pubs.acs.org/Langmuir © 2010 American Chemical Society
Role of Marangoni Instability in Fabrication of Axially and Internally Grooved Hollow Fiber Membranes Jun Yin, Nicole Coutris, and Yong Huang* Department of Mechanical Engineering, Clemson University, Clemson, South Carolina 29634-0921 Received May 28, 2010. Revised Manuscript Received September 2, 2010 Hollow fiber membranes (HFMs) are extensively used in different industrial applications. Under some controlled fabrication conditions, axially aligned grooves can be formed on the HFM inner surface during typical immersion precipitation-based phase inversion fabrication processes. Such grooved HFMs are found to be promising for nerve repair and regeneration. The axially aligned grooves appearing on the inner surface of the membrane are considered as hydrodynamic instability patterns. During the immersion precipitation process, a transfer of solvent takes place across the interface between a polymer solution and a nonsolvent. This solvent transfer induces gradients of interfacial tension that are considered to be the driving mechanism for Marangoni instability. The onset of the stationary instability is studied by means of a linear instability theory, and the critical and maximum wavenumbers are determined and discussed in terms of the dimensionless groups characterizing the system: viscosity ratio, diffusivity ratio, Schmidt number, crispation number, adsorption number, Marangoni number, and the polymer bulk concentration. A good agreement is found between the predicted wavelength of the most dangerous wave and the experimental groove width. Consequently, solutal Marangoni instability can explain the groove formation mechanism in HFM fabrication.
1. Introduction Hollow fiber membranes (HFMs) are commonly used in many processes as filtration, gas separation, dialysis, desalination due to their semipermeability, and high surface area to volume ratio. Recently, such semipermeable HFMs have been found to be promising in promoting axonal outgrowth for nerve repair and regeneration.1-3 HFM is generally fabricated using an immersion precipitationbased phase inversion process,3 which is a dry-wet solution spinning process. For an HFM to be an effective nerve conduit in nerve regeneration, there is a need for a 3D textured surface microtopology on the order of 10 μm as the physical guidance to better stimulate axonal outgrowth.4-6 It was found that aligned grooves (∼50 μm in groove width) can be formed on the inner surface of HFM by carefully controlling the operating conditions, and such grooved HFMs have demonstrated encouraging applications in stimulating the rat dorsal root ganglion (a nodule on a dorsal root containing cell bodies of neurons in afferent spinal nerves) in vitro regeneration.3 As the HFM grooved texture should be carefully controlled to be a better physical guidance cue for nerve regeneration, a better understanding of the groove formation mechanism is necessary to control HFM fabrication for such neural engineering applications. The objective of this study is to understand the groove formation mechanism in HFM fabrication and model the grooves as the pattern resulting from a solutal Marangoni instability taking place at the beginning of the phase inversion process. The Article *To whom correspondence should be addressed. Tel: 864-656-5643. Fax: 864-656-4435. E-mail:
[email protected]. (1) Zhang, N.; Yan, H. H.; Wen, X. J. Brain Res. Rev. 2005, 49, 48–64. (2) Lu, J.; Waite, P. Spine 1999, 24, 926–930. (3) Long, Y.; Zhang, N.; Huang, Y.; Wen, X. J. Manuf. Sci. Eng. 2008, 130, 021011, 1–8. (4) Miller, C.; Shanks, H.; Witt, A.; Rutkowski, G.; Mallapragada, S. Biomaterials 2001, 22, 1263–1269. (5) Tai, H. C.; Buettner, H. M. Biotechnol. Prog. 1998, 14, 364–370. (6) Taylor, A. M.; Blurton-Jones, M.; Rhee, S. W.; Cribbs, D. H.; Cotman, C. W.; Jeon, N. L. Nat. Methods 2005, 2, 599–605.
Langmuir 2010, 26(22), 16991–16999
is organized as follows. First, observations of corrugated inner surface in HFM fabrication are briefly reviewed and discussed. Then, a linear stability analysis is performed to study the onset of interfacial instability between the polymer solution and the nonsolvent. The dispersion equation is obtained and analyzed in terms of nondimensional groups. The effects of different material properties and the initial composition of the polymer solution in HFM fabrication are discussed. Finally, conclusions are summarized, and future work is further outlined.
2. Scientific and Technical Background Phase inversion is a process whereby a homogeneous polymer solution is transformed into two phases: a polymer-rich phase and a polymer-lean phase. The polymer-rich phase develops into a polymer dense matrix, whereas the polymer-lean phase develops into a porous structure containing macrovoids (Figure 1). Because the polymer glass-transition temperature is greater than the ambient temperature, the polymer solution precipitates. The immersion precipitation process is a common phase inversion practice. The polymer solution prepared with a good solvent is in contact with a nonsolvent, which has a high affinity with the solvent and no affinity with the polymer. The nonsolvent phase induces the solvent extraction from the polymer solution, which is then invaded by the nonsolvent. The quality of the solvent with respect to the polymer decreases and induces the phase separation. Such an immersion precipitation-based phase inversion mechanism has been widely implemented to fabricate HFM, as shown in Figure. 2. The polymer solution (polymer dissolved in a solvent) is coextruded with an inner pure nonsolvent through the spinneret.3 Then, both flows pass through an air gap before entering a coagulation bath. HFMs usually have a uniformly thick and smooth inner surface layer, as shown in Figure 1a. Instabilities can develop in the longitudinal (along the spinning direction) or circumferential directions during HFM fabrication. The longitudinal instability can be in form of melt fracture or
Published on Web 10/05/2010
DOI: 10.1021/la102173f
16991
Article
Yin et al.
Figure 1. Inner surface of HFM: (a) HFM with a smooth inner surface and (b) HFM with an axially grooved inner surface.3
Figure 2. Schematic drawing of spinneret and hollow fiber spinning process (S: solvent, NS: nonsolvent). The spinneret is characterized by its inner diameter (ID) of 1.64 mm and outer diameter (OD) of 2.29 mm and the air gap of 10 mm.3
draw resonance,7 which may lead to fiber breakage during the spinning process. The circumferential instability can lead to a deformed HFM cross section, nonuniform wall thickness, or grooved inner surface,3,8-13 as shown in Figure 1b. The instability in the circumferential direction is considered to play a significant role in groove formation on the HFM inner surface, but there are only a few works on the modeling of groove formation based on the material properties and fabrication conditions. Bonyadi et al.13 recently have pioneered two different instability mechanisms: the hydrodynamic instability and the buckling mechanisms. The hydrodynamic instability mechanism (7) Su, Y. Theoretical Studies of Hollow Fiber Spinning. Ph.D. Thesis, The University of Toledo: Toledo, OH, 2007. (8) Roesink, H. D. W. The Influence of Spinning Conditions on the Morphology of Microporous Capillary Membranes. Ph.D. Thesis, Twente University, 1989. (9) McKelvey, S. A.; Clausi, D. T.; Koros, W. J. J. Membr. Sci. 1997, 124, 223– 232. (10) Pereira, C. C.; Nobrega, R.; Borges, C. P. Braz. J. Chem. Eng. 2000, 17, 599–606. (11) Santoso, Y. E.; Chung, T. S.; Wang, K. Y.; Weber, M. J. Membr. Sci. 2006, 282, 383–392. (12) Shi, L.; Wang, R.; Cao, Y. M.; Feng, C. S.; Liang, D. T.; Tay, J. H. J. Membr. Sci. 2007, 305, 215–225. (13) Bonyadi, S.; Chung, T. S.; Krantz, W. B. J. Membr. Sci. 2007, 299, 200–210.
16992 DOI: 10.1021/la102173f
is related to the onset of instability when the structure is not completely solidified, whereas the buckling mechanism results from the polymer shrinkage-induced buckling of a cylindrical elastic shell located between the bore fluid, which is nonsolvent in this study, and the coagulation bath. The interfacial instability between two immiscible liquid phases has been widely studied. When a third component diffuses between two immiscible liquid phases, the interface may become unstable, and spontaneous convection sets in.14 A theoretical understanding of the interfacial instability observed during mass transfer across an interface has been developed by Sternling and Scriven15 who reported high fluctuations in the vicinity of the interface (also known as interfacial turbulence) between two incompressible fluid phases, which in some cases accompany mass transfer. The essence of their explanation is that the interfacial instability is due to concentration-induced variation of interfacial tension, known as the Marangoni effect, but they assumed no deformation of the interface. Hennenberg and Sørensen et al.14,16 have carried out a stability analysis on isothermal, deformable interfaces with transfer of surface-active substances. They formulated the balance laws for a moving and deformable interface in local equilibrium with immiscible bulk liquids in which a third component is distributed. von Gottberg et al.17 studied interfacial instabilities due to a chemical reaction taking place at the interface. The stability of the system was examined with respect to small perturbations with the normal mode method. Slavtchev et al.18 studied Marangoni instability in partially miscible liquid-liquid systems in the presence of surface-active solutes. The surface-active solute, soluble in both phases, is transferred across the interface between the two liquids, introducing interfacial tension gradients that drive a solutal Marangoni instability. In all of these works, a linear stability analysis was carried out to study the condition of stability and the influence of different parameters such as the diffusivity ratio and the viscosity ratio of the two phases. Marangoni instability19 is known as an interfacial instability driven by interfacial tangential forces due to interfacial tension gradients. Solvent extraction introduces concentration gradients in the solution that lead to these interfacial tension gradients. (14) Sørensen, T. S.; Hennenberg, M.; Sanfeld, A. J. Colloid Interface Sci. 1977, 61, 62–76. (15) Sternling, C. V.; Scriven, L. E. AIChE J. 1959, 5, 514–523. (16) Hennenberg, M.; Sørensen, T. S.; Sanfeld, A. J. Chem. Soc., Faraday Trans. 2 1977, 73, 48–66. (17) von Gottberg, F. K; Hatton, T. A.; Smith, K. A. Ind. Eng. Chem. Res. 1995, 34, 3368–3379. (18) Slavtchev, S.; Kalitzova-Kurteva, P.; Mendes, M. A. Colloids Surf., A 2006, 282, 37–49. (19) Davies, T. V.; Rideal, E. K. In Interfacial Phenomena; Academic Press: New York, 1963.
Langmuir 2010, 26(22), 16991–16999
Yin et al.
Article
Figure 3. Fluid domain with the unperturbed interface between two phases: nonsolvent and polymer solution. (The inset shows the perturbed adsorbed polymer layer.)
An interfacial instability can cause the inner surface pattern of HFM; the number of grooves depends on the wavelength of the most dangerous wave. In the present study, it is assumed that the groove formation occurs during the extraction process and is due to the onset of an instability developing at the interface. Consequently, phase inversion or solidification-induced shrinkage effects are decoupled from the Marangoni instability and not of interest here. For the first time, Marangoni instability is used to explain the formation of axially and internally grooved HFMs. Mass transfer starts with the spontaneous extraction of solvent between the two immiscible phases: solution and nonsolvent. Then, the nonsolvent invades the solution. To study the onset of instability, only the extraction process is considered.
3. Assumptions and Problem Formulation 3.1. Modeling Assumptions. In HFM fabrication, two flows, polymer solution (dope) and nonsolvent (bore), discharge from the spinneret together through the air gap (Figure 2). Once these two flows are brought into contact in the air gap, several steps occur: first, a spontaneous extraction of the solvent from the polymer solution; then, an invasion of the polymer solution by the nonsolvent leading to phase inversion in the ternary mixture polymer/solvent/nonsolvent and solidification. The interfacial instability is supposed to occur during the first step of the process.20 Different assumptions are considered for this first step and summarized as follows: (1) The time scale of the first step, extraction of the solvent, is smaller than the characteristic diffusion time t0 associated with the interfacial region estimated as 1 ms (thickness of the zone 1 μm and diffusivity 5 10-10 m2/s) and smaller than the characteristic advective time associated with the vertical flows of the two phases on the order of 0.1 s (volume flow rate 4 mL/min for a 1 mm diameter). The two phases are assumed to be at rest; consequently, for the onset of instability, the basic solution is the classical diffusion solution when two incompressible semi-infinite fluid layers are in contact along a deformable interface. (2) During the first step, only the solvent mass transfer is considered. Therefore, polymer solution and nonsolvent can be treated as two immiscible liquid phases while the solvent is transferred across the interface. (3) Thermal and rheological effects are neglected. (4) The region of interest is located close to the interface. Its thickness on the order of 1 μm is far smaller than the inner radius or the thickness of the HFM, so the problem can be studied in a 2D rectangular geometry with two semi-infinite layers on each side of a planar interface. Because the wavenumber of the most dangerous wave is k ≈ 0.3, as seen in Section 5, the associated wavelength is far larger than the length scale H, justifying the planar interface assumption. (20) Cohen Addad, J. P.; Panine, P. Polym. Bull. 1999, 42, 345–352.
Langmuir 2010, 26(22), 16991–16999
3.2. Problem Formulation. Consider a 2D system of two semi-infinite layers of a polymer solution, phase 2, in contact with a nonsolvent, phase 1, along an interface. The governing equations of mass, momentum and solvent diffusion are written in both phases as
r 3 vBi ¼ 0
Dv Fi Bi þ vBi 3 rvBi Dt
ð1Þ
!
¼ - rpi þ μi r2 vBi
Dci þ v rci ¼ Di r2 ci Dt Bi 3
ð2Þ
ð3Þ
where B v *i is the velocity, c*i is the solvent concentration distribution, p*i is the pressure, Fi is the density, μi is the viscosity, and Di is the solvent diffusivity in phase i (i = 1 for nonsolvent and i = 2 for polymer solution). It should be pointed out that variables with a superscript “*” are dimensional. The densities, viscosities, and diffusivities of each phase are assumed to be constant. The fluid domain is divided into three regions, as shown in Figure 3: the two bulk regions separated by the third one, which is a thin adsorption layer adjacent to the interface, the adsorbed polymer layer. The solvent transfer is due to diffusion in each bulk up to the adsorption layer and due to adsorption-desorption of polymer inside the adsorption layer in phase 2. Polymer adsorption is a reversible process, but polymer desorption21 is a very slow process; the characteristic time is far larger than the diffusion time t0. It is assumed that only adsorption is concerned herein. The Langmuir model is used to describe a polymer adsorption coupled to bulk diffusion.21 The interfacial excess polymer concentration Γ* is expressed in terms of the concentration of polymer c*p/s in the solution evaluated just above the interface
Γ ¼ Γmax
K l c p=s 1 þ K l c p=s
ð4Þ
where Γmax is the maximum interfacial polymer concentration and Kl is the Langmuir equilibrium constant measuring the ability to adsorb. The interfacial tension σ* depends on the polymer concentration, and this dependence is expressed by the LangmuirSzyskowski isotherm σ ¼ σ solv -
RT Γmax lnð1 þ K l c p=s Þ Mw
ð5Þ
(21) Somasundaran, P.; Hubbard, A. T. In Encyclopedia of Surface and Colloid Science; Taylor & Francis: Boca Raton, FL, 2006; Vol. 3.
DOI: 10.1021/la102173f
16993
Article
Yin et al.
where σ*solv is the interfacial tension of pure solvent, T is the absolute temperature, R is the gas constant, and Mw is the molecular weight of the polymer. Equation 5 shows the drop of the interfacial tension due to the presence of a polymer such as polyurethane (PU) in a solvent dimethyl sulfoxide (DMSO) and the surfactant property of the polymer. 3.3. Basic Solution. During the solvent extraction, the concentration profiles in each phase are obtained from the diffusion equations, the bulk fluid being considered to be at rest with a planar interface. The diffusion time scale t0 is assumed to be far larger than the advective time scale associated with the perturbation. Consequently, the concentration profiles are frozen at time t0 to study the perturbation, as considered by Sterling and are solutions of the Scriven.15 The solvent concentrations c*(0) i 1D diffusion equation in two semi-infinite domains ð0Þ
ð0Þ
Dci D2 ci ¼ D , i Dt Dy2
(
- ¥ < y e 0 0 e y < ¥
i ¼ 1, i ¼ 2,
ð6Þ
where y* is the y coordinate. The concentration profiles c*(0) i (t*, y*) are deduced from eq 6 complemented by the following equations: (A1) Interfacial conditions: (a) Concentration continuity: ð0Þ
ð0Þ
c1 ðt, 0Þ ¼ c2 ðt, 0Þ
ð7Þ
(b) Equality of the solvent fluxes, that is, no accumulation of solvent: ð0Þ
D1
ð0Þ
Dc1 Dc ðt , 0Þ ¼ D2 2 ðt, 0Þ Dy Dy
ð8Þ
(A2) Initial conditions: uniform concentration profiles in each phase: (
ð0Þ
c1 ð0, y Þ ¼ c1¥ ð0Þ
c2 ð0, y Þ ¼ c2¥
ð9Þ
Figure 4. Concentration profiles: ci*(0) (solid line), ci*(00) (dashed line), and cp*(0) (dotted line) of the basic polymer solution in bulk 2 at time t0 = 10-3 s deduced from eqs 10, 13, and 11, respectively, with D1 = 5 10-9 m2/s, D2 = 5 10-10 m2/s, c1¥ = 0, and c2¥ = 990 kg/m3.
written at time t*=t0, then eq 12 is used to obtain the approximation of eq 10 as follows ( ð00Þ ð0Þ -¥ < ye0 c1 ðyÞ ¼ c1¥ þ Δc1 expðβ1 yÞ, ð13Þ ð00Þ ð0Þ c2 ðyÞ ¼ c2¥ - Δc2 expð - β2 yÞ, 0 e y < ¥ √ √ (0) (0) where Δc (0) 1 = (c 2¥ - c 1¥ )/( d þ √ 1), Δc 2 = d Δc 1 1/2 β1 = 1.444/(2(dD2t0) ), and β2 = β1 d.
4. Stability Analysis in HFM Fabrication 4.1. Governing Equations for the Perturbation. Different scales are first introduced to perform the stability analysis. The length scale H is the diffusion length (D2t0)1/2 associated with the diffusion process in phase 2 at time t0. Time, velocities, pressures, and solvent concentrations are scaled with H2/D2, D2/H, F2(D2/H)2, and Δc2(0), respectively. The dimensionless governing equations of mass, linear momentum, and solvent diffusion read now
where c1¥ and c2¥ are the initial solvent concentrations in phases 1 and 2, respectively. The solutions can then be written as
r 3 vBi ¼ 0
ð14Þ
8 pffiffiffi c1¥ d þ c2¥ ðc2¥ - c1¥ Þ - y > ð0Þ > > - pffiffiffi erf pffiffiffiffiffiffiffiffiffiffi , - ¥ < y e 0 c1¥ d þ c2¥ ðc2¥ - c1¥ Þ d y > > pffiffiffi pffiffiffi : cð0Þ þ erf pffiffiffiffiffiffiffiffiffiffi , 0 e y < ¥ 2 ðt , y Þ ¼ 2 D1 t d þ1 d þ1
DvBi F μ þ v rv ¼ - 2 rpi þ i r2 vBi Dt Bi 3 Bi Fi Fi D2
ð15Þ
Dci Di 2 þ v rci ¼ r ci Dt Bi 3 D2
ð16Þ
ð10Þ √ R -z2 dz. The solvent where d = D1/D2 and erf(Y) = (2/ π) Y 0 e concentration profiles are shown in Figure 4 for t0 = 10-3 s, and is defined as follows the polymer concentration c*(0) p cp ð0Þ ðyÞ ¼ F2 - c2 ð0Þ ðyÞ
ð11Þ
The concentration profiles are approximated by using an approximation of the error function14 1 - erfðYÞ ¼ 1 þ erfð - YÞ expð - 1:444YÞ
ð12Þ
The corresponding concentration profiles after approximation, , define the basic solution for the solvent concentration in c*(00) i each phase. The basic solution c*(0) i (t*, y*) given by eq 10 is first 16994 DOI: 10.1021/la102173f
where in dimensionless form, B v i is the velocity, pi is the pressure, and ci is the solvent concentration in phase i. Some dimensionless parameters are introduced such as the density ratio f = F1/F2, , the viscosity ratio m = μ1/μ2, and the Schmidt number, Sc = μ2/F2D2. The velocities, pressures and concentrations are then expressed as the sums of the corresponding value for the basic solution and a perturbation vBi ¼ vBi
ð0Þ
þ δvi yB,
pi ¼ pi ð0Þ þ δpi ,
ci ¼ ci ð00Þ þ δci ð17Þ
where B v i(0) = 0 and pi(0) is uniform. The location of the deformed interface yS is defined as yS ¼ δhðx, tÞ
ð18Þ
Langmuir 2010, 26(22), 16991–16999
Yin et al.
Article
where δh is the interfacial deformation. Equations 14-16 are then linearized and read r 3 δvBi ¼ 0
ð19Þ
8 Dδv 1 mSc 2 > > r δvB1 < B1 ¼ - rδp1 þ f f Dt ð20Þ > > DδvB2 2 : ¼ - rδp2 þ Scr δvB2 Dt 8 Dδc1 ð00Þ > < þ δvB1 3 rc1 ¼ dr2 δc1 Dt ð21Þ > : Dδc2 þ δv rcð00Þ ¼ r2 δc 2 B2 3 2 Dt The perturbation is generated at the interface, and the perturbation vanishes far from the interface ( δvB1 , δp1 , δc1 f 0 as y f - ¥ ð22Þ δvB2 , δp2 , δc2 f 0 as y f ¥
where DS is the surface diffusion coefficient (C2) Interfacial normal linear momentum balance: Γ0 Dðδv1 Þ F2 H Dt
y¼0
Dðδv1 Þ Dðδv2 Þ ¼ ðδp1 - δp2 Þy ¼ 0 - 2Sc m Dy Dy y ¼ 0 þ
Γ0 D2 D Dðδv1 Þ μ2 H Dt Dy
"
y¼0
(b) Continuity of the tangential component of the velocities. This equation can be expressed because of eq 19 in terms of the normal components δνi (i = 1, 2) only D D ðδv1 Þjy ¼ 0 ¼ ðδv2 Þjy ¼ 0 Dy Dy (c) Continuity of the solvent concentrations ð00Þ ð00Þ Dc1 Dc2 δc1 þ δh ¼ δc2 þ Dy Dy y¼0
ð24Þ
δh
ð25Þ
y¼0
Dδh Dt
ð26Þ
The interfacial excess polymer concentration Γ* and the interfacial tension σ* are scaled, respectively, with Γmax and σ*, 0 the interfacial tension at the interface in the basic state. The dimensionless interfacial excess polymer concentration Γ and interfacial tension σ are such that Γ ¼
Γ0 þ δΓ, Γmax
σ ¼ 1 þ δσ
ð27Þ
where Γ*0 is the value of Γ* for the basic state. Because of polymer adsorption, three more interfacial equations are needed as follows (C1) Interfacial mass balance: Γmax DðδΓÞ Dðδv2 Þ DS Γmax D2 ðδΓÞ Dt Dy Γ0 D2 Γ0 Dx2 y¼0
2 ð0Þ Δc2 H 4Dðδc2 - dδc1 Þ ¼ Dy Γ0
y¼0
D2 ðc2 ð00Þ - dc1 ð00Þ Þ þ Dy2
3 δh5 y¼0
ð28Þ Langmuir 2010, 26(22), 16991–16999
- Ma y¼0
D2 ðδσÞ Dx2
ð30Þ where
Dσ DΓ
y¼0
Γmax H
μ2 D2
is the Marangoni number. The perturbations δΓ and δσ can be expressed as follows on the basis of eqs 4 and 5 0
1
ð00Þ
Dc δΓ ¼ - K2 @δc2 jy ¼ 0 þ 2 Dy δσ ¼ - RΓ
(B2) Kinematic condition: δv1 jy ¼ 0 ¼
# D2 ðδv2 Þ D2 ðδv1 Þ ¼-m Dx2 Dx2
Ma ¼ ð23Þ
ð29Þ
where Cr is the crispation number defined as (μ2D2)/(σ0*H) and the last term of the right side includes the deformation of the interface (δh) (C3) Interfacial tangential linear momentum balance:
The interfacial conditions at the interface are: (B1) Condition for the bulks: (a) Continuity of the normal component of the velocities: δvB1 3 yBjy ¼ 0 ¼ δvB2 3 yBjy ¼ 0
Sc D2 ðδhÞ Cr Dx2
δhA
ð31Þ
y¼0
Γmax δΓ σ 0
ð32Þ
(0) 2 þ K l Δc (0) and where K 2 = (K l Δc (0) 2 )/(1 2 c p/s ) (0) (0) l R Γ = (RT)/Mw(1 þ K Δc2 cp/s). 4.2. Normal Mode Method for Stationary Instability. The stability analysis is carried out in terms of normal modes. The different unknowns are defined as
δvi ¼ ~vi ðyÞeikxþωt , ~ ikxþωt , δΓ ¼ Γe
δpi ¼ p~ i ðyÞe
δσ ¼ σ~ eikxþωt ,
ikxþωt
,
δci ¼ c~i ðyÞeikxþωt ,
~ ikxþωt δh ¼ he
ð33Þ
where k is the wavenumber, ω is the growth rate of the perturbation, and v~i, p~i, c~i, Γ~, σ~, and h~ are the amplitudes of each perturbation. First, the different fields p~i, v~i, and c~i are determined as follows and then are introduced to the interfacial balances to obtain the dispersion equation. Combining the continuity and NavierStokes equations in each phase, a differential equation for the pressures p~i is deduced Δ~ pi ¼ 0,
i ¼ 1, 2
ð34Þ
The boundary conditions (eq 22) lead to the solutions (
p~1 ðyÞ ¼ p~10 eky , - ¥ < ye0 p~ ðyÞ ¼ p~ e - ky , 0ey < ¥ 2
ð35Þ
20
where p~10 and p~20 are two constants to be determined. Substituting the two expressions for p~1 and p~2 given by eq 35 into the DOI: 10.1021/la102173f
16995
Article
Yin et al.
Navier-Stokes equations (eq 20), the velocities in phases 1 and 2 take the form ( -¥ < ye0 ~v1 ðyÞ ¼ A1 eky þ B1 eq1 y , ð36Þ ~v2 ðyÞ ¼ A2 e - ky þ B2 e - q2 y , 0 e y < ¥ where q1 = (k2 þ ((fω)/(mSc)))1/2, q2 = (k2 þ ω/Sc)1/2, p~10 = -((fω)/k)A1, p~20=(ω/k)A2, and A1, A2, B1, and B2 are four constants to be determined. The two diffusion equations eq 21 can now be solved with c1¥ 1 β2 H ð00Þ p ffiffiffi p ffiffiffi þ exp y c1 ¼ ð0Þ d d Δc2 c 2¥ ð00Þ - expð - β2 HyÞ ð37Þ c2 ¼ ð0Þ Δc2 Because of the choice of diffusion time t0 associated with H, the length H can be such that β2H = 1. The perturbations c~1 and c~2 are consequently given by 8 A1 B1 > r1 y ðk þ bÞy > þ eðq1 þ bÞy , - ¥ < y e 0 > < c~ 1 ðyÞ ¼ l1 e þ d 2 ½ðb þ kÞ2 - r 2 e d 2 ½ðb þ q1 Þ2 - r1 2 1 > A2 B2 > c~ ðyÞ ¼ l e - r2 y þ > e - ðk þ 1Þy þ e - ðq2 þ 1Þy , 0 e y < ¥ : 2 2 ½ð1 þ kÞ2 - r2 2 ½ð1 þ q2 Þ2 - r2 2
ð38Þ √ where r1 = (k þ ω/d) , r2 = (k þ ω) , b = β1/β2 = 1/ d, and l1 and l2 are two new constants to be determined. The perturba~ Γ~, tion solution depends on nine constants, A1, A2, B1, B2, l1, l2, h, ~ and σ. The perturbation is the solution of the linear system of equations deduced from eqs 23-26 and 28-32. The dispersion equation is obtained by writing that the determinant det(M) of the linear system defined by its matrix M has to be zero. detðMÞ ¼ Fðω, kÞ ¼ 0 ð39Þ 2
1/2
2
1/2
The function F contains several parameters such as the diffusivity, viscosity, and density ratios, the Schmidt number Sc = μ2/F2D2, the adsorption number Na = Γmax/Δc(0) 2 H, the interfacial diffusion number ID = (DSΓmax)/(μ2H), the crispation number Cr=(μ2D2)/(σ0*H), the Marangoni number Ma=(RΓΓmaxH)/(μ2D2), and the mass fraction of solvent in phase 2. All of these dimensionless numbers can be expressed as the ratios of characteristic times, as shown in Table 1. 4.3. Onset of Instability. To study the onset of instability in the neighborhood of marginal stability corresponding to ω=0, when the perturbation does not grow or decay with time, the different elements of the matrix M are expanded in powers of ω. Keeping only the first-order terms in ω in the determinant, the characteristic equation is now written as A - Bω ¼ 0 ð40Þ where A and B are expressed in terms of the dimensionless groups as ( " #) Γ0 1 1 1 2 ð41Þ þ A ¼ Mak Γmax Na dð2k þ bÞ2 ð2k þ 1Þ2 " # ð0Þ Δc2 ðd þ 1Þ Crðf - 1Þ 2 B ¼ - k ID - k ðb þ 1 - 2kÞ þ Ma 2 2k NaSc F2 ScK2 2 3 1 fd 1 þ 1 þ k6 2Sc 2mSc 7 6 7 - Ma Na4ð2k þ 1Þ3 d 2 ð2k þ bÞ3 5
l > p ffiffiffi K F < 2 d þ1 NumðωÞ ¼ k > c > :1 þ K l F2 - pffiffiffi2¥ d þ1 # pffiffiffi " c2¥ H d 1 1 pffiffiffi þ Γmax ð d þ 1Þ dð2k þ bÞ2 ð2k þ 1Þ2
Langmuir 2010, 26(22), 16991–16999
g
g
2 3 fd pffiffiffi 1 þ c2¥ H d 6 1 2mSc 7 67 pffiffiffi þ þ 4 5 3 2 Γmax ð d þ 1Þ ð2k þ 1Þ d ð2k þ bÞ3 where ðA1Þ
σ0
¼
σsolv
ðA2Þ
" # RT c2¥ l Γmax ln 1 þ K F2 - pffiffiffi Mw d þ1
DOI: 10.1021/la102173f
16999
