Langmuir 1998, 14, 5929-5945
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Adsorption and Desorption Kinetics of Mixtures on Planar Surfaces under Flow Conditions Nadezhda L. Filippova Russian Branch RTD Corporation, Moscow 111538, Russia Received May 8, 1998. In Final Form: July 20, 1998 The adsorption isotherms and the adsorption/desorption kinetics of a mixture of nonionic water-soluble associative polymers with different molecular weights on silicon wafers and polystyrene films under flow conditions were studied. Equations were derived to calculate the parameters characterizing the kineticdiffusive-convective-controlled adsorption and desorption processes under flow conditions: (a) the equilibrium adsorption, (b) the thickness of the adsorbed layer, (c) the activation energy of adsorption, (d) the rate constant, (e) the effective coefficients of diffusion in the adsorbed layer, and (f) the time needed to attain the equilibrium state for the adsorption and desorption processes for individual components and mixtures. We also studied the competitive adsorption kinetics of water-soluble nonionic polyurethane polymer based on poly(ethylene glycol), with an average molecular weight of 12 000, 62 000, and 120 000 g/mol and with a C16H33 linear alkyl group on each end of the molecule, from aqueous solution onto the silica (SiO2) and polystyrene films by ellipsometry.
Introduction Competitive adsorption kinetics of macromolecules (surfactants, polymers, and so on) is an important concern in different applications: colloidal stabilization/destabilization, mineral flotation, lubrication, biomaterial, separation, and so on. A knowledge of the competitive adsorption kinetics behavior of adsorbed macromolecule layers under flow conditions at fluid interfaces is of great importance for optimal application of macromolecules. Numerous authors1-11 have tried to determine the general macromolecule mechanism under flow conditions during adsorption. The most extensive study on the competitive mechanism for macromolecule adsorption kinetics was performed by applying different experimental techniques such as dynamic light scattering, neutron scattering, total internal reflection fluorescence, ellipsometry, etc.1-11 But even these accurate measurements could not give an unambiguous answer on the competitive adsorption mechanisms of these macromolecules. In our opinion, a key to this understanding is the knowledge of the timedependent composition and conformation of macromolecules comprising the adsorbed layer. The equilibrium and adsorption kinetics behavior in the adsorbed layer have been studied extensively; however, there has not been a satisfactory theoretical explanation to quantita(1) Watkins, R. W.; Robertson, C. B. J. Biomed. Mater. Res. 1977, 11, 915. (2) Burghardt, T. P.; Axelrod, D. Biophys. J. 1981, 33, 455. (3) Lok, B. K.; Cheng, Y. L.; Robertson, C. R. J. Colloid Interface Sci. 1983, 81, 87. (4) Darst, S. A.; Robertson, C. R.; Berzofsky, B. J. Colloid Interface Sci. 1986, 111, 466. (5) Adamson, A. W. Physical Chemistry of Surfaces; Interscience: New York, 1986. (6) Rondelez, F.; Ausserre, D.; Hervet, H. Annu. Rev. Phys. Chem. 1988, 38, 317. (7) Young, B. R.; Pitt, W. G.; Cooper, S. L. J. Colloid Interface Sci. 1988, 125, 246. (8) Caucheteux, I.; Hervet, H.; Jerome, R.; Rondelez, F. J. Chem. Soc., Faraday Trans. 1990, 86, 1369. (9) Shibata, C. T.; Lenhoff, A. M. J. Colloid Interface Sci. 1992, 148, 485. (10) Kim, D.; Cha, W.; Beissinger, R. L. J. Colloid Interface Sci. 1993, 159, 1. (11) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymer at Interfaces; Chapman & Hall: London, 1993.
tively describe the equilibrium adsorption and competitive adsorption kinetics of macromolecules under flow conditions on a planar surface. We examine the analytical approaches1-11 and the simplifying assumptions of the earlier investigations. Despite several attempts by other researchers, there is no treatment of the problem which produces a general analytical approach for solving the system of nonlinear equations for polymers in solutions, without making numerous simplifying assumptions. To describe the adsorption and desorption processes under flow conditions, a number of investigators1-10 have applied the approach of Leveque12 for steady-state heat/mass transfer under flow conditions. In our opinion this approach cannot be used to describe adsorption in a flow cell since it is a nonstationary process. In view of this fact, we consider this problem in detail. As will be shown below, the competitive adsorption kinetics of macromolecules under flow conditions is described by (for coi * 0, cisurf * 0)
dΓi(t)/dt ) Ki(coi - cisurf), i ) 1, 2, ..., n
(1a)
cisurf ) f-1 i (Γ1,Γ2,...,Γn), i ) 1, 2, ..., n
(1b)
( )
Di2γs Ki ) β L
1/3
, β ≈ 0.65
(1c)
where Γi is the amount of macromolecule adsorbed of the ith species, Ki is the rate constant of adsorption of the ith species, coi is the bulk concentration of the ith species, cisurf is the surface macromolecular concentration of the ith species, fi(c) is a function for the adsorption isotherm of the ith species, fi-1(Γ1,Γ2,...,Γn) is a reciprocal function for the adsorption isotherm of the ith species, γs is the wall shear rate, Di is the diffusion coefficient of the ith species, n is the number of species in the solution, and L is the distance from the entrance of the flow channel to the detection point. From eqs 1a and 1b it follows that the competitive adsorption process (for coi * 0, cisurf * 0) under flow (12) Leveque, M. Ann. Mines 1928, 13, 284.
S0743-7463(98)00545-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/29/1998
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Filippova
Theory of Competitive Adsorption and Desorption Processes in Flow Cell
conditions is described by
dΓi(t) ) Ki[coi - fi-1(Γ1,Γ2,...,Γn)], i ) 1, 2, ..., n (2a) dt Γi(t)0) ) 0, i ) 1, 2, ..., n
(2b)
The diffusive-convective-controlled desorption process (for coi ) 0, cisurf * 0) under flow conditions is described by
dΓi(t) ) -Kicisurf, i ) 1, 2, ..., n dt
(3a)
dΓi(t) ) -Kifi-1(Γ1,Γ2,...,Γn), i ) 1, 2, ..., n (3b) dt Γi(t)0) ) Γoi, i ) 1, 2, ..., n
(3c)
Now we compare the previous approach of eqs 1a-3c and the Leveque approach12 to describe the adsorption/ desorption process under flow conditions. According to the Leveque approach, the diffusive-convective-controlled adsorption process (for coi * 0, cisurf ) 0) under flow conditions is described by
dΓi(t) ) Kicoi, i ) 1, 2, ..., n dt
( )
K i ) βo
Di2γs L
(4a)
1/3
, βo ≈ 0.54
(4b)
From eq 4a the amount of macromolecule adsorbed for the ith species is given by
Γi(t) )
{
}
Kicoit, 0 e t e ti, ti ) Γoi/(Kicoi) , Γoi, t g ti i ) 1, 2, ..., n (5)
From the previous solutions it follows that in the framework of the Leveque approach the adsorption of each ith species is independent, i.e., competitive adsorption does not take place. Next we consider the desorption process (for coi ) cisurf ) 0) under flow conditions. In the framework of the Leveque approach,12 the diffusive-convective-controlled desorption process is described by
dΓi(t) ) 0, i ) 1, 2, ...., n dt surf
Γi(t) ) constant ) 0 (for ci
(6a)
) 0), i ) 1, 2, ..., n
For the plane Poiseuille flow in a rectangular channel, the axial velocity, Vx(x,y,t), has a parabolic profile across the thickness of the channel when a full flow is developed.13-16 The transient convective diffusion equation governing transport of macromolecules in a rectangular channel under flow conditions is given by
∂ci(x,y,t) ∂2ci(x,y,t) ∂ci(x,y,t) + γsy ) Di(y) , 1 e i e n, ∂t ∂x ∂y2 y g 0 (7a) ∂Γi(x,t) ∂ci(x,0,t) ) Di(y) , 1eien ∂t ∂y
(7b)
ci ) coi, y g 0, 0 e x e L, t g tm, tm ) L/Vav (7c) ∂Γi(x,t) ) R+i - R-i, 1 e i e n ∂t
(8a)
R+i ) Riad[c1(x,0,t),...,cn(x,0,t),Γ1(x,t),Γ2(x,t),...,Γn(x,t)] (8b) R-i ) Rides[Γ1(x,t),Γ2(x,t),...,Γn(x,t)]
(8c)
where ci(x,y,t) and Di(y) are the concentration and the diffusion coefficient in the direction (y) of the ith species, respectively, x is a coordinate in the direction of flow, y is a coordinate in the direction normal to the interface, t is time, (Vx)av is the average linear velocity, n is the number of species in the solution, γs is the wall shear rate, coi is the concentration of the ith species in the bulk, ci(x,0,t) is the surface macromolecular concentration of the ith species, Γi(x,t) is the amount of adsorbed macromolecule of the ith species at y ) 0, and R+i and R-i are the rates of adsorption and desorption of the ith species, respectively, on a planar surface. Adsorption Kinetics on Planar Surfaces. In the framework of the Arrehenius and Eyring approaches5,17,18 and Lucassen-Reynders convection,19-22 the adsorption kinetics for macromolecules onto a planar surface (eq 8a) reduces to15,16
∂Γi(x,t) ) Kiadci(x,0,t)[Γ∞ - Γi(x,t)] × ∂t γi exp θ 2(x,t) - KidesΓi(x,t) × 2 γi γi exp [1 - θ∑(x,t)]2 + 2 2
[ ( )∑ ] {()
}
(9a)
n Γi(x,t) (-∆H)i , θ∑(x,t) ) , γi ) θk(x,t) θi(x,t) ) RT k)1 Γ∞ (9b)
∑
(6b)
where Kiad and Kides are the rate constants for the
From eqs 5 and 6b it follows that the Leveque approach cannot be used to describe the competitive adsorption and desorption processes. Therefore, the motivation for the present research is to develop a theory for competitive adsorption processes taking into account the nonequilibrium processes in the adsorbed layer and showing the dominant role of different mechanisms in interpreting adsorption kinetics data.
(13) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (14) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962. (15) Filippov, L. K.; Filippova, N. L. J. Colloid Interface Sci. 1997, 189, 1. (16) Filippova, N. L. Langmuir 1998, 14, 1162. (17) Eyring, H. Basic Chemical Kinetics; Wiley: New York, 1980. (18) Rosenberg, R. M. Principles of Physical Chemistry; Oxford University Press: New York, 1977.
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adsorption and desorption processes on a planar surface of the ith species, θi(x,t) is the surface coverage of the ith species, θ∑(x,t) is the total coverage, Γ∞ is the maximum amount of macromolecule adsorbed, T is the absolute temperature, R is the gas constant, and γi and (-∆H)i are the interaction parameter and the activation energy of adsorption of the ith species, respectively, characterizing the interaction between macromolecule/interface, macromolecule/macromolecule, and macromolecule/solvent. The system of equations 7a-9b may be used to describe the kinetic-diffusive-convective-controlled adsorption and desorption processes on a planar surface under flow conditions. For the kinetic-diffusive-convective-controlled adsorption and desorption processes on a planar surface, when the constant rate of adsorption and desorption on a planar surface is infinite (Kiad f ∞, Kides f ∞), eq 9a reduces to the equations for the adsorption isotherms as15,16
coi ) fi-1(θ1,θ2,...,θn) )
θi (p)
Ki (1 - θ∑)
exp(γiθ∑)
(10a)
Γk(coi) ) fk(co1,co2,...,con), 1 e k, i e n (10b) where Ki(p) ()Kiad/Kides) is the equilibrium constant for the ith species, fi is a function describing the adsorption isotherm for the ith species, and fi-1 is a reciprocal function. Thus, the system of equations 7a-8a and 10a is the complete system of equations by which one may describe the diffusive-convective-controlled adsorption and desorption processes for a mixture of macromolecules on a planar surface under flow conditions. The diffusive-convective-controlled adsorption and desorption processes for a mixture of macromolecules on a planar surface under flow conditions is described by the equation15
()
1 dΓi(L,t) ) coi-ci(L,0,t), 1 e i e n Ki dt
(11)
[ ]
(
Φi(θi) ) Γ∞Kiad[1 - θi(L,t)] exp fi-1(θ1,θ2,...,θn) )
θi Ki(p)(1 - θ∑)
γi 2 θ (L,t) 2 ∑
)
exp(γiθ∑), Ki(p) )
(12a)
(12b) Kiad Kides (12c)
θi(L,t) )
{
}
dΓi(L,t) Ki(total)(Γi)[coi - fi-1(Γ)], (adsorption) ) , dt (desorption) -Ki(total)(Γi)fi-1(Γ), 1 e i e n (13a) Ki(total)(Γi) )
KiΦi(Γi) Ki + Φi(Γi)
(13b)
where Ki(total)(Γi) is the total rate constant for the adsorption and desorption processes taking into account the adsorption and desorption kinetics onto a planar surface and convective-diffusive mass transfer for the ith species. It should be noted that, in order to describe the kineticdiffusive-controlled adsorption and diffusive-convectivecontrolled adsorption of macromolecules on a planar surface under flow conditions over a wide range of times for multicomponent mixtures, the suggested adsorption and desorption model of eq 13a may be used. Our experimental study of the behavior of nonionic water-soluble polymers in the adsorbed layer utilizes a flow cell in combination with an ellipsometric technique which allows us to test the behavior of nonionic watersoluble polymers on planar interfaces. The advantages of these techniques are that these techniques allow direct measurement of the structure and architecture of the adsorbed layer and parameters characterizing the competitive adsorption/desorption processes over a wide range of times. The focus of our experimental research is the study of competitive adsorption/desorption processes for nonionic water-soluble polymer over a wide range of concentrations and wall shear rates on planar surfaces of silica (SiO2) and polystyrene films. A quantitative interpretation of the interfacial behavior of polymers allows one to use these materials successfully in different commercial areas. Experimental Materials and Methods
where ci(L,0,t) is the surface concentration of the ith species. It is reasonable to rewrite eq 9a in the following convenient form:
1 dΓi(L,t) ) ci(L,0,t) - fi-1(Γ), 1 e i e n dt Φi(θi)
sorption processes over a wide range of times,
n Γi(L,t) , θ∑(L,t) ) θk(L,t), 1 e i e n k)1 Γ∞ (12d)
∑
Eliminating the surface concentration, ci(L,0,t), from eqs 11 and 12a, one writes the equation which describes the kinetic-diffusive-convective-controlled adsorption and de(19) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (20) Lucassen, J.; Hollway, F.; Buckingham, J. H. J. Colloid Interface Sci. 1978, 67, 423. (21) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (22) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1986, 41, 199.
We have been developing a theory of the kinetic-diffusiveconvective-controlled adsorption to describe the adsorption and desorption kinetics of polymers in a flow cell on planar surfaces, and we have experimentally studied the adsorption and desorption kinetics of water-soluble nonionic polymers with different molecular weights and different concentrations on a planar silica (SiO2) substrate as well as polystyrene films. The goal of this experimental research is to make a fundamental contribution to the understanding and prediction of the bulk and interfacial properties of new nonionic polymers in a flow cell and also to describe the adsorption and desorption kinetics of nonionic polymers in a flow cell on planar surfaces. Materials. The flow cell was used to measure the values of the refractive index of the silica (SiO2) in DDI (distilleddeionized) water and the ellipsometric values for polymer solutions on these substrates during the adsorption processes at room temperature. The silica surfaces were cleaned by using sonification in DDI water for 5 h after the adsorption runs were complete when the equilibrium state was reached. Fresh polymer solutions were prepared for each run. Ellipsometric measurements were performed with an ellipsometer with λ ) 546.1 nm, and at an incident angle of 70°. A solution of polystyrene (with a molecular weight of 110 000 g/mol) in toluene (0.02% by weight) was spin-coated on 5.2 cm polished silicon wafers; the thickness of SiO2 was 123 nm; the resulting thickness of the polystyrene film was 15 nm. The polystyrene films were dried in a vacuum oven at 80 °C. The model watersoluble associative polymers used in the kinetic study were obtained from nonionic polyurethanes based on poly(ethylene glycol) and have average molecular weights of 12 000, 62 000, and 120 000 g/mol with a C16H33 linear alkyl group on each end of the molecule. We used a special flow cell to study the adsorption
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and desorption kinetics of polymer mixtures under flow conditions by ellipsometry. The polymer solution flow in a flow cell is perpendicular to the direction of the laser beam. To calculate the amount of adsorption in the flow cell from the ellipsometric data we used optical systems in the form of two layers for the silica surface and in the form of four layers for the polystyrene surface, which consists of the following layers: bulk silicon (Si) with a complex refractive index, n*4 ) n4 - ik4, a layer of silica (SiO2) with a refractive index of n3 and a thickness of d3, a polystyrene layer with a refractive index of n2 and a thickness of d2, the adsorbed polymer layer with a refractive index of n1 ) nad.layer and a thickness of d1 ) dad.layer, and a surrounding solution with a refractive index of no. The thickness, dad.layer, and the refractive index, nad.layer, of the adsorbed layer were found simultaneously by using the equations23
Re(δ) ) (2π/λ)dad.layer[nad.layer2 - nsol2 sin θsol]1/2 (14a) Im(δ) ) 0
(14b)
where Re(δ) and Im(δ) are the real and imaginary parts of the phase shift of the polarized light, respectively, λ is the wavelength (λ ) 546.1 nm), θsol is the incident angle of the incident laser beam, dad.layer is the thickness of the adsorbed layer, and nad.layer and nsol are the refractive indices of the adsorbed layer and the polymer solution, respectively. First, the values of the adsorbed layer, dad.layer, and the refractive index of the adsorbed layer, nad.layer, from eqs 14a and 14b are calculated by using the ellipsometric experimental data (ψ and ∆). The weight fraction of polymer in the adsorbed layer, Xad.layer, and the adsorption, Γ, are given by16
Xad.layer )
Rpol )
6nad.layer(nad.layer - nsol) (Rpol - Rsol)(nad.layer2 + 2)2
(npol2 - 1) , (npol2 + 2)
Rsol )
(nsol2 - 1) (nsol2 + 2)
Γ ) dad.layerFpolymerXad.layer
(15a)
(15b) (15c)
The weight fraction of polymer in the adsorbed layer, Xad.layer, is then calculated by using eqs 15a and 15b. Finally, the amount of polymer adsorbed, Γ, is determined from eq 15c.
Discussions The above theory for the kinetic-diffusive-convectivecontrolled adsorption/desorption processes for multicomponent mixture of polymers under flow conditions was developed in order to understand and explain (i) the competitive adsorption kinetics for polymer mixtures in the adsorbed layer on planar surfaces of various substrates [i.e., the silica (SiO2) and the polystyrene film], (ii) the desorption kinetics for polymer mixtures for water-soluble polymer in the adsorbed layer on various planar surfaces, and (iii) the difference between the behavior of adsorption and desorption kinetics for water-soluble polymers in the adsorbed layer on planar surfaces under flow conditions. The adsorption and desorption of water-soluble nonionic associative polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol from aqueous solution onto the silica (SiO2) and the polystyrene film [with a polystyrene molecular weight of 110 000 g/mol] were studied in the flow cell for a polymer concentration range of 1 ppm (mg/ kg) to 500 ppm by ellipsometry at room temperature. First, we studied the adsorption of water-soluble nonionic associative polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol in the flow cell from aqueous solution onto the silica and polystyrene film by ellipsometry. Figures 1A and 2A show the thicknesses of (23) So, S. S.; Vedam, K. J. Opt. Soc. Am. 1972, 62, 16.
Figure 1. (A) The thickness of the adsorbed layer, dad.layer, (B) the amount of polymer adsorbed, Γ(c), versus the polymer concentration, (C) the function ln(c/Γ) versus amount of polymer adsorbed, Γ, and (D) the function (c/Γ) versus polymer concentration, c, for the water-soluble associative polymer with a molecular weight of 12 000, 62 000, and 120 000 g/mol on a SiO2 substrate at a wall shear rate, γs, of 15 s-1 and L ) 1.8 cm.
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Langmuir, Vol. 14, No. 20, 1998 5933
the adsorbed layer for polymer with different molecular weights onto the silica and the polystyrene film, respectively, which were calculated from the experimental ellipsometric data by using eq 14. The weight fraction, Xad.layer, and the amount of adsorbed polymer, Γ, were calculated by using eqs 15a-c. The weight fractions of water-soluble polymer in the adsorbed layer on SiO2 and the polystyrene film, respectively, were found to be approximately 0.12 and 0.14; i.e., the adsorbed layer consisted of 12 and 14 wt % water-soluble polymer and 88 and 86 wt % water, respectively. Figures 1B and 2B show the adsorption isotherms of the water-soluble polymer onto SiO2 and polystyrene film substrates, respectively. To estimate the parameters K(p), Γ∞, and γ which characterize the adsorption isotherm it is reasonable to rewrite eq 10a for an individual polymer species (n ) 1) in more convincing forms (eqs 16c and 17b),
Γ Γ ) cK(p) exp -γ Γ∞ Γ -Γ
(
∞
ln
()
)
(16a)
( )
c Γ ) -ln(K(p)) + γ ∞ - ln(Γ∞ - Γ) Γ Γ ln
(Γc ) ) I
o
s
Iso ) -ln(K(p)Γ∞), Sso )
(16c)
1+γ ∂ c , Ss ) ln , ∂Γ Γ Γ∞ Sso ) Ss(Γf0) (16d)
[ ( )]
c 1 Γ c ) + exp Γ ∞ Γ Γ∞ K(p)Γ∞ Γ
( )
(17a)
(Γc ) ) I
+ Ss∞c, c f co
(17b)
∞
s
Ss∞ )
+ SsoΓ, Γ f 0
(16b)
1 ∂ c , Ss ) , Ss∞ ) Ss(cfco) ∂c Γ Γ∞
()
o (-∆H) Ss γ) ) ∞-1 RT S
(17c)
(17d)
s
Figure 2. (A) The thickness of the adsorbed layer, dad.layer, (B) the amount of polymer adsorbed, Γ(c), versus the polymer concentration, (C) the function ln(c/Γ) versus amount of polymer adsorbed, Γ, and (D) the function (c/Γ) versus polymer concentration, c, for the water-soluble associative polymer with a molecular weight of 12 000, 62 000, and 120 000 g/mol on a polystyrene film substrate at a wall shear rate, γs, of 15 s-1 and L ) 1.8 cm.
where K(p) is the equilibrium constant, co is the bulk polymer concentration, (-∆H) is the activation energy characterizing the interaction between polymer/interface, polymer/polymer, and polymer/solvent, Iso and Sso are the intercept and slope, respectively, of the straight line of ln(c/Γ) versus Γ for Γ f 0, and Is∞ and Ss∞ are the intercept and slope, respectively, of the straight line of (c/Γ) versus c for c f co. Figures 1C and 2C show the function of ln(c/Γ) versus the amount of polymer adsorbed, Γ, and Figures 1D and 2D show the function of (c/Γ) versus the polymer concentration which was calculated from experimental data represented in Figures 1B and 2B for polymer adsorbed onto SiO2 and polystyrene film, respectively. The values of Iso, Sso, and Ss∞ were found from data represented in Figures 1C,D and 2C,D; the values of K(p), Γ∞, γ, and (∆H) were calculated by using eqs 16d, 16c, and 17d, respectively. These values are listed in Table 1. The surface area occupied on the SiO2 and polystyrene film
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Table 1. Characteristics of the Adsorbed Layers of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in a Flow Cella MWpol (kg/mol)
Γ∞ (mg/m2)
12 62 120
0.28 0.81 0.99
12 62 120
0.38 0.85 1.2
K(p) (1/ppm)
σm (nm2)
dad.layer (nm)
Xad.layer
Adsorption of Associative Polymer in a Flow Cell onto SiO2 0.13 66.4 2.5 0.12 0.17 128.7 6.4 0.12 0.59 209.7 7.8 0.12 Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 0.12 49.8 2.7 0.14 0.21 121.3 6.6 0.13 0.86 181.1 7.9 0.14
χArch
γ
(-∆H)
3.7 2.0 2.1
2.9 3.7 4.2
7.0 9.0 10.2
2.9 1.9 1.9
2.6 3.3 3.8
6.3 8.0 9.2
a MW ∞ pol ) molecular weight of polymers; Γ ) amount of polymer adsorbed per unit area corresponding to the adsorption isotherm plateau; K(p) ) equilibrium constant characterizing the adsorption isotherm; σm ) area occupied by one polymer molecule; dad.layer ) thickness of the adsorbed layer corresponding to the adsorption isotherm plateau; Xad.layer ) weight fraction of polymers in the adsorbed layer corresponding to the adsorption isotherm plateau; χArch ) parameter characterizing the architecture of the adsorbed layer; γ ) parameter characterizing the interaction in the adsorbed layer; (-∆H) ) activation energy of adsorption; γs ) 15 s-1; and L ) 1.8 cm.
substrates by one polymer molecule, σm, in the adsorbed state corresponding to the plateau is given by
σm(nm2) )
MWpol ΓmaxNA
(18)
where MWpol is the polymer molecular weight, Γmax is the amount of adsorbed polymer on a planar surface corresponding to the isotherm plateau, and NA is Avogadro’s number. The values of σm for the polymer calculated from the experimental data in Figures 1B and 2B by using eq 18 are presented in Table 1. The length, Lsur, characterizing the surface area on a planar surface occupied by one polymer molecule is estimated as
Lsur ≈
( ) 4σm π
1/2
(19a)
The architecture of the adsorbed layer may be characterized by using the ratio χArch, which is equal to
χArch )
Lsur dad.layer
(19b)
where dad.layer is the thickness of the adsorbed layer in the equilibrium state corresponding to the isotherm plateau. The architecture of the adsorbed layer may be estimated by using the ratio χArch, since the architecture of the adsorbed layer is like that of a polymer brush when χArch e 0.5 and is like that of a polymer pancake when χArch g 2. The ratio χArch was calculated from the experimental data presented in Figures 1B and 2B by using eqs 18-19b and is shown in Table 1. Figure 3A shows the thicknesses of the adsorbed layer for a mixture of polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol with a bulk concentration, (co)mix, of 500 ppm (or 500 mg/kg), respectively, onto the silica and the polystyrene film which were calculated from the experimental ellipsometric data by using eq 14. The weight fraction, Xad.layer, and the amount of adsorbed polymer, Γ, were calculated by using eqs 15a-15c. For a mixture of polymers with different molecular weights, the weight fractions of water-soluble polymer in the adsorbed layer onto SiO2 and polystyrene film, respectively, were found to be approximately 0.13 and 0.14, respectively; i.e., the adsorbed layer consists of 13 and 14 wt % water-soluble polymer and 87 and 86 wt % water, respectively. Figure 3B shows the adsorption isotherm of the water-soluble polymer onto SiO2 and polystyrene
Figure 3. (A) The thickness of the adsorbed layer, (dad.layer)mix, and (B) the amount of polymer adsorbed, Γmix(Cmix), versus the polymer concentration, for the mixture of the water-soluble associative polymer with a molecular weight of 12 000, 62 000, and 120 000 g/mol on a SiO2 substrate at a wall shear rate, γs, of 15 s-1 and L ) 1.8 cm.
film substrates for a mixture of polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol with a bulk concentration, (co)mix, of 500 ppm, respectively. As shown in Figure 3A and Table 2, the thickness of the adsorbed layer, (dad.layer)∑, and the total polymer adsorbed, (Γmax)∑, for a mixture of polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol with a bulk concentration, (co)mix, of 500 ppm, respectively, onto the silica and the polystyrene film are less than those for individual polymer with a molecular weight of 64 600 g/mol due to strong interactions between individual polymers in the adsorbed layer. The surface area occupied on the SiO2 and polystyrene film substrate by one average molecule, (σm)∑, in the adsorbed state corresponding to the plateau for a three-component mixture and the architecture parameter, (χArch)∑, were calculated by using eqs 18-19b for the
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Langmuir, Vol. 14, No. 20, 1998 5935
Table 2. Characteristics of the Adsorbed Layers for a Three-Component Mixture of Water-Soluble Nonionic Polymer onto Planar Polystyrene and SiO2 Substrates in a Flow Cella (MWpol)av (kg/mol)
(Γmax)∑ (mg/m2)
(σm)∑ (nm2)
(Xad.layer)∑
(χArch)∑
64.6
Adsorption of Associative Polymer in a Flow Cell onto SiO2 0.82 130.6 6.2
(dad.layer)∑ (nm)
0.13
2.08
64.6
Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 1.0 107.2 7.1
0.14
1.64
(MWpol)av ) average molecular weight of the three-component mixture of polymers; (Γmax)∑ ) amount of polymer adsorbed per unit area corresponding to the adsorption isotherm plateau for a three-component mixture of polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol, respectively, for the bulk concentration of each component of mixture of 500 ppm; (σm)∑ ) area occupied by one average polymer molecule for a three-component mixture of polymers; (dad.layer)∑ ) thickness of the adsorbed layer corresponding to the adsorption isotherm plateau for a three-component mixture of polymers; (Xad.layer)∑ ) weight fraction of polymers in the adsorbed layer corresponding to the adsorption isotherm plateau for a three-component mixture of polymers; (χArch)∑ ) parameter characterizing the architecture of the adsorbed layer for a three-component mixture of polymers; γs ) 15 s-1; and L ) 1.8 cm. a
average molecular weight of the mixture (MWpol)av ) (12 000 + 62 000 + 120 000)/3 (g/mol) ) 64 600 g/mol. As shown in Figures 1-3 and Tables 1 and 2, the strong interaction between polymer molecules in the polymer mixture takes place in the adsorbed layer at the interface and causes compression of the adsorbed layer due to arrangements of polymer molecules with different molecular weights. These results may quantitatively be estimated by using the architecture parameter χArch. For a mixture of polymers with an average molecular weight of 64 600 g/mol the architecture parameter, (χArch)∑, in the adsorbed layer on SiO2 and polystyrene film was found to be 2.08 and 1.64, respectively. However, for an individual polymer with a molecular weight of 64 600 g/mol, the architecture parameter, χArch, in the adsorbed layer onto SiO2 and polystyrene film was found to be 2.0 and 2.08, respectively. In fact, the architecture of the adsorbed layer may be evaluated by using the ratio χArch, since the architecture of the adsorbed layer is like that of a polymer brush when χArch < 0.5 and is like that of a polymer pancake when χArch > 2. From the data shown in Figures 1-3 and Tables 1 and 2, it follows that the architecture of the adsorbed layer of an individual polymer with a molecular weight of 62 000 g/mol onto SiO2 and polystyrene films and a mixture of polymers with an average molecular weight of 64 600 g/mol on SiO2 is like that of a polymer pancake. However, the architecture of the adsorbed layer of a mixture of polymers with an average molecular weight of 64 600 g/mol onto polystyrene films is like that of a polymer pancake-brush. Next, we studied the adsorption and desorption kinetics in a flow cell of water-soluble polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol, and also a mixture of these polymers from aqueous solution onto the silica and polystyrene films over a wide range of polymer concentrations from 2 to 200 ppm by ellipsometry. Figures 4 and 5 show the time-dependence of the relative amount of polymer adsorbed, Uk(t), onto SiO2 and polystyrene films, respectively, over a wide range of times for a polymer concentration from 2 to 200 ppm for the adsorption and desorption processes, respectively. The diffusion coefficient in the adsorbed layer, (Dk(y)(θ∑), depends on the total surface coverage, θ∑, as5
(Dk(y))(θ∑) ) (Dk(y))(θ∑ f 0) exp(-Rkθ∑), Rk )
Figure 4. The relative amount of polymer adsorbed, U(t), versus time for the adsorption process at polymer concentrations of 2 (A), 20 (B), and 200 ppm (C) for the water-soluble associative polymer with a molecular weight of 12 000, 62 000, and 120 000 g/mol on silica (dotted curves) and polystyrene film (solid curves) substrates at a wall shear rate, γs, of 15 s-1 and L ) 1.8 cm.
where Rk and Qk are the activation parameter and activation energy, respectively, for the diffusion coefficient in the adsorbed layer for the kth species.
According to the developed theory of the kineticdiffusive-convective-controlled adsorption process for polymers under flow conditions, from eqs 13a and 13b, for
Qk , RT 1 e k e n (20)
5936 Langmuir, Vol. 14, No. 20, 1998
Filippova
[χ(θ)] ok )
(tdif rel)k )
o (tkin rel )k o (tdif rel)k
)
(tkin rel )k
× (1 - θok/4)(tdif rel)k γk 1 θ k2 - Rkθok (21d) exp 32 o 6
(
Γok cokKk(θkf0)
)
Γok
, (tkin rel )k )
{
}
[Dk(y)(θkf0)]2γs Kk(θkf0) ) β L
, θk )
ad ∞
cokKk Γ
Γk Γ∞
(21e) 1/3
, β ≈ 0.65
(21f)
diff kin where (ttotal rel )k, (trel )k, and (trel )k are the relaxation times for the kth species due to the adsorption kinetics and diffusion simultaneously, due to the adsorption kinetics, and also due to the diffusion, respectively, for short and intermediate times, (Soad)k is the slope of the straight line of the relative amount of polymer adsorbed versus time for short and intermediate times for the kth species, and [χ(θ)] ok is the ratio characterizing which of the mechanisms for the kth species is governed by the adsorption process for short and intermediate times. The ratio [χ(θ)] o k may be used as the criterion to estimate which of the mechanisms for the kth species is governed by the adsorption process for short and intermediate times. For short and intermediate times the adsorption process for the kth species is governed (a) by the adsorption kinetics on a planar surface for [χ(θ)] ok . 1, (b) by the diffusiveconvective mass transfer for [χ(θ)] ok , 1, and (c) by the adsorption kinetics on a planar surface and the diffusiveconvective mass transfer simultaneously for [χ(θ)] ok ≈ 1. For the kinetic-convective-diffusive controlled adsorption process under flow conditions from eqs 13a and 13b for long times, the relative amount of polymer adsorbed for the kth species is given by
Fkad(t) ≡ -ln[1 - Uk(t)] ≈ (S∞ad)kt, (S∞ad)k ) Figure 5. The amount of polymer adsorbed, Γ(t), versus time for the desorption process at polymer concentrations of 2 (A), 20 ppm (B), and 200 ppm (C) for the water-soluble associative polymer with a molecular weight of 12 000, 62 000, and 120 000 g/mol on silica (dotted curves) and polystyrene films (solid curves) substrates at a wall shear rate, γs, of 15 s-1 and L ) 1.8 cm.
short and intermediate times, the relative amount of polymer adsorbed for the kth species is given by
Γk(t) 1 ≈ (Soad)kt; (Soad)k ) total , Uk(t) ) Γok (trel )(θk)θok/4) 1 e k e n (21a)
(
(ttotal rel )k θk)
)
θok o kin o ) (tdif rel)k + (trel )k, 1 e k e n 4
(
)
(21b)
1 o dif kin o (tdif rel)k ) (trel)k exp Rkθok , (trel )k ) 6 γk (tkin rel )k exp θ k2 (21c) 1 - θok/4 32 o
(
)
a∞k
(ttotal rel )k(θk)θok)
dif ∞ kin ∞ (ttotal rel )k(θk)θok) ) (trel)k + (trel )k , 1 e k e n
(
)
2 ∞ dif (tdif rel)k ) (trel)k exp Rkθok , 3 ∞ (tkin rel )k )
[χ(θ)]∞k )
(tdif rel)k )
∞ (tkin rel )k ∞ (tdif rel)k
)
Γok cokKk(θkf0)
ak∞ )
(22b)
(
)
(tkin γk rel )k exp θok2 (22c) 1 - θok 2
(tkin rel )k (1 -
(22a)
∞ θok)(tdif rel)k
, (tkin rel )k )
exp
(
)
γk 2 2 θ k - Rkθok 2 o 3 (22d)
Γok
, θk )
ad ∞
cokKk Γ
(
Γk
Γ∞ (22e)
∂fk-1(Uk)1) γk ) 1 + b∞k 1 + ∂Uk 1 + b∞ k
)
(22f)
dif kin where (ttotal rel )k, (trel)k, and (trel )k are the relaxation times for
Adsorption/Desorption Kinetics of Mixtures
Langmuir, Vol. 14, No. 20, 1998 5937
the kth species due to the adsorption kinetics and diffusion simultaneously, due to the adsorption kinetics, and also due to the diffusion, respectively, for long times, (S∞ad)k is the slope of the straight line of a function of -ln(1 -the relative amount of polymer adsorbed) versus time for long times for the kth species, [χ(θ)]∞k is the ratio characterizing which of the mechanisms for the kth species is governed by the adsorption process for long times, and b∞k is the value which is found from the following algebraic equation (since bk is a known equilibrium constant):
b∞k ) bk exp
[
]
γk(1 - b∞k ) 2(1 -
b∞k )
(22g)
The ratio [χ(θ)]∞k may be used as the criterion to estimate which of the mechanisms for the kth species is governed by the adsorption process for long times. For long times the adsorption process for the kth species is governed (a) by the adsorption kinetics on a planar surface for [χ(θ)]∞k . 1, (b) by the diffusive-convective mass transfer for [χ(θ)]∞k , 1, and (c) by the adsorption kinetics on a planar surface and the diffusive-convective mass transfer simultaneously for [χ(θ)]∞k ≈ 1. From eqs 21a and 22a, it follows that the time dependence of the relative adsorption, Uk(t), over a wide range of time for the kinetic-diffusive-convective-controlled adsorption is given by
/ (tkin rel )ko
[χ(θ)] /o k )
(tdif rel)k )
/ (tdif rel)ko
)
(tkin rel )k
× (1 - 3θok/4)(tdif rel)k 9γk 2 1 θ k - Rkθok (24e) exp 32 o 2
(
Γok cokKk(θkf0)
)
, (tkin rel )k )
(Soad)k
1
g∞k )
(S∞ad)k
)
∞ kin ∞ (tdif rel)k + (trel )k
a∞k
(S∞des)k )
(
b∞k ) bk exp
des
Uk(t) ≈ 1 - (So
des
)kt, (So
)k )
(
)
b∞k ) bk exp
(
(ttotal rel )k θk)
)
[
]
γk(1 - b∞k ) 2(1 + b∞k )
3θok /o kin /o ) (tdif rel)k + (trel )k , 1 e k e n 4
(
)
(25b)
[
]
γk(1 - b∞k ) 2(1 + b∞k )
, bk ) K(p) k cok
(25c)
(24c)
)
[χ(θ)] /∞ k )
/∞ (tkin rel )k /∞ (tdif rel)k
)
(tkin Γok rel )k , (tdif , rel)k ) dif cokKk(θkf0) (trel)k Γok (25e) (tkin rel )k ) cokKkadΓ∞
(24b)
1 /o dif (tdif rel)k ) (trel)k exp Rkθok , 2 9γk 2 (tkin rel )k /o exp θ k (24d) (tkin ) ) rel k 1 - 3θok/4 32 o
(
)
/∞ dif kin /∞ kin (tdif rel)k ) (trel)k, (trel )k ) (trel )k (25d)
, (ttotal rel )k(θk)3θok/4) 1 e k e n (24a)
k
(25a)
dif /∞ kin /∞ (ttotal rel )k(θkf0) ) (trel)k + (trel )k ,
a∞k
∂fk-1(Uk)1) γk ∞ ) 1 + b∞k 1 + , ak ) ∂Uk 1 + b∞
aok total (trel )k(θkf0)
∂fk-1(Uk)0) γkb∞k 1 ) exp ∂Uk 1 + b∞k 1 + b∞k
(23b)
For the kinetic-diffusive-convective-controlled desorption process under flow conditions from eqs 13a and 13b for short and intermediate times, the relative amount of polymer adsorbed for the kth species is given by
Γ∞
Fkdes(t) ≡ -ln[Uk(t)] ≈ (S∞des)kt,
aok )
o kin o ) (tdif rel)k + (trel )k,
cokKk Γ
Γk
dif kin where (ttotal rel )k, (trel)k, and (trel )k are the relaxation times for the kth species due to the desorption kinetics and diffusion simultaneously, due to the desorption kinetics, and also due to the diffusion, respectively, for short and intermediate times, (Sodes)k is the slope of the straight line of the relative amount of polymer adsorbed versus time for short and intermediate times for the kth species, and [χ(θ)]/o k is the ratio characterizing which of the mechanisms for the kth species is governed by the desorption process for short and intermediate times. For the reversible kinetic-diffusive-convective-controlled desorption process under flow condition from eqs 13a and 13b, for long times, the relative amount of polymer adsorbed for the ith species is given by
(23a) 1
, θk )
ad ∞
(24f)
t ≈ (gok - g∞k )Uk(t) - g∞k ln[1 - Uk(t)], 1 e k e n
gok )
Γok
dif kin where (ttotal rel )k, (trel)k, and (trel )k are the relaxation times for the kth species due to the adsorption kinetics and diffusion simultaneously, due to the desorption kinetics, and also due to the diffusion, respectively, for long times, (S∞des)k is the slope of the straight line of the relative amount of polymer adsorbed versus time for long times for the kth species, and [χ(θ)] /o k is the ratio characterizing which of the mechanisms for the kth species is governed by the desorption process for long times. From eqs 24a and 25a, it follows that the time dependence of the relative
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Table 3. Characteristics of the Adsorption Processes of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in a Flow Cell for Short and Intermediate Timesa MWpol (kg/mol)
co (ppm)
12
2 20 200 2 20 200 2 20 200 2 20 200 2 20 200 2 20 200
62 120
12 62 120
Kadco [10-2(1/s)]
o (tkin rel ) (min)
o (tdif rel) (min)
[χ(θ)]o
Soad (min-1)
Adsorption of Associative Polymer in a Flow Cell onto SiO2 0.18 3.2 0.5 1.8 0.6 0.3 18 0.07 0.07 0.32 2.1 5.1 3.2 0.28 1.5 32 0.05 0.6 0.4 1.8 10.8 4 0.3 2.8 40 0.04 0.45
6.4 2.0 1.0 0.4 0.19 0.08 0.17 0.1 0.09
0.27 1.1 7.1 0.14 0.56 2.5 0.08 0.32 2.0
Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 0.21 4.3 0.6 2.1 0.65 0.4 21 0.07 0.09 0.35 2.9 6.3 3.5 0.42 1.9 35 0.05 0.76 0.45 4.5 9.5 4.5 0.4 2.9 45 0.06 0.5
7.1 1.6 0.78 0.46 0.22 0.06 0.47 0.14 0.12
0.2 0.95 6.25 0.11 0.43 2.1 0.07 0.3 1.78
a MW ad ) adsorption rate constant; (tkin)o and (tdif )o ) pol ) molecular weight of polymers; co ) polymer concentration in the bulk; K rel rel relaxation times for the adsorption process for short and intermediate times due to the adsorption kinetics and the diffusion, respectively; ad o [χ(θ)] ) ratio characterizing which of the mechanisms governs the adsorption process for short and intermediate times; So ) slope of the straight line of the relative amount of polymer adsorbed versus time for short and intermediate times; γs ) 15 s-1; and L ) 1.8 cm.
desorption, Uk(t), over a wide range of times for the kineticdiffusive-convective-controlled desorption is given by
t ≈ (hok - h∞k )[1 - Uk(t)] - h∞k ln[Uk(t)], 1 e k e n (26a) hok
/o kin /o (tdif rel)k + (trel )k ) , (Sodes)k a∞k
1
h∞k
/∞ kin /∞ (tdif rel)k + (trel )k ) ) (26b) aok (S∞des)k
1
Equations 23a and 26a are useful to describe over a wide range of time the kinetic-diffusive-convective-controlled adsorption and the kinetic-diffusive-convective-controlled desorption processes obeying arbitrary adsorption isotherms. To calculate the parameters characterizing the adsorption process it is reasonable to represent the experimental kinetic data (i) as the time-dependent adsorption, Γ(t), in the form of the relative adsorption, U(t), versus t for short and intermediate times and (ii) in the form of the relaxation functions, Fad(t) and Fdes(t) versus t for long times. According to eqs 21a-f, the slope Soad from the straight line of the relative amount of adsorption, U(t), versus time (for short and intermediate times) characterizes the different mechanisms controlling the adsorption process for short and intermediate times. According to eqs 22af, the slope S∞ad from the straight line of the relaxation function, Fad(t), versus time (for long times) characterizes the different mechanisms controlling the adsorption process for long times. Therefore, the values of the slopes Soad and S∞ad found from the experimental kinetic data for the adsorption process for the water-soluble associative polymers with different molecular weights at polymer concentrations ranging from 2 to 200 ppm are summarized in Tables 3 and 4. The adsorption process for polymers with high molecular weights (62 000 g/mol and more) at high polymer concentrations of 20 ppm and more (or a high surface coverage) is governed by the diffusiveconvective mass transfer for short and intermediate times;
therefore, from eqs 21a-f we write
1 1 o dif ≈ (tdif rel)k ) (trel)k exp Rθok , 6 (Sad ) o k Γok Γok (tdif , θok ) ∞ (27a) rel)k ) Kk(θf0)cok Γ
(
{
)
}
1/3
2 [D(y) k (θf0)] γs Kk(θf0) ) β L
, β ≈ 0.65
(27b)
where cok is the bulk concentration, Γok is the amount of polymer adsorbed with a polymer concentration of cok, and θok is surface coverage. To estimate the parameter R, which takes into account the activation energy for the diffusion coefficient, D(y)(θ), in the adsorbed layer, it is reasonable to rewrite eq 27a in a more convincing form, ad
H[cok,Γok,(So )k] )
cokΓo1(Soad)1
1 ) exp R(θok - θo1) , 6 )k Γo1 θo1 ) ∞ (28a) Γ
ad
co1Γok(So
[
1 ln{H[cok,Γok,(Soad)k]} ) R(θok - θo1) 6 SR )
ln{H[cok,Γok,(Soad)k]} R ) dθok 6 R ) 6SR )
Q RT
]
(28b)
(28c) (28d)
where SR is the slope of the straight line of the function ln{H[cok,Γok,(Soad)k]} versus θok and Q is the activation energy for the diffusion coefficient, D(y), in the adsorbed layer. From the experimental data represented in Table 3 for polymer concentrations co of 20 and 200 ppm the slope SR was calculated and then the parameter R and the activation energy Q were found for the polymers with
Adsorption/Desorption Kinetics of Mixtures
Langmuir, Vol. 14, No. 20, 1998 5939
Table 4. Characteristics of the Adsorption Processes of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in a Flow Cell for Long Timesa MWpol (kg/mol)
co (ppm)
12
2 20 200 2 20 200 2 20 200 2 20 200 2 20 200 2 20 200
62 120
12 62 120
[χ(θ)]∞
S∞ad (min-1)
Adsorption of Associative Polymer in a Flow Cell onto SiO2 0.18 10.5 4.7 1.8 4.8 1.7 18 2.8 0.42 0.32 8.2 15.6 3.2 4.5 7.2 32 2.9 3.2 0.4 5.5 34.2 4 2.9 13.2 40 2.5 6.3
2.2 2.8 6.7 0.52 0.62 0.9 0.16 0.22 0.4
0.31 1.02 3.5 0.24 0.72 2.1 0.18 0.52 1.8
Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 0.21 9.1 3.4 2.1 3.4 1.1 21 2.2 0.28 0.35 9.8 12.8 3.5 5.1 6.3 35 3.3 2.4 0.45 5.3 27.7 4.5 3.1 10.9 45 2.3 5.4
2.7 3.1 7.8 0.76 0.81 1.37 0.19 0.28 0.42
0.27 0.9 3.2 0.22 0.66 2.5 0.16 0.42 1.5
Kadco [10-2(1/s)]
∞ (tkin rel ) (min)
∞ (tdif rel) (min)
a MW ad ) adsorption rate constant; (tkin)∞ and (tdif )∞ ) pol ) molecular weight of polymers; co ) polymer concentration in the bulk; K rel rel relaxation times for the adsorption process for long times due to the adsorption kinetics and the diffusion, respectively; [χ(θ)]∞ ) ratio ad characterizing which of the mechanisms governs the adsorption process for long times; S∞ ) slope of the straight line of the relaxation function versus time for long times; γs ) 15 s-1; and L ) 1.8 cm.
Table 5. Characteristics of the Adsorption Process of Water-Soluble Nonionic Polymers onto Planar Polystyrene at SiO2 Substrates in a Flow Cella D(y)(θf0) [10-7(cm2/s)]
MWpol (kg/mol)
Q (kJ/mol)
R
Adsorption of Associative Polymer in a Flow Cell onto SiO2 12 3.1 2.6 6.3 62 1.7 3.2 7.8 120 1.2 3.8 9.2 Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 12 2.7 2.5 6.1 62 1.5 3.1 7.5 120 1.0 3.5 8.5 a MW (y) pol ) molecular weight of polymers; D (θf0) ) diffusion coefficient for polymer in the bulk; R ) parameter characterizing the surface coverage dependence of the activation energy of diffusion in the adsorbed layers; Q ) activation energy of diffusion in the adsorbed layer; γs ) 15 s-1; and L ) 1.8 cm.
molecular weights of 12 000, 62 000, and 120 000 g/mol by using eq 28c. Finally, the diffusion coefficient, D(y)(θf0), in the adsorbed layer for SiO2 and polystyrene film substrates was calculated by using eqs 27a and 27b. These values are listed in Table 4. The adsorption process for polymers with low polymer concentrations of 2 ppm and less (or a low surface coverage) is governed by the adsorption kinetics on a planar surface for short and intermediate times; therefore, from eq 21a-e we write
1 (Soad)k
o ≈ (tkin rel )k )
(
(tkin rel )k )
{
)
(tkin γk 2 rel )k exp θ , 1 - θok/4 32 ok Γok KkadcokΓ
}
2 [D(y) k (θf0)] γs Kk(θf0) ) β L
, θok ) ∞
Γok Γ∞
(29a)
1/3
, β ≈ 0.65
(29b)
where cok is the bulk concentration, Γok is the amount of polymer adsorbed with a polymer concentration of cok, θok is surface coverage, and Kkad is the rate constant of the adsorption process for the kth species. By using the values
of γ from Tables 3 and the experimental slope, Soad, we calculated the rate constant, Kad, for water-soluble polymers on planar surface SiO2 and polystyrene films by using eqs 29a and 29b; this value is listed in Table 3. By using the values of Γo, Γ∞, Kad, b, γ, R, and D(y)(θf0) from Tables dif o o 3-5, we calculated the relaxation times, (tkin rel ) and (trel) o ∞ and the ratios, [χ(θ)] and [χ(θ)] , for water-soluble polymers on planar surface SiO2 and polystyrene films by using eqs 21c,d, 22c-f; these values and the experimental values of slope, Soad are shown in Table 3. By using the ratios, [χ(θ)]o and [χ(θ)]∞, we may estimate which of the mechanisms governs the adsorption process for short, intermediate, and long times, since
{
(AK) >4 [χ(θ)]o, [χ(θ)]∞ ) 0.25/4 (AK + DCMT) 4 [χ(θ)]* ) 0.25/4 (DK + DCMT) 1, the algebraic additivity of resistances due to the different mechanisms controlling the adsorption process does not take place since the interactions between species of the mixture occur. The values of ξoad, calculated from experimental kinetic data by using eqs 34a,b and 35, are shown in Table 9. The
ratios ξoad were 1.2 for a polymer concentration of 2 ppm and 1.6 for a polymer concentration of 200 ppm. Thus, the interactions between species of the mixture in the adsorbed layer take place and increase with increasing polymer concentrations. From eq 22a for long times, the relative amount of polymer adsorbed in a mixture is given by ad ∞ Fad mix(t) ≡ -ln[1 - Umix(t)] ≈ (Srel)mixt, ∞ (Sad rel)mix )
1 (36) ∞ (tad rel)mix
∞ where (tad rel)mix is the relaxation time for a polymer mixture resulting from to the adsorption kinetics and diffusiveconvective mass transfer simultaneously for long times ∞ and (Sad rel)mix is the slope of the straight line of the relative amount of polymer adsorbed in a mixture versus time for long times. From eqs 34a and 36 it follows that the time dependence of the relative adsorption for a polymer mixture, Umix(t), over a wide range of times for the kinetic-diffusiveconvective-controlled adsorption is given by
t ≈ (gomix - g∞mix)Umix(t) - g∞mix ln[1 - Umix(t)] gomix )
1 1 , g∞mix ) ad ∞ o (Sad ) (S rel mix rel)mix
(37a) (37b)
The time, (tΓad)mix, needed to reach the quasi-equilibrium state for the kinetic-diffusive-convective adsorption process can be approximated by using eq 36 as ad ∞ ad ∞ (tad Γ )mix ≈ ln(1/)(trel)mix, (trel)mix )
1
, ) 0.01
∞ (Sad rel)mix
(38)
The values, (tΓad)mix, calculated by using eq 38 are listed in Table 9. For the kinetic-diffusive-convective-controlled desorption process under flow conditions for a polymer mixture from eq 24a for short and intermediate times, the relative amount of polymer adsorbed in a mixture is given by n
Umix(t) )
Γi ∑ i)1 n
∑ i)1
o des o ≈ 1 - (Sdes rel )mixt, (Srel )mix )
Γoi
1 o (tdes rel )mix
(39)
o where (tdes rel )mix is the relaxation time for a polymer mixture resulting from the desorption kinetics and diffusive-convective mass transfer simultaneously for short o and intermediate times and (Sdes rel )mix is the slope of the straight line of the relative amount of polymer desorbed in a mixture versus time for short and intermediate times. For the reversible kinetic-diffusive-convective-controlled desorption process under flow conditions from eq 25a, for long times, the relative amount of polymer adsorbed in a polymer mixture is given by
des ∞ des ∞ Fdes mix(t) ) -ln[Umix(t)] ≈ (Srel )mixt, (trel )mix )
1 ∞ (Sdes rel )mix
(40)
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Table 10. Architecture of the Adsorbed Layer and Times Needed To Reach the Quasi-Equilibrium State for Adsorption and Desorption Processes of the Individual Water-Soluble Polymer and Polymer Mixture onto Planar Polystyrene and SiO2 Substrates in a Flow Cella MWpol (kg/mol) 62 (individual) 64.6 (mixture)
62 (individual) 64.6 (mixture)
co (ppm)
χArch
architecture of the adsorbed layers
tad Γ (min)
tdes Γ (days)
Adsorption/Desorption of Associative Polymer in a Flow Cell onto SiO2 2 22.2 20 6.4 200 2.0 pancake 2.2 2 28.5 20 8.9 200 2.08 pancake 2.5
109 138 152 138 201 283
Adsorption/Desorption of Associative Polymer in a Flow Cell onto Polystyrene Films 2 23.3 20 7 200 1.89 pancake 1.8 2 31.7 20 11 200 1.64 pancake-brush 3
102 122 140 121 122 273
a MW pol ) molecular weight of polymers; co ) polymer concentration in the bulk; χArch ) parameter characterizing the architecture of des the adsorbed layer; tad Γ and tΓ ) times needed to reach the quasi-equilibrium state for the adsorption and desorption processes for the individual polymer and polymer mixture, respectively; γs ) 15 s-1; and L ) 1.8 cm.
where (tdes rel )∞mix is the relaxation time for a polymer mixture due to the desorption kinetics and diffusiveconvective mass transfer simultaneously and (Sdes rel )∞mix is the slope of the straight line of the relative amount of polymer adsorbed in a polymer mixture versus time for long times. From eqs 39 and 40, it follows that the time dependence of the relative desorption in a polymer mixture, Umix(t), over a wide range of times for the kinetic-diffusiveconvective-controlled desorption is given by
t ≈ (homix - h∞mix)[1 - Umix(t)] - h∞mix ln[Umix(t)] 1 1 homix ) des o , h∞mix ) des ∞ (Srel )mix (Srel )mix
polymer and a polymer mixture onto polystyrene films under flow conditions is different due to the competitive effect in the adsorbed layer. From data presented in Table 10, it follows that the rate of adsorption processes for a polymer mixture is greater (4 and more times the order of magnitude) than that of the desorption processes for a polymer mixture which are irreversible due to the strong interactions between polymer molecules and interface. Conclusions
(41a) (41b)
Equations 37a and 41a are useful to describe over a wide range of times the kinetic-diffusive-convective-controlled adsorption and the kinetic-diffusive-convective-controlled desorption processes in a polymer mixture obeying arbitrary adsorption isotherms. Finally, we consider the architecture of the adsorbed layer for an individual polymer and a polymer mixture des and the times (tad Γ )mix and (tΓ )mix needed to reach the quasi-equilibrium state under flow conditions for the adsorption and desorption processes. From the experimental data presented in Tables 1, 2, and 10, the architecture parameter, χArch, was found to be 2.0 for the water-soluble associative polymer with a molecular weight of 62 000 g/mol and was found to be 2.08 for a polymer mixture with an averaged molecular weight of 64 600 g/mol onto the SiO2 substrate. Thus, the architecture of the adsorbed layer for an individual polymer and a mixture of water-soluble associative polymers onto SiO2 under flow conditions is like that of a polymer pancake. The architecture parameter, χArch, was found to be 1.89 for the water-soluble associative polymer with a molecular weight of 62 000 g/mol and was found to be 1.64 for a polymer mixture with an averaged molecular weight of 64 600 g/mol onto polystyrene films. Thus, the architecture of the adsorbed layer for an individual polymer onto polystyrene films under flow conditions is like that of a polymer pancake. However, the architecture of the adsorbed layer for a polymer mixture onto polystyrene films under flow conditions is like that of a polymer pancake-brush. Thus, the architecture of the adsorbed layer for an individual
We have developed a theory describing the kineticconvective-diffusive-controlled adsorption and desorption processes for water-soluble associative polymers and polymer mixtures on planar surfaces in a flow cell by using a new approach. We showed that the rates of the adsorption and desorption processes on a planar surface for individual polymers and polymer mixtures depend on the rate of the adsorption and desorption kinetics and convective-diffusive-mass transfer simultaneously. Equations were derived to calculate the parameters which characterize the adsorption and desorption processes for individual polymers and polymer mixtures under flow conditions. Using a theoretical approach and analysis of the ellipsometric experimental data, it is shown that the adsorption and desorption processes for individualpolymers and polymer mixtures under flow conditions for water-soluble associative polymers onto the silica and polystyrene film on a planar surface are governed (i) by the kinetics of polymer adsorption and desorption for short and intermediate times and (ii) simultaneously by the adsorption and desorption kinetics of polymers and the diffusive-convective mass transfer in the adsorbed layer for long times, as shown in Tables 6 and 7. Tables 1-10 present the parameters characterizing an individual polymer and a polymer mixture: (I) the adsorbed layers, (a) the adsorption, Γmax, of the water-soluble associative polymers on planar polystyrene film and SiO2 surfaces, (b) the structure (i.e., the thickness, dad.layer, the area occupied by one polymer molecule, σm, and the weight fraction, Xad.layer) and architecture (the parameter, γ) of the adsorbed layer, and (c) the activation energy of adsorption (-∆H), and (II) the adsorption process, i.e., (d) the adsorption rate constant, Kad, (e) the polymer diffusion coefficient, D(cf0), (f) the activation energy of polymer diffusion in the adsorbed layers, Q, and (g) the times (tΓad,
Adsorption/Desorption Kinetics of Mixtures
tΓdes) needed to reach the quasi-equilibrium state for adsorption and desorption processes in the adsorbed layer. It is shown that because of the competitive effect in the adsorbed layer under flow conditions, the behavior of the adsorption and desorption processes for individual polymers and mixtures of the water-soluble associative polymer under flow conditions is significantly different. The architecture of the adsorbed layer for individual polymers and polymer mixtures of the water-soluble associative polymers onto SiO2 and for individual polymers
Langmuir, Vol. 14, No. 20, 1998 5945
onto polystyrene films is like that of a polymer pancake under flow conditions, but the adsorbed layer for polymer mixtures onto polystyrene films is like that of a polymer pancake-brush under flow conditions. The rate of adsorption under flow conditions is greater than the rate of desorption; therefore, the equilibrium state for the adsorption process is reached significantly faster than that for desorption, as shown in Tables 9 and 10. LA980545V
