = 0.1 · A · t −0.9 according to the previous and v sh is the velocity of shrinkage, cased by the penetration of the droplet directly into the underlying water film. The latter rate is proportional to the area of the droplet base, v sh = C · π L 2 , where C is a constant. Hence, the dependency of the radius of the droplet base on time can be described by the following differential equation: dL /dt = 0.1 · A · t −0.9 − C π L 2 . Solution of the latter equation shows that the droplet radius goes via a maximum value and disappears with a characteristic time scale, t 1 ≈ 1/(π L 0 C ), where L 0 is the initial value of the droplet base. Assumption (iii) can be justified using the consideration presented in [3]. It means that the characteristic time scale of dissolution of surfactant molecules should be substantially bigger than the time of spreading and disappearance of the small droplet. Our experimental data with two surfactants of different solubility confirm the validity of this assumption for those two surfactants. The slower stage was observed when the concentration in the small droplet was below CMC and fast first stage was not detected by the regular camera used in [3]. Below we used a high speed video camera, which allowed us to detect the presence of the fast first stage even in the case of surfactant concentration below CMC. Past t 1 , a second slower stage starts. During the second stage the total mass of surfactant remains constant in the case of non-soluble surfactants and decreases in the case of soluble surfactants.
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
We can follow consideration presented in [3] to deduce the following power laws for the first stage: R (t ) = A 1 · t 0.5 ,
t < t1
(3)
Table 1 Properties and concentrations of surfactants used and experimentally obtained spreading exponents. Materials
CMC (mmol/l)
C/CMC
SDS DTAB
8.3 [3] 15.1 [8]
0.1 0.1
and R (t ) = A 2 · t
0.25
t > t1 ,
,
(4)
during the second slower stage, where A 1 and A 2 are constants, which can be deduced using consideration similar to that presented in [3,4]. Note, the two power law equations (3) and (4) were deduced for the case of insoluble surfactants and we try to modify those two laws for the case of soluble surfactants. 2.1. First stage of spreading In [4] a very simple semi-empirical derivation of the dependency of the moving front on time was suggested. Those derivations were based on the boundary conditions on the moving front. The concentration gradient on the moving front can be estimated as
Γeq (c 0 ) Γ (0) − Γ ( R ) Γ (0) ∂Γ ( R ) ∼− =− =− , ∂r R R R
(5)
where the boundary condition on the moving front was used [3,4]:
Γ ( R ) = 0.
(6)
Note, the latter boundary condition is not “the real boundary condition” but an effective condition for the inner solution: that is we ignore the existence of a narrow region where the concentration changes very rapidly (see [3,4] for details). The next boundary condition should be satisfied on the moving front [3,4]:
∂Γ ( R ) R˙ (t ) = 2h0 γ (0) . ∂r
(7)
The latter condition determines the front motion. Substitution of the estimation (5) into the boundary condition (7) results in R˙ (t ) ∼ −2h0 γ (0) or
Γm R
Γm = 2h0 γ (0) R
363
Spreading exponent 1st stage/duration
2nd stage
0.48 ± 0.02/0.15 s 0.48 ± 0.01/0.2 s
0.21 ± 0.01 0.03 ± 0.01
Now the concentration gradient on the moving front can be estimated similar to the previous qualitative consideration as [4]:
Γ (0) − Γ ( R ) ∂Γ ( R ) Q 0 e −αt ∼− =− . ∂r R π R3 Substitution of the latter expression into boundary condition (7) gives R˙ (t ) ∼ −2h0 γ (0)
Q 0 e −αt
π
R3
Q 0 e −αt = 2h0 γ (0) . 3
πR
Solution of the latter differential equation with the boundary condition R (t 1 ) = R 1 , is
Q0
R (t ) ∼ 8h0 γ (0)
απ
1 − e −α (t −t1 ) + R 41
0.25
.
(10)
The latter equation shows that: (i) at slow dissolution rate, that is, at t − t 1 1/α we recover the power law (4) in the case of insoluble surfactants, (ii) at t − t 1 1/α the radius of spreading tends to a limiting value, R ∞ :
Q0
R ∞ ∼ 8h0 γ (0)
απ
0.25
+ R 41
,
(11)
that is the spreading front gradually tends to the limiting position R ∞ and stops after this position is reached. The conclusion is as follows: the dissolution of surfactant molecules results in a slower motion of the front as compared with the prediction (4) for insoluble surfactants and it results in a full stop of the motion in the end of the process. 3. Experimental
R R˙ (t ) ∼ 2h0 γ (0)Γm .
3.1. Materials
Solution of the latter differential equation with initial condition R (0) = 0 results in
The latter power law coincides with Eq. (3), which was deduced for insoluble surfactants. Hence, according to the simplified derivation used the deduced power law equation (8) does not depend on the solubility of the surfactant molecules. The latter conclusion was satisfied in our experiments (see below).
Surfactants used in the experiments were analytical grade sodium dodecyl sulphate (SDS) purchased from Fisher Scientific UK, 99% pure dodecyltrimethylammonium bromide (DTAB) from Sigma–Aldrich, UK. The CMC and molecular weights of surfactants are given in Table 1. The solid substrate used in the spreading experiments is a 20 cm diameter borosilicate Petri dish. Aqueous solutions of these surfactants were prepared at fractions of critical micelle concentration (CMC). The surfactant solutions were subjected to an ultrasonic bath to ensure a well mixed concentration.
2.2. Second stage of spreading
3.2. Methodology
It was shown in [4] that the influence of solubility of surfactant molecules results in the following dependency of the total amount of the surfactant molecules on the film surface:
10 ml of distilled water were used to create a uniform thin aqueous layer, which covers the bottom of the Petri dish. Dewetting of the aqueous layer does not occur because the Petri dish was washed before each experiment according to the following protocol. The Petri dish was cleaned by (i) soaking it in chromic acid for 50 min, (ii) extensively rinsing with distilled and deionised water after that, and (iii) dried in an oven. The cleaning procedure was repeated after each experiment. By spreading of 10 ml of distilled water, a thin water layer with thickness 0.32 mm was deposited evenly across the surface of the Petri dish. A small amount of talc powder was homogeneously smeared over the surface of the aqueous layer to trace the motion of the liquid front under the
R (t ) ∼ 2 h0 γ (0)Γm t
Q (t ) = Q 0 e −αt ,
0.5
.
(8)
(9)
where Q 0 is the initial amount of surfactants at the beginning of the second stage and 1/α is a characteristic time scale of surfactant molecules dissolution. Equation (9) shows that (i) the total amount of surfactants on the film surface decays over time exponentially and (ii) the spreading process finally stops because all surfactant molecules dissolve into the bulk of the film.
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K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
Fig. 3. Screenshot extracted from the surfactant spreading over thin water layer experiment. Formation of the moving front is clearly seen.
Fig. 4. Log–log plot of spreading front against time for surfactants at concentrations below the CMC.
action of surfactants. High precision 5 μl Hamilton syringe (Hamilton GB Ltd., UK) was used to inject 3 μl droplets of aqueous surfactant solutions on the top of the aqueous layer at room temperature (25 ◦ C) and humidity (around 50%). A mechanical manipulator was structured to enable placement of the surfactant droplet on the thin aqueous layer whilst minimising the kinetic impact of the surfactant droplet and capillary waves on the thin aqueous film. The entire spreading process was captured using a high speed video camera (Olympus i-Speed) at a rate of 500 frames per second. The recorded video was then analysed using Olympus i-Speed software, tracking the position of the moving front/dry spot for the spreading process. Three tracking points of the spreading front radius were measured to obtain an average. The pixel/length calibration was done using a known length. For each sample, the experiment was repeated to produce at least 5 sets of data and each experimental point below is averaged over 5 experimental points. 4. Results and discussions In all experiments we observed a formation of a circular moving liquid front (Fig. 3). The front moved from the centre to the
periphery of the Petri dish. The time evolution of this front, R (t ), was monitored. All experimental dependences of R (t ) were plotted using log–log co-ordinate system. In all cases considered, we found the motion of the front to be in two stages: a fast first stage and a slower second stage. The power law exponents were obtained by fitting experimental data (with a minimal fit of 95%) by straight lines. Initial experiment using a pure water droplet was performed to verify the inertness of the talc powder particles used as a tracer. We found that a drop of pure water placed on the surface did not produce any spreading process, thus, allowing us to neglect the influence of talc powder particles on our spreading experiments. We assumed that the lower CMC the lower solubility of the corresponding surfactant is (see Table 1). The latter means that DTAB has a higher solubility compared to SDS. For the concentration studied for DTAB, we observed two stages, with the second stage reaching a limiting position as the surfactant starts to dissolve into the liquid layer underneath. Comparison of SDS and DTAB (see Fig. 4) at concentrations 0.1 CMC shows similar first stage spreading exponents (0.48 ± 0.02). The latter exponent is very close to the theoretically pre-
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
dicted value 0.5 according to Eqs. (3) and (8). These exponents do not depend on the surfactant solubility as predicted by Eq. (8). During the second stage, SDS continues to spread at a slower rate (0.21 ± 0.01), which is close to the theoretical value 0.25 deduced for insoluble surfactants. The latter means that at the time scale used in our experiments (1 s) SDS can be considered as insoluble surfactant. It is the reason why surfactants with lower solubility as compared with SDS were not used: SDS behaviour turns out to be very close to the predicted behaviour of insoluble surfactants. In the case of highly soluble DTAB the moving spreading front reached a limiting position (Fig. 4) as predicted by Eq. (11). It is necessary to note that we did not observe the dry spot formation in any cases considered.
A, B , C h L Q R r t u v z
5. Conclusions
Greek symbols
The time evolution of the spreading front on the surface of thin water film was monitored. The spreading front was caused by the deposition of a small surfactant droplets and the formation of the non-uniform distribution of surfactants over the surface of the water film. Concentration of surfactants in small deposited droplets was below CMC. Two surfactants of different solubility were used: SDS (moderate solubility) and DTAB (high solubility). According to the theoretical predictions (i) the spreading process proceeds in two stages: the fast first stage, which is followed by the lower second stage; (ii) during the first stage the evolution of the spreading front proceeds according to the power law with the exponent 0.5, which does not depend on the surfactant solubility; (iii) during the second stage the front in the case of insoluble surfactant should develop according to the power law with the exponent 0.25, while in the case of the soluble surfactant the spreading front should move slower and reach the final position. Our experiments showed that SDS on the time scale used (1 s) behaves as insoluble surfactant. Our experimental data confirmed the above theoretical predictions. It is necessary to note that SDS and DTAB are rather different in chemical nature and, thus, other aspects might be responsible for the differences found. Probably a more extensive study would be necessary to establish solubility of surfactant as the main reason for the differences found. Acknowledgment This research was supported by Engineering and Physical Sciences research Council, UK (Grant EP/D077869).
365
Appendix A. Nomenclature Roman
η γ 1/α
Γ
constants film thickness droplet base radius total amount of surfactants on film surface position of the moving front radial coordinates time tangential velocity on the liquid surface velocity vertical coordinates
dynamic shear viscosity of a liquid liquid–air interfacial tension characteristic time scale of surfactant molecules dissolution surface excess concentration
Subscripts 0 1
∞ eq m sh sp
initial time when first stage ends and second stage begins limiting position equilibrium maximum shrinkage spreading
References [1] A.-M. Cazabat, F. Heslot, P. Carles, S.M. Troian, Adv. Colloid Interface Sci. 39 (1992) 61. [2] H.M. Princen, A.M. Cazabat, M.A. Cohen Stuart, F. Heslot, S. Nicolet, J. Colloid Interface Sci. 126 (1) (1998) 84. [3] V.M. Starov, A. de Ryck, M.G. Velarde, J. Colloid Interface Sci. 190 (1997) 104. [4] K.S. Lee, V.M. Starov, J. Colloid Interface Sci. 314 (2007) 631. [5] K.S. Lee, V.M. Starov, T.J.P. Muchatuta, S.I.R. Srikantha, Colloid J. (Russian Academy of Sciences, English Translation) (2008), in press. [6] V.G. Levich, Physicochemical Hydrodynamics, Prentice–Hall, Englewood Cliffs, NJ, 1962. [7] V.M. Starov, M.G. Velarde, C.J. Radke, in: Wetting and Spreading Dynamics, in: Surfactant Science Series, vol. 138, CRC Press, Taylor&Francis Group, Boca Raton, 2007, pp. 1–515. [8] N.M. van Os, J.R. Haak, L.A.M. Rupert, Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactant, Elsevier, Amsterdam, 1993.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Spreading of surfactant solutions over thin aqueous layers at low concentrations: Influence of solubility K.S. Lee, V.M. Starov ∗ Department of Chemical Engineering, Loughborough University, Loughborough, LE11 3TU, UK
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 7 July 2008 Accepted 4 October 2008 Available online 29 October 2008 Keywords: Surfactant solutions Marangoni effect
A small droplet of aqueous surfactant solution at concentration below CMC was deposited on a thin water layer. A moving circular wave in the centre was formed. The time evolution of the radius of the wave was monitored. Two surfactants of different solubility were used. It was shown that the time evolution of the moving front (i) proceeds in two stages: a fast first stage and slower second stage; (ii) the time evolution of the front motion substantially depends on the surfactant solubility. We suggest a qualitative explanation of the phenomenon, which reasonably agrees with our experimental observations. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Spreading behaviour in the presence of surface tension gradients frequently shows instabilities [1,2]: fingering instability of the liquid films driven by temperature gradients [1], stick-slip phenomena when a liquid containing surface-active molecules advances on a hydrophilic solid results in an instability of the moving front [2]. The presence of instabilities is an interesting phenomenon in its own rights. However, on the current stage it is difficult to extract information on properties of surfactants from experimental data on instabilities. Hence, it is important to select a method, which allows investigating the dynamic properties of surfactant solutions on the liquid–air interface without instabilities occurring. It has been shown earlier [3,4] that investigation of the motion of a front after deposition of a small droplet of aqueous surfactant solution on a thin aqueous layer allows investigating separately the influence of Marangoni phenomenon on the hydrodynamic flow. In this case the capillary and gravitational forces can be neglected [3]. The suggested method results in a stable motion of the wetting front. The latter allows extracting properties of surfactants on a liquid–air interface as described in [3,4]. The same method was used for an investigation of properties of trisiloxane solutions [5]. Let us consider a schematic of the experimental procedure (Fig. 1): a small droplet of an aqueous surfactant solution is deposited on the top of the initially uniform thin aqueous film (thickness h0 ) as in Fig. 1a. The latter results in a formation of a depression in the centre of the film and a moving front forms (Fig. 1b). When a droplet of a surfactant solution is deposited on a clean liquid–air interface tangential stress develops on the liquid surface.
*
Corresponding author. E-mail address: [email protected] (V.M. Starov).
0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.10.031
©
2008 Elsevier Inc. All rights reserved.
The latter is caused by the non-uniform distribution of the surfactant concentration, Γ , over a part of the liquid surface covered by the surfactant molecules. The latter leads to surface stresses and a flow (Marangoni effect) [6]:
η
∂ u (r , h) dγ ∂Γ = , ∂z dΓ ∂ r
(1)
where η and u are the dynamic shear viscosity of the liquid and tangential velocity on the liquid surface h, respectively; (r , z) are radial and vertical coordinates; γ (Γ ) is the dependency of the liquid–air interfacial tension on surface excess concentration of the surfactant. The surface tension gradient-driven flow induced by the Marangoni effect moves surfactant molecules along the surface and a dramatic spreading process takes place. The liquid–air interface deviates from an initially flat position to accommodate the normal stress occurring in the course of the motion. This process is referred below to as a spreading process. The consideration in [3] was restricted to the case of “insoluble” surfactants, that is, those solubility can be neglected over duration of the process (referred to below as “insoluble surfactants”). The current state of art in the area was presented in [3] with the proper references therein. Surface diffusion was neglected in [3] as compared to convective transfer. The latter assumption was proven to be valid [3,4]. Surfactants of different solubility at concentrations above CMC [4] were used to investigate the influence of surfactant solubility on the spreading process. It was shown that the time evolution of the moving front proceeds in two stages: a rapid first stage associated with disintegration of micelles, which is followed by a slower second stage, when all micelles were already disintegrated. The time evolution of the moving front substantially depended on the surfactant solubility. An exact solution for the evolution of the moving front was deduced for the case of insoluble surfactants at
362
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
(a)
(b) Fig. 1. (a) A small droplet of aqueous surfactant solution is deposited on a top of thin aqueous layer of thickness h0 ; (b) cross section of the system: R (t )—radius of a circular moving front, r (t )—radius of dry spot in the centre, H —the height of the moving front.
Fig. 2. The first stage. Small aqueous surfactant solution droplet deposited on the surface of pure water film. Concentration of surfactant inside the small droplet is c 0 . Radius of the droplet base is L (t ).
concentration above CMC. A qualitative theory was developed to account for the influence of the solubility on the front motion. Experimental observations [4] are in a good agreement with the theory predictions. The duration of that stage is fixed by the initial amount of micelles in the deposited droplet. The fast stage is followed by the second slower stage when the surfactant concentration decreases in the film centre but the total mass of surfactant remains constant over the whole film in the case of insoluble surfactants and decreases over time in the case of soluble surfactants. In both cases a solution provides a power law dependency predicting the position of the moving front, R (t ), as time proceeds [4]. 2. Theory In our experiments below a droplet of surfactant solution with concentration below CMC was placed in the centre of a thin aqueous film (Fig. 1) which the initial concentration of surfactant c 0 < CMC. Our experimental observations showed that as in the case of concentration above CMC two distinct stages of spreading take place: (i) a first fast stage, and (ii) a second slower stage. This is similar to the case of the concentration above CMC [4]. However, the physical reason behind those stages is substantially different from the case of high concentrations above CMC. During the first stage the deposited droplet serves as a reservoir of the surfactant molecules, hence, surface concentration of single molecules is kept constant during that stage in the centre. That is, the following boundary condition are satisfied during the first stage of the moving droplet contact line, L (t ) (Fig. 2):
Γ (t , 0) = Γeq (c 0 ),
(2)
where Γeq (c 0 ) is the equilibrium adsorption isotherm and c 0 is the initial concentration inside the droplet. The following assumptions have been adopted in the boundary condition (2): (i) there is an equilibrium on the edge of the surfactant droplet, L (t ), and the edge of the liquid film (Fig. 2); (ii) the radius of the small liquid droplet, L (t ), is much smaller than the radius of the moving front, R (t ), that is L (t ) R (t ); (iii) the dissolution of surfactant molecules from the small droplet into the underlying pure water can be neglected during the first stage. Assumption (i) is precisely the same, which was adopted earlier [3,4]. It is not in contradiction with our previous and current
experimental data. However, this can be violated in the case of surfactants, which undergo a reorientation on the liquid–air interface. Assumption (ii) allows using the boundary condition (2) at the centre of the liquid film. The spreading of liquid droplets on the surface of the film of the same liquid [7] obeys the same power law as the spreading over dry solid substrate in the case of the complete wetting (which is obviously the case under consideration) and proceeds according to the following power law dependency, L (t ) = A · t 0.1 , where r is the droplet base radius and the constant A was calculated in [7]. It will be shown below that R (t ) ≈ B · t 0.5 during the first stage, where B is a constant. That is, the development of the spreading front, R (t ), proceeds much faster than the spreading of the small droplet. The latter was confirmed by our direct experimental observations. Note, the surfactants solution from the droplet, penetrates into the underlying water film and, hence, dL /dt = v sp − v sh , where v sp is the velocity of spreading, which is equal to v sp = 0.1 · A · t −0.9 according to the previous and v sh is the velocity of shrinkage, cased by the penetration of the droplet directly into the underlying water film. The latter rate is proportional to the area of the droplet base, v sh = C · π L 2 , where C is a constant. Hence, the dependency of the radius of the droplet base on time can be described by the following differential equation: dL /dt = 0.1 · A · t −0.9 − C π L 2 . Solution of the latter equation shows that the droplet radius goes via a maximum value and disappears with a characteristic time scale, t 1 ≈ 1/(π L 0 C ), where L 0 is the initial value of the droplet base. Assumption (iii) can be justified using the consideration presented in [3]. It means that the characteristic time scale of dissolution of surfactant molecules should be substantially bigger than the time of spreading and disappearance of the small droplet. Our experimental data with two surfactants of different solubility confirm the validity of this assumption for those two surfactants. The slower stage was observed when the concentration in the small droplet was below CMC and fast first stage was not detected by the regular camera used in [3]. Below we used a high speed video camera, which allowed us to detect the presence of the fast first stage even in the case of surfactant concentration below CMC. Past t 1 , a second slower stage starts. During the second stage the total mass of surfactant remains constant in the case of non-soluble surfactants and decreases in the case of soluble surfactants.
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
We can follow consideration presented in [3] to deduce the following power laws for the first stage: R (t ) = A 1 · t 0.5 ,
t < t1
(3)
Table 1 Properties and concentrations of surfactants used and experimentally obtained spreading exponents. Materials
CMC (mmol/l)
C/CMC
SDS DTAB
8.3 [3] 15.1 [8]
0.1 0.1
and R (t ) = A 2 · t
0.25
t > t1 ,
,
(4)
during the second slower stage, where A 1 and A 2 are constants, which can be deduced using consideration similar to that presented in [3,4]. Note, the two power law equations (3) and (4) were deduced for the case of insoluble surfactants and we try to modify those two laws for the case of soluble surfactants. 2.1. First stage of spreading In [4] a very simple semi-empirical derivation of the dependency of the moving front on time was suggested. Those derivations were based on the boundary conditions on the moving front. The concentration gradient on the moving front can be estimated as
Γeq (c 0 ) Γ (0) − Γ ( R ) Γ (0) ∂Γ ( R ) ∼− =− =− , ∂r R R R
(5)
where the boundary condition on the moving front was used [3,4]:
Γ ( R ) = 0.
(6)
Note, the latter boundary condition is not “the real boundary condition” but an effective condition for the inner solution: that is we ignore the existence of a narrow region where the concentration changes very rapidly (see [3,4] for details). The next boundary condition should be satisfied on the moving front [3,4]:
∂Γ ( R ) R˙ (t ) = 2h0 γ (0) . ∂r
(7)
The latter condition determines the front motion. Substitution of the estimation (5) into the boundary condition (7) results in R˙ (t ) ∼ −2h0 γ (0) or
Γm R
Γm = 2h0 γ (0) R
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Spreading exponent 1st stage/duration
2nd stage
0.48 ± 0.02/0.15 s 0.48 ± 0.01/0.2 s
0.21 ± 0.01 0.03 ± 0.01
Now the concentration gradient on the moving front can be estimated similar to the previous qualitative consideration as [4]:
Γ (0) − Γ ( R ) ∂Γ ( R ) Q 0 e −αt ∼− =− . ∂r R π R3 Substitution of the latter expression into boundary condition (7) gives R˙ (t ) ∼ −2h0 γ (0)
Q 0 e −αt
π
R3
Q 0 e −αt = 2h0 γ (0) . 3
πR
Solution of the latter differential equation with the boundary condition R (t 1 ) = R 1 , is
Q0
R (t ) ∼ 8h0 γ (0)
απ
1 − e −α (t −t1 ) + R 41
0.25
.
(10)
The latter equation shows that: (i) at slow dissolution rate, that is, at t − t 1 1/α we recover the power law (4) in the case of insoluble surfactants, (ii) at t − t 1 1/α the radius of spreading tends to a limiting value, R ∞ :
Q0
R ∞ ∼ 8h0 γ (0)
απ
0.25
+ R 41
,
(11)
that is the spreading front gradually tends to the limiting position R ∞ and stops after this position is reached. The conclusion is as follows: the dissolution of surfactant molecules results in a slower motion of the front as compared with the prediction (4) for insoluble surfactants and it results in a full stop of the motion in the end of the process. 3. Experimental
R R˙ (t ) ∼ 2h0 γ (0)Γm .
3.1. Materials
Solution of the latter differential equation with initial condition R (0) = 0 results in
The latter power law coincides with Eq. (3), which was deduced for insoluble surfactants. Hence, according to the simplified derivation used the deduced power law equation (8) does not depend on the solubility of the surfactant molecules. The latter conclusion was satisfied in our experiments (see below).
Surfactants used in the experiments were analytical grade sodium dodecyl sulphate (SDS) purchased from Fisher Scientific UK, 99% pure dodecyltrimethylammonium bromide (DTAB) from Sigma–Aldrich, UK. The CMC and molecular weights of surfactants are given in Table 1. The solid substrate used in the spreading experiments is a 20 cm diameter borosilicate Petri dish. Aqueous solutions of these surfactants were prepared at fractions of critical micelle concentration (CMC). The surfactant solutions were subjected to an ultrasonic bath to ensure a well mixed concentration.
2.2. Second stage of spreading
3.2. Methodology
It was shown in [4] that the influence of solubility of surfactant molecules results in the following dependency of the total amount of the surfactant molecules on the film surface:
10 ml of distilled water were used to create a uniform thin aqueous layer, which covers the bottom of the Petri dish. Dewetting of the aqueous layer does not occur because the Petri dish was washed before each experiment according to the following protocol. The Petri dish was cleaned by (i) soaking it in chromic acid for 50 min, (ii) extensively rinsing with distilled and deionised water after that, and (iii) dried in an oven. The cleaning procedure was repeated after each experiment. By spreading of 10 ml of distilled water, a thin water layer with thickness 0.32 mm was deposited evenly across the surface of the Petri dish. A small amount of talc powder was homogeneously smeared over the surface of the aqueous layer to trace the motion of the liquid front under the
R (t ) ∼ 2 h0 γ (0)Γm t
Q (t ) = Q 0 e −αt ,
0.5
.
(8)
(9)
where Q 0 is the initial amount of surfactants at the beginning of the second stage and 1/α is a characteristic time scale of surfactant molecules dissolution. Equation (9) shows that (i) the total amount of surfactants on the film surface decays over time exponentially and (ii) the spreading process finally stops because all surfactant molecules dissolve into the bulk of the film.
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Fig. 3. Screenshot extracted from the surfactant spreading over thin water layer experiment. Formation of the moving front is clearly seen.
Fig. 4. Log–log plot of spreading front against time for surfactants at concentrations below the CMC.
action of surfactants. High precision 5 μl Hamilton syringe (Hamilton GB Ltd., UK) was used to inject 3 μl droplets of aqueous surfactant solutions on the top of the aqueous layer at room temperature (25 ◦ C) and humidity (around 50%). A mechanical manipulator was structured to enable placement of the surfactant droplet on the thin aqueous layer whilst minimising the kinetic impact of the surfactant droplet and capillary waves on the thin aqueous film. The entire spreading process was captured using a high speed video camera (Olympus i-Speed) at a rate of 500 frames per second. The recorded video was then analysed using Olympus i-Speed software, tracking the position of the moving front/dry spot for the spreading process. Three tracking points of the spreading front radius were measured to obtain an average. The pixel/length calibration was done using a known length. For each sample, the experiment was repeated to produce at least 5 sets of data and each experimental point below is averaged over 5 experimental points. 4. Results and discussions In all experiments we observed a formation of a circular moving liquid front (Fig. 3). The front moved from the centre to the
periphery of the Petri dish. The time evolution of this front, R (t ), was monitored. All experimental dependences of R (t ) were plotted using log–log co-ordinate system. In all cases considered, we found the motion of the front to be in two stages: a fast first stage and a slower second stage. The power law exponents were obtained by fitting experimental data (with a minimal fit of 95%) by straight lines. Initial experiment using a pure water droplet was performed to verify the inertness of the talc powder particles used as a tracer. We found that a drop of pure water placed on the surface did not produce any spreading process, thus, allowing us to neglect the influence of talc powder particles on our spreading experiments. We assumed that the lower CMC the lower solubility of the corresponding surfactant is (see Table 1). The latter means that DTAB has a higher solubility compared to SDS. For the concentration studied for DTAB, we observed two stages, with the second stage reaching a limiting position as the surfactant starts to dissolve into the liquid layer underneath. Comparison of SDS and DTAB (see Fig. 4) at concentrations 0.1 CMC shows similar first stage spreading exponents (0.48 ± 0.02). The latter exponent is very close to the theoretically pre-
K.S. Lee, V.M. Starov / Journal of Colloid and Interface Science 329 (2009) 361–365
dicted value 0.5 according to Eqs. (3) and (8). These exponents do not depend on the surfactant solubility as predicted by Eq. (8). During the second stage, SDS continues to spread at a slower rate (0.21 ± 0.01), which is close to the theoretical value 0.25 deduced for insoluble surfactants. The latter means that at the time scale used in our experiments (1 s) SDS can be considered as insoluble surfactant. It is the reason why surfactants with lower solubility as compared with SDS were not used: SDS behaviour turns out to be very close to the predicted behaviour of insoluble surfactants. In the case of highly soluble DTAB the moving spreading front reached a limiting position (Fig. 4) as predicted by Eq. (11). It is necessary to note that we did not observe the dry spot formation in any cases considered.
A, B , C h L Q R r t u v z
5. Conclusions
Greek symbols
The time evolution of the spreading front on the surface of thin water film was monitored. The spreading front was caused by the deposition of a small surfactant droplets and the formation of the non-uniform distribution of surfactants over the surface of the water film. Concentration of surfactants in small deposited droplets was below CMC. Two surfactants of different solubility were used: SDS (moderate solubility) and DTAB (high solubility). According to the theoretical predictions (i) the spreading process proceeds in two stages: the fast first stage, which is followed by the lower second stage; (ii) during the first stage the evolution of the spreading front proceeds according to the power law with the exponent 0.5, which does not depend on the surfactant solubility; (iii) during the second stage the front in the case of insoluble surfactant should develop according to the power law with the exponent 0.25, while in the case of the soluble surfactant the spreading front should move slower and reach the final position. Our experiments showed that SDS on the time scale used (1 s) behaves as insoluble surfactant. Our experimental data confirmed the above theoretical predictions. It is necessary to note that SDS and DTAB are rather different in chemical nature and, thus, other aspects might be responsible for the differences found. Probably a more extensive study would be necessary to establish solubility of surfactant as the main reason for the differences found. Acknowledgment This research was supported by Engineering and Physical Sciences research Council, UK (Grant EP/D077869).
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Appendix A. Nomenclature Roman
η γ 1/α
Γ
constants film thickness droplet base radius total amount of surfactants on film surface position of the moving front radial coordinates time tangential velocity on the liquid surface velocity vertical coordinates
dynamic shear viscosity of a liquid liquid–air interfacial tension characteristic time scale of surfactant molecules dissolution surface excess concentration
Subscripts 0 1
∞ eq m sh sp
initial time when first stage ends and second stage begins limiting position equilibrium maximum shrinkage spreading
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