2884
Langmuir 1999, 15, 2884-2897
Activity Coefficients for Equilibrium and Adsorption Kinetics of Mixtures on Planar Surfaces under Flow Conditions Nadezhda L. Filippova* Russian Branch RTD Corporation, Bethlehem, Pennsylvania 18015 Received July 27, 1998. In Final Form: October 9, 1998 The adsorption isotherms and adsorption kinetics for mixtures of triblock water-soluble polymers having poly(ethylene glycols) with a molecular weights of 12 000, 62 000, and 120 000 g/mol as the middle block and C16H33 linear alkyl groups on each end of the molecule onto silica (SiO2) and polystyrene film substrates were studied by ellipsometry. Equations for activity coefficients were derived (a) to calculate the adsorption isotherm for a mixture by using the experimental adsorption isotherms for individual species, (b) to calculate the adsorption isotherm for individual species of a mixture by using the experimental adsorption for a mixture, and (c) to predict behaviors of individual species in a mixture by using the experimental kinetic data for a mixture onto a planar surfaces under flow conditions.
Introduction A knowledge of both the adsorption equilibrium and adsorption kinetics of a mixture onto planar surfaces under flow conditions is of great importance for optimal application of polymers.1-3 Theoretical and experimental studies have been carried out to investigate the structure of polymeric materials near an interface. Knowledge of the adsorption process of polymers is of great importance to a complete understanding of mass transfer in the adsorbed layer.3-19 Many important phenomena, such as interfacial turbulence, thin film stability, and so on, are consequences of the fact that the adsorbed layer varies with surface concentration. In hydrodynamic modeling, these phenomena are coupled with mass transfer through the surface stress boundary conditions.5 Polymer adsorption is the subject of increasing attention owing to the * Address for correspondence: Nadezhda L. Filippova, 634 Broadway (second floor), Bethlehem, PA 18015. (1) Polymer Adsorption and Dispersion Stability; Goddard, E. D., Vincent, B., Eds.; American Chemical Society: Washington, DC, 1984. (2) Adamson, A. W. Physical Chemistry of Surfaces; Interscience: New York, 1986. (3) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Corgrove, T.; Vincent, B. Polymer Interfaces; Chapmann & Hall: London, 1993. (4) Lucassen, J.; Hollway, F.; Buckingham, J. H. J. Colloid Interface Sci. 1978, 67, 423. (5) Cero, R. L.; Whitaker, S. J. Colloid Interface Sci. 1980, 37, 33. (6) Joos, P.; Rillaers, E. J. Colloid Interface Sci. 1981, 79, 96. (7) Lee, J. J.; Fuller, G. G. J. Colloid Interface Sci. 1985, 103, 569. (8) Stuart, M. A.; Cosgrove, T.; Vincent, B. Adv. Colloid Interface Sci. 1986, 23, 143. (9) Young, B. R.; Pitt, W. G.; Cooper, S. L. J. Colloid Interface Sci. 1988, 125, 246. (10) Caucheteux, I.; Hervet, H.; Jerome, R.; Rondelez, F. J. Chem. Soc., Faraday Trans.1990, 86, 1369. (11) Munch, M. R.; Gast, A. P. J. Chem. Soc., Faraday Trans. 1990, 86, 1341. (12) Pefferkorn, E.; Elaissari, A. J. Colloid Interface Sci. 1990, 138, 187. (13) Serrien, G.; Joos, P. J. Colloid Interface Sci. 1990, 139, 149. (14) Halperin, A.; Tirell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 31. (15) Serrien, G.; Geeraets, G.; Ghosh, L.; Joos, J. J. Colloids Surf. 1992, 68, 219. (16) Shibata, C. T.; Lenhoff, A. M. J. Colloid Interface Sci. 1992, 148, 485. (17) Dorgan, J. R.; Stamm. M.; Toprakcioglu, C.; Jerome, R.; Fetters, L. J. Macromolecules 1993, 26, 5321. (18) Kim, D.; Cha, W.; Beissinger, R. L. J. Colloid Interface Sci. 1993, 159, 1. (19) Amiel, C.; Sikka, M.; Schneider, W.; Yi Hua Tsao; Tirrell, M.; Mays, J. W. Marcomolecules 1995, 28, 3125.
many technological needs for macromolecules at surfaces. Surface modification for lubrication of bulk polymer films, biocompatibility, adhesion, and stabilization of colloidal particles are all examples of applications of adsorbed polymers. The stabilization of colloidal particles by anchored polymers is an attractive approach to improving paints, coating, inks, and another media.1,8,11 Despite this activity, little work to date has focused on the kinetics of the adsorption process owing to the difficulty in measuring dynamic process at an interface. Kinetic studies yield important information on how the adsorbed amount is affected by phenomena such as macromolecular rearrangement, desorption exchange with bulk solution, and flow-induced conformational changes.7,19 Mixtures of several nonionic polymers are used in different polymers applications. Therefore, the study of the adsorption of mixtures of polymers on planar surfaces under flow conditions is important. Numerous authors6-18 have tried to determine the mechanism that controlled the adsorption processes under flow conditions for nonionic polymers. Most isotherms of the adsorption of individual polymers and mixtures are curves that reach saturation at given concentrations. However, the thickness of the adsorbed layers and the amount of polymer adsorbed from the polymer mixture cannot be calculated from data for individual polymers according to the classical concept. Therefore, it is important to develop approaches which allow us to predict (a) the adsorption isotherm for a polymer mixture using the experimental isotherms obtained for individual species and (b) the adsorption kinetics for a polymer mixture using the experimental kinetic data obtained for individual species. Our theoretical study concerning the scientific aspects of the exchange kinetics of water-soluble polymers under flow conditions on planar surfaces has focused on the new approaches and allows us (a) to estimate the effect of interactions between polymer molecules and interface for the adsorption isotherm of polymer mixture, (b) to calculate the adsorption isotherm for a polymer mixture by using the experimental isotherms obtained for individual species, and (c) to predict the behaviors of the exchange kinetics by using the experimental kinetics data for individual species. Our experimental research is the study of adsorption kinetics for mixtures of nonionic water-soluble polymer over a wide
10.1021/la980950m CCC: $18.00 © 1999 American Chemical Society Published on Web 03/23/1999
Adsorption Kinetics of Mixtures on Planar Surfaces
Langmuir, Vol. 15, No. 8, 1999 2885
range of concentrations under flow conditions on planar surfaces of silica and polystyrene films by ellipsometry. Activity Coefficients for Equilibrium and Adsorption Kinetics on Planar Surfaces under Flow Conditions First, we consider the adsorption isotherm for a mixture of nonionic polymers on a planar surface under flow conditions. In the framework of the Arrehenius and Eyring approaches, and the Lucassen-Reynders convention, the adsorption isotherm for mixtures of nonionic polymers onto a planar surface under flow conditions for the individual species (eq 1a) and a mixture (eq 1b) are described by using the following system of equations20
θk Ckbk exp(-γkθΣ) ) 1 - θΣ
Ck )
Ck
θk )
Cok
bk ) Kk(p) Cok,
Γk
(1b)
Γ∞
∑
θk,
Ck ) RkC
k)1
1eken
(2b)
where Γk is the amount of nonionic polymer adsorbed of the kth species, Γ∞ is the maximum amount of polymer adsorbed, cok is the nonionic polymer concentration in the bulk of the kth species, Kk(p) is the equilibrium constant for the kth species, bk is the relative equilibrium constant for the kth species, γk and (-∆Hk) are the interaction parameter and the activation energy of adsorption of the kth species, respectively, characterizing the interaction between polymer/interface, polymer/polymer, and polymer/ solvent, θk is the surface coverage of the kth species, θΣ is the total surface coverage, n is the number of species in the solution, Ck and C are the relative concentration of the kth species and the total concentration, respectively, in the solution, Rk is the concentration fraction of the kth species in the solution, R is the gas constant, and T is the absolute temperature. To estimate the effect of interactions between polymer species and interface, it is reasonable to rewrite eqs 1a and 2b by using the following new variables
Ukeq )
θk θok
)
Γk
n
,
Γok
UΣeq )
dkUkeq, ∑ k)1
dk )
θok n
θk ) φk(UΣeq),
θΣ ) φΣ(UΣeq)
,
(5)
Γk ) ψk(U1eq, U2eq, ..., Uneq)
(6)
1eken For the two-component mixture from eqs 1a and 2b it follows that
1 e k e r (3)
∑ θom
m)1
where Ukeq and UΣeq are the relative amount of polymer adsorbed for the kth species and the total relative amount of polymer adsorbed, respectively, in the equilibrium state. In the system of eqs 1-3 for the two-component mixture (n ) 2) it is convenient to analyze in the phase planes (U1eq, U2eq) and (U1eq, UΣeq) to estimate the effect of interactions between polymer species and interface in the equilibrium state. The adsorption isotherms from eqs 1a (20) Filippova, N. L. Chem. Eng. Commun. 1998, 167, 181.
(7a)
ΓΣ(C) ) Γ1(C) + Γ2(C) k ) 1, 2
{ ( )
β1eq(UΣeq) ) 1 +
R2b2 exp[(γ1 - γ2) (θo1U1eq + R1b1
}
-1
(2a)
(-∆H)k , γk ) RT
(4)
However, to estimate the interactions between polymer molecules and interface, the adsorption isotherms are more convenient to represent in the following form
Σ
n
θΣ )
,
θΣ ) φΣ(C)
Γk(C) ) βkeq(UΣeq)ΓΣ(C)
∑ Rkbk exp(-γkθΣ) ) 1 - θ k)1
,
θk ) φk(C);
(1a)
θΣ
n
C
and 1b may be rewritten as
θo2U2eq)] β1eq(UΣeq) + β2eq(UΣeq) ) 1
(7b) (7c)
where βkeq(UΣeq) is the activity coefficient characterizing the interactions between polymer molecules in the adsorbed layer for the equilibrium state. Equations 7a and 7b may be used in order to find the amount of polymer adsorbed for the kth species from the experimental data for the total amount of polymer adsorbed, ΓΣ(C). The value of θoΣ is found from the following algebraic equation
R1b1 exp(-γ1θoΣ) + R2b2 exp(-γ2θoΣ) )
θoΣ 1 - θoΣ
(8a)
and then the values of θo1 and θo2 are found from the following equations
θo1 ) (1 - θoΣ)R1b1 exp(-γ1θoΣ)
(8b)
θo2 ) θoΣ - θo1 For the Langmuir adsorption isotherms, when the interactions between polymer molecules and interface are ignored (γk ) 0), from eqs 7a and 7b, it follows that the activity coefficients are constant, i.e.
[ ( )]
β1eq(UΣeq) ) 1 +
R2b2 R1b1
-1
) constant
(9)
β2eq(UΣeq) ) 1 - β1eq(UΣeq) Therefore, the deviation of the adsorption isotherm, taking into account the interactions between polymer molecules and an interface (γk * 0), from the Langmuir adsorption isotherms (γk ) 0) may be used to estimate the effect of the interactions between polymer molecules and an interface. Second, we consider the adsorption kinetics for mixtures on a planar surface under flow conditions. In the framework of the Arrehenius and Eyring approaches and the Lucassen-Reynders convention, the exchange kinetics
2886 Langmuir, Vol. 15, No. 8, 1999
Filippova
for mixtures of nonionic polymers onto a planar surface under flow conditions is described by using the following system of equations20-22
dθk(t) ) Kk*(θk)[1 - fk-1(θ1, θ2, ..., θn)] dt
()[ -1
{
1 1 + Kk(θk) Φk(θk) (y)
]
}
2
[Dk (θk)] γs Kk(θk) ) 0.65 L
U1(0) ) U2(0) ) 0 dUΣ(t) dU1(t) (10b)
[() ]
θ(t) ) Γk(t)/Γ∞,
) d1 + d2(U1, U2) 1 - f1-1[U1(t), U2(t)]
(11b)
UΣ(0) ) U1(0) ) 0
(10c)
θk(t) Γk(t) ) θok Γok
(11c)
2
UΣ(t) )
∑ dkUk(t)
k)1
γk 2 θ (t) 2 Σ
Φk(θk) ) Γ∞Kkad[1 - θk(t)] exp -
bk ) Kk(p) Cok,
1 - f2-1[U1(t), U2(t)]
Uk(t) )
1/3
1eken
fk-1(θ1, θ2, ..., θn) )
(11a)
(10a)
1eken cok 1 ) Γ∞ Kk*(θk)
1 - f2-1[U1(t), U2(t)] ) [U1(t), U2(t)] dU1(t) 1 - f1-1[U1(t), U2(t)]
dU2(t)
θk bk(1 - θΣ)
θok θo1 + θo2
k ) 1, 2
exp(γkθΣ) (10e)
Kk(p) ) Kkad/Kkdes
dk )
(10d)
(U1, U2) )
∑ θk(t) k)1
where Γk is the amount of nonionic polymer adsorbed of the kth species, Γ∞ is the maximum amount of polymer adsorbed, cok is the nonionic polymer concentration in the bulk of the kth species, Kk*(θk) is the total rate constant for the adsorption process taking into account the adsorption kinetics onto a planar surface and convective-diffusive mass transfer for the kth species, fk(c) is a function describing the adsorption isotherm of the kth species, fk-1(Γ) is the reciprocal function, Kk(θk) and Φk(θk) are the rate constants due to the convective-diffusive mass transfer and the adsorption kinetics onto a planar surface, respectively, for the kth species, n is the number of species in the solution, γs is the wall shear rate, L is the point of measurement of adsorption, Kkad and Kkdes are the constant rate of adsorption and desorption, respectively, for the kth species, Kk(p) is the equilibrium constant for the kth species, γk and (-∆Hk) are the interaction parameter and the activation energy of adsorption of the kth species, respectively, characterizing the interactions between polymer/interface, polymer/polymer, and polymer/solvent, θk is the surface coverage of the kth species, θΣ is the total surface coverage, and Dk(y) is the diffusion coefficient of the kth species in the direction (y) perpendicular to the flux direction. To estimate both the role of rates of exchange kinetics and interactions between polymer species on a planar interface for the kinetic-convective-controlled adsorption for the system of eqs 10a through 10f for the twocomponent mixture (n ) 2), it is convenient to analyze in the phase planes (U1, U2) and (U1, UΣ). In this case the system of eqs 10a-f may be rewritten as (21) Filippov, L. K.; Filippova N. L. J. Colloid Interface Sci. 1997, 189, 1. (22) Filippova, N. L. Langmuir 1998, 14, 1162.
d2K1*(U1)
(11d)
d1 + d2 ) 1
(10f)
n
θΣ(t) )
d1K2*(U2)
fk-1[U1(t), U2(t)] )
θo1U1(t) bk[1 - θo1U1(t) - θo2U2(t)] exp{γk[θo1U1(t) + θo2U2(t)]} (11e)
where Uk(t) and UΣ(t) are the relative amount of polymer adsorbed for the kth species and total relative amount of polymer adsorbed, respectively. Now, we consider how from curve fit eq 11a. For the case when o g 1, from eq 11a for U1(t) f 0 (eq 12a) and U1(t) f 1 (eq 13a) we find that
U2(t) ≈ SoU1(t)
(12a)
[U1(t) f 0] So ) o ) (U1)0, U2)0)
(12b)
1 - U2(t) ≈ S1[1 - U1(t)]
(13a)
[U1(t) f 1] S1 ) A1 - (A12 + A2)1/2 < 0
(13b)
1 ≡ (U1)U2)1) A1 )
1a22 - a11 2a12
A2 ) akm )
1a21 a12
∂fk-1(U1, U2) ∂Um
1 e k, m e 2
(13c)
Adsorption Kinetics of Mixtures on Planar Surfaces
a11 ) 1 +
θo1 + γ1θo1 1 - θo1 - θo2
Langmuir, Vol. 15, No. 8, 1999 2887
(13d)
θo2 + γ1θo2 a12 ) 1 - θo1 - θo2 a22 ) 1 + a21 )
θo2 + γ2θo2 1 - θo1 - θo2
Γk(t) ) βk[UΣ(t)]ΓΣ(t)
(19a)
ΓΣ(t) ) Γ1(t) + Γ2(t) k ) 1, 2 β1[UΣ(t)] + β2[UΣ(t)] ) 1
(13e)
θo1 + γ2θo1 1 - θo1 - θo2
where So is the slope of the straight line of U2(t) versus U1(t) for U1(t) f 0 and S1 is the slope of the straight line of [1 - U2(t)] versus [1 - U1(t)] for U1(t) f 1. From eqs 12a through 13a we find that
U2(t) ≈ SoU1(t) + (3 - 2So - S1) [U1(t)]2 + (So + S1 - 2)(U1(t)]3 (14)
(19b)
where βk[UΣ(t)] is the activity coefficient characterizing both the interactions between polymer molecules in the adsorbed layer and the rate of kinetics on a planar surface under flow conditions. Equations 19a and 19b may be used in order to find the amount of polymer adsorbed for the kth species from the experimental kinetic data for the total amount of polymer adsorbed, ΓΣ(t). Equations 18 through 19b may be used to find the activity coefficient, βk[UΣ(t)]. The above developed approaches may be used to study the adsorption isotherms for individual species and mixtures and also the exchange kinetics of water-soluble nonionic polymers in a flow cell on planar silica (SiO2) and polystyrene film substrates by ellipsometry.
The maximum value of U*2 is given by
U2* ) maxU2(U1)U1*)
(15a)
U1* ) -B1 + (B12 - B2)1/2 B1 < 0; So g 1 B1 ) B2 )
3 - S1 - 2So 3(So + S1 - 2)
(15b)
So 3(So + S1 - 2)
Next, we consider how from curve fit eq 11b. From eq 11b for U1(t) f 0 (eq 16a) and U1(t) (eq 17a) we find that
UΣ(t) ≈ So*U1(t);
[U1(t) f 0]
So* ) d1 + d2o 1 - UΣ(t) ≈ S1*[1 - U1(t)],
(16a) (16b)
[U1(t) f 1]
(17a)
S1* ) d2(λ˜ - 1)/a12
(17b)
λ˜ ) R1 - (R12 - R2)1/2
(17c)
R1 ) (a11 + 1a22)/2
(17d)
R2 ) 1(a11a22 - a12a21) where akm is the coefficient which is found from eqs 13d and 13e, S0* is the slope of the straight line of UΣ(t) versus U1(t) for U1(t) f 0, and S1* is the slope of the straight line of [1 - UΣ(t)] versus [1 - U1(t)] for U1(t) f 1. From eqs 16a and 17a we find that
UΣ(t) ≈ So*U1(t) + (3 - 2So* - S1*)[U1(t)]2 + (So* + S1* - 2)[U1(t)]3 (18) For the two-component mixture from eq 10a for the adsorption kinetics on a planar surface under flow conditions, the adsorption of the kth species is given by
Experimental Materials and Methods We have experimentally studied the adsorption isotherms and adsorption for mixtures of triblock water-soluble 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 (a) to describe the adsorption isotherm for a mixture by using the experimental adsorption isotherms for individual species, (b) to describe the adsorption isotherm for individual species by using the experimental adsorption isotherm for a mixture, and (c) to predict behaviors of individual species in a mixture by using the experimental kinetic data for a polymer mixture onto a planar surfaces under flow conditions. The flow cell20-22 was used to measure the values of the refractive index of the silica (SiO2) in DDI (distilled, deionized) water and the ellipsometric values for polymer solutions on the silica during the adsorption processes at room temperature. The silica 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 water-soluble associative polymers used in the kinetic study were obtained from nonionic polyurethanes based on poly(ethylene glycol) and have 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 both the adsorption isotherms and adsorption 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 the 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, n4* ) n4 - ik4, a layer of silica (SiO2) with a refractive index of n3 and a thickness d3, a polystyrene layer with a refractive index of n2 and a thickness d2, the adsorbed polymer layer with a refractive index n1 ) nad.layer and a thickness d1 ) dad.layer, and a surrounding solution with a refractive index no. As shown in our previous papers,20-22 the
2888 Langmuir, Vol. 15, No. 8, 1999
Filippova nad.layer, from eq 20 are calculated by using the ellipsometric experimental data (ψ and ∆). As shown in our previous papers,20-22 the weight fraction of polymer in the adsorbed layer, Xad.layer, and the adsorption, Γ, are given by
Xad.layer )
Rpol )
6nad.layer(nad.layer - nsol) 2 (Rpol - Rsol)(nad.layer + 2)2
(n2pol - 1) , (n2pol + 2)
Rsol )
2 - 1) (nsol 2 (nsol + 2)
Γ ) dad.layerFpolymerXad.layer
(21a)
(21b)
(21c)
The weight fraction of polymer in the adsorbed layer, Xad.layer, is then calculated by using eqs 21a and 21b. Finally, the amount of polymer adsorbed, Γ, is determined from eq 21c.
Discussions
Figure 1. Amount of polymer adsorbed, Γk(ck), versus the polymer concentration, ck, for (A) the individual water-soluble associative polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol, (B) the two-component mixture with molecular weights of 12 000 and 62 000 g/mol, and (C) the twocomponent mixture with molecular weights of 62 000 and 100 000 g/mol [1 and 2 are calculated by eq 7a, 3 (circles) is from experimental data] on a SiO2 substrate at a wall shear rate, γs, of 25 s-1 and L ) 2.2 cm.
The above approaches for the kinetic-convective-diffusive-controlled adsorption kinetics for a multicomponent mixture of nonionic polymers under flow conditions were developed in order to understand and explain behavior the exchange kinetics for nonionic polymer mixtures in the adsorbed layer on planar surfaces of various substrates, i.e., the silica and the polystyrene film. The adsorption of water-soluble nonionic associative polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol from aqueous solution onto the silica and the polystyrene film (with a polystyrene molecular weight of 110 000 g/mol) was studied in the flow cell for a polymer concentration range of 4 ppm (mg/kg) to 400 ppm by ellipsometry at room temperature. First, we studied the adsorption isotherms for watersoluble nonionic associative polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol and also the adsorption of a mixture of these polymers in the flow cell from aqueous solution onto the silica and polystyrene film by ellipsometry. Figures 1A and 2A show the adsorption isotherm of the water-soluble polymer onto SiO2 and polystyrene film substrates, respectively. The thickness of the adsorbed layer, dad.layer, the weight fraction, Xad.layer, and the amount of adsorbed polymer, Γ, were calculated by using eqs 20 through 21c. To estimate the parameter b which characterizes the adsorption isotherm, K(p), Γ∞, and γ, it is reasonable to rewrite eq 1a for an individual polymer species in more convincing forms
Ki(p)ci )
ln thickness, dad.layer, and the refractive index, nad.layer, of the adsorbed layer were found simultaneously by using the following equations 2 2 Re(δ) ) (2π/λ)dad.layer[nad.layer - nsol sin θsol]1/2
(20)
Im(δ) ) 0 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,
()
Γi ∞
Γ - Γi
( )
exp γi
Γi
Γ∞
()
c1 Γ1 ) -ln(K1(p)) + γ1 ∞ - ln(Γ∞ - Γ1) Γ1 Γ
( )
c1 c1 Γ1 1 ) ∞ + (p) ∞ exp γ1 ∞ Γ1 Γ Γ K1 Γ
(22a)
(22b)
(22c)
where Ki(p) the equilibrium constant for the ith species of polymer, Γ∞ is the maximum amount of polymer adsorbed, and γ1 is the parameter characterizing the interaction between polymer/interface, polymer/polymer, and polymer/ solvent. From eqs 22b and 22c we write, respectively, the following relations
Adsorption Kinetics of Mixtures on Planar Surfaces
ln
()
c1 ) Iso + SsoΓ1 Γ1
Langmuir, Vol. 15, No. 8, 1999 2889
(23a)
Γ1 f 0 Iso ) -ln(K1(p)Γ∞) Sso ) Ss )
(23b)
1 + γ1 Γ∞
[ ( )]
c1 ∂ ln ∂Γ1 Γ1
Sso ) Ss(Γ1f0)
()
c1 ) Is∞ + Ss∞c1 Γ1
(23c)
c f co1 Ss∞ ) Ss )
1 Γ∞
(23d)
()
∂ c1 ∂c1 Γ1
Ss∞ ) Ss(c1fco1) γ1 )
(-∆H)1 Sso ) ∞-1 RT S
(23e)
s
where Iso and Sso are the intercept and slope, respectively, of the straight line of ln(c1/Γ1) versus Γ1 for low and intermediate amount of polymer adsorbed (Γ1 f 0), co1 is the bulk polymer concentration, (-∆H)1 is the activation energy characterizing the interaction between polymer/ interface, polymer/polymer, and polymer/solvent, and Is∞ and Ss∞ are the intercept and slope, respectively, of the straight line of (c1/Γ1) versus c1 for intermediate and high polymer concentration (c1 f co1). The function of ln(c1/Γ1) versus the amount of polymer adsorbed, Γ1, was calculated from experimental data represented in Figures 1A and 2A for polymer adsorbed onto SiO2 and polystyrene films substrate, respectively. The values of Iso and Sso were found from experimental data. The equilibrium constant, K1(p), and the value of Sso ) (1 + γ1)/Γ∞ were calculated by using eq 23b. The function of (c1/Γ1) versus the polymer concentration, c1, was calculated from experimental data represented in Figures 1A and 2A for polymer adsorbed onto SiO2 and polystyrene films substrate, respectively. The function of (c1/Γ1) versus the polymer concentration, c1, is the straight line for the intermediate and high polymer concentration for adsorption onto SiO2 polystyrene films substrate, respectively. The value of Ss∞ was found from experimental data. The maximum amount of polymer adsorbed, Γ∞, was calculated by using eq 23e. The parameter, γ1, and the activation energy of adsorption, (-∆H)1, were calculated by using eqs 23b, 23d, and 23e. The values Γ∞, γ1, and (-∆H)1, characterizing the adsorption isotherms onto SiO2 and polystyrene films substrate for water-soluble associative polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol are listed in Table 1. The surface area occupied by one polymer molecule on the SiO2 and polystyrene films substrate, σm, in the adsorbed state corresponding to the plateau is given by
σm(nm2) ) MWpol/(ΓmaxNA)
(24)
Figure 2. Amount of polymer adsorbed, Γk(ck), versus the polymer concentration, ck, for (A) the individual water-soluble associative polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol, (B) the two-component mixture with molecular weights of 12 000 and 62 000 g/mol, and (C) the twocomponent mixture with molecular weights of 62 000 and 120 000 g/mol [1 and 2 are calculated from eq 7a, 3 (circles) is from experimental data] on a polystyrene film substrate at a wall shear rate, γs, of 25 s-1 and L ) 2.2 cm.
where MWpol is the polymer molecular weight, Γmax is the amount of the 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 1A and 2A by using eq 24 are represented in Table 1. Figures 1B,C and 2B,C show the adsorption isotherm for mixtures of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000, 62 000, and 120 000 g/mol, respectively) onto SiO2 and polystyrene film substrates, respectively. The thickness of the adsorbed layer, dad.layer, the weight fraction, Xad.layer, and the amount of adsorbed polymer, Γ, were calculated by using eqs 20
2890 Langmuir, Vol. 15, No. 8, 1999
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Table 1. Characteristics of the Adsorbed Layers of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in Flow Cella MWpol Γmax K(p) σm dad.layer (kg/mol) (mg/m2) (ppm) (nm2) (nm) Wtplat 12 62 120
θo
γ
(-∆H) (kJ/mol)
Adsorption of Associative Polymer in Flow Cell onto SiO2 0.28 0.17 66.4 2.5 0.12 0.85 2.9 7.0 0.81 0.42 128.7 6.4 0.12 0.87 3.7 9.0 0.99 0.65 209.7 7.8 0.12 0.87 4.2 10.2
Adsorption of Associative Polymer in Flow Cell onto Polystyrene 12 0.38 0.14 49.8 2.7 0.10 0.86 2.6 6.3 62 0.85 0.29 121.3 6.6 0.11 0.87 3.3 8.0 120 1.2 0.40 181.1 7.9 0.14 0.87 3.9 9.5 a MW pol ) molecular weight of polymers; Γmax ) 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; Wtplat ) weight fraction of polymers in the adsorbed layer corresponding to the adsorption isotherm plateau; θo (co) 400 ppm) ) surface coverage; γ ) parameter characterizing the interaction in the adsorbed layer; (-∆H) ) activation energy of adsorption; γs ) 25 s-1; L ) 2.2 cm.
Table 2. Characteristics of the Adsorbed Layers for a Mixture of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in Flow Cella (MWpol)av (kg/mol)
(Γmax)Σ (mg/m2)
(σm)Σ (nm2)
(dad.layer)Σ (nm)
θo1
θo2
θoΣ
Adsorption of Associative Polymer in Flow Cell onto SiO2 37 0.61 100.7 5.1 0.51 0.35 0.86 (12/62) 91 0.86 175.7 7.0 0.45 0.42 0.87 (62/120) Adsorption of Associative Polymer in Flow Cell onto Polystyrene 37 0.62 99.1 4.6 0.44 0.42 0.86 (12/62) 91 0.97 155.7 7.2 0.45 0.41 0.86 (62/120) a (MW ) ) average molecular weight of the two-component pol av. mixture of polymers; (Γmax)Σ ) amount of polymer adsorbed per unit area corresponding to the adsorption isotherm plateau for a two-component mixture of polymers with molecular weight of 12 000 and 62 000 g/mol (12/62), and 62 000 and 120 000 g/mol (62/120), respectively, for the bulk concentration of each component of mixture of 200 ppm; (σm)Σ ) area occupied by one average polymer molecule for a two-component mixture of polymers; (dad.layer)Σ ) thickness of the adsorbed layer corresponding to the adsorption isotherm plateau for a two-component mixture of polymers; θok (co ) 400 ppm) ) surface coverage for the kth species in a mixture; θoΣ (co ) 400 ppm) ) surface coverage for the two component mixture; γs ) 25 s-1; and L ) 2.2 cm.
through 21c. These values are represented in Table 2. The surface coverage, θok, for the kth species and the total surface coverage, θoΣ, were calculated by using the experimental data represented in Figures 1B, 1C, 2B, 2C, Tables 1 and 2, and eqs 8a and 8b. These values are represented in Table 2. It is of interest to estimate the effects of interactions between polymer molecules and interface by using the activity coefficient βkeq(UΣeq). The activity coefficient, βkeq(UΣeq), for mixtures of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000, 62 000, and 120 000 g/mol, respectively) were calculated by using data represented in Tables 1 and 2 and eqs 7a and 7c. Figure 3 shows the activity coefficients, βkeq(UΣeq), which characterize interactions between polymer molecules and interface onto SiO2 and polystyrene film substrates. The amounts of individual polymers adsorbed, Γk(c), for a mixture of individual water-soluble polymers were calculated by using data represented in Figure 3 and eq 7a.
Figure 3. Activity coefficients, βkeq(UΣeq) versus the relative total amount of polymer adsorbed, UΣeq, for mixtures of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol and 62 000 and 120 000 g/mol, respectively] onto (A) SiO2 and (B) polystyrene film substrates, respectively, calculated by using data represented in Tables 1 and 2 and eqs 7b and 7c at a wall shear rate, γs, of 25 s-1 and L ) 2.2 cm.
Figures 1B,1C, 2B, and 2C show the amounts of individual polymer adsorbed, Γk(c), for a mixture of individual watersoluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000, 62 000, 62 000, and 120 000 g/mol, respectively) onto SiO2 and polystyrene film substrates. From data in Figure 3, it follows that the activity coefficient for the first strongly adsorbed polymer, β1eq(UΣeq), for mixtures of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol) is greater than that for a mixture of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol). Thus, the activity coefficient for the first strongly adsorbed polymer, β1eq(UΣeq), increases with increasing ratio of molecular weights in a two-component polymer mixture due to increase of exchange between polymer molecules with different molecular weights. For a mixture (A) of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol), the ratio of molecular weights of components in a mixture is to be 62/12 ) 5.17, and for a mixture (B) of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol), the ratio of molecular weights of components in a mixture is to be 120/62 ) 2. The amount adsorbed for the first strongly adsorbed component in mixture A is greater than that for mixture B, as shown in Figures 1B,C and 2B,C onto SiO2 and polystyrene film substrates, respectively.
Adsorption Kinetics of Mixtures on Planar Surfaces
Figure 4. Relative amount of polymer adsorbed, Uk(t), versus time for the adsorption process for the water-soluble associative polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol on a SiO2 at polymer concentrations of 4, 40, and 400 ppm, with a wall shear rate, γs, of 25 s-1, and L ) 2.2 cm.
The above developed approach allows us (a) to estimate the effect of interactions between polymer molecules and an interface at the equilibrium state and (b) to calculate the adsorption isotherms for individual component in a mixture from the experimental data for the total amount of polymer adsorbed. Next, we studied the adsorption kinetics of water-soluble polymers with molecular weights of 12 000, 62 000, and 120 000 g/mol and also a mixture of water-soluble nonionic associative polymers with molecular weights of 12 000 and 62 000 g/mol and 62 000 and 120 000 g/mol in the flow cell from aqueous solution onto the silica and polystyrene film substrates over a wide range of polymer concentrations from 4 to 400 ppm by ellipsometry. Figures 4 and 5 show the time-dependence of the relative amount of polymer adsorbed, U(t), onto SiO2 and polystyrene films, respectively, over a wide range of times for a polymer concentration from 4 to 400 ppm. According to the developed theory of the kinetic-convective-diffusive-controlled adsorp-
Langmuir, Vol. 15, No. 8, 1999 2891
Figure 5. Relative amount of polymer adsorbed, Uk(t), versus time for the adsorption process for the water-soluble associative polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol on a polystyrene film substrate at polymer concentrations of 4, 40, and 400 ppm, a wall shear rate, γs, of 25 s-1, and L ) 2.2 cm.
tion process20-22 for polymers under flow conditions, from eqs 10a and 10b, for short times, the relative amount of polymer adsorbed for the ith individual species is given by
Ui(t) )
Γi(t) ≈ Soiadt Γoi
(25a)
1 ) (trelad)oitotal ) (trelkin)oi + (treldif)oi Soiad (trelkin)oi )
Γoi ad
Ki Coi (1 - θoi/2)
(treldif)oi )
exp
Γoi coiKi(θi)θoi/2)
( ) γi 2 θ 8 oi
(25b)
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Filippova
where (trelad )oitotal, (trelkin)oi, and (treldif)oi are the relaxation times for the ith species due to the adsorption kinetics and diffusion simultaneously, due to the adsorption kinetics, and also due to the diffusion, respectively, for short times and Soiad is the slope of the straight line of the relative amount of polymer adsorbed versus time for short times for the ith individual species. For the kinetic-convective-diffusive-controlled adsorption process under flow conditions from eqs 10a and 10b for long times, the relative amount of polymer adsorbed for the ith species is given by
Fiad(t) ≡ -ln[1 - Ui(t)] ≈ S∞iadt
(26a)
1/S∞iad ) (trelad)∞itotal (trelad)∞itotal ) (trelkin)∞i + (treldif)∞i (treldif)∞i )
Γoi a∞icoiKi(θi)θoi)
(26b) (26c)
dU2(t)
-1
a∞i )
dfi (Ui)1) θoi )1+ + γiθoi dUi 1 - θoi
[
] [( ) ]
γi exp (trelkin)∞i ) θ 2 ∞ ad 2 oi a∞iΓ coiKi (1 - θoi) ad
total
Γoi
kin
rene film substrates. We studied the adsorption kinetics for the mixture of the individual water-soluble nonionic associative polymer (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol and 62 000 and 120 000 g/mol) from aqueous solution onto the silica and the polystyrene film substrates (with a polystyrene molecular weight of 110 000 g/mol) in a flow cell for a polymer concentration range of 4 ppm (mg/kg) to 400 ppm by ellipsometry at room temperature. Figures 6 through 9 show the time dependence of the amount of polymer adsorbed in the mixture, ΓΣ(t), onto SiO2 and polystyrene films, respectively, over a wide range of times for polymer concentrations ranging from 4 to 400 ppm. The amount of polymer adsorbed, ΓΣ(t), was calculated by using eqs 20 through 21c. Next, we applied the above developed approach to calculated the adsorption isotherms for individual component in a mixture from the experimental data for the total amount of polymer adsorbed. For the two-component mixture eqs 10a through 10f and 11a reduce to
dU1(t)
) [U1(t), U2(t)]
1 - f1-1[U1(t), U2(t)]
(28a)
U1(0) ) U2(0) ) 0 (26d) [U1(t), U2(t)] )
dif
where (trel )oi , (trel )oi, and (trel )oi are the relaxation times for the ith species due to the adsorption kinetics and diffusion simultaneously, due to the adsorption kinetics, and also due to the diffusion, respectively, for long times and S∞iad is the slope of the straight line of the relative amount of polymer adsorbed versus time for long times for the ith species. The diffusion coefficient, Di(y)(θ), which depends on the polymer concentration (and correspondingly, the surface coverage, θ), is given by2,20-22
Di(y)(θ) ) Di(y)(θ f 0) exp(-Rθ)
1 - f2-1[U1(t), U2(t)]
o ≡ 1 ≡
where Di(y)(θ f 0) is the polymer diffusion coefficient when
θ f 0 and R and Q are the parameter and activation energy of the diffusion process characterizing the interactions of the polymer molecules in the adsorbed layer. The slope, Soiad, of the straight line of the relative adsorption, Γi(t)/Γo, versus time and the slope S∞iad, of the straight line of a function -ln[1 - Γi(t)/Γo] versus time were calculated from the experimental ellipsometric kinetic data for different polymer concentration represented in Figures 4 and 5. The values of the rate adsorption constant, Ki(p), the polymer diffusion coefficient, Di(y)(θ f 0), and the parameter R were found by using eqs 26a through 27 for the individual polymer with molecular weights of 12 000, 62 000, and 120 000 g/mol. These values are listed in Tables 1 and 3. The data represented in Tables 1 and 3 are useful to calculate the rate of the exchange kinetics for the kineticconvective-diffusive-controlled adsorption processes on a planar surface of SiO2 and polystyrene film substrate under flow conditions. Next, we applied a theory for the kinetic-convectivediffusive-controlled adsorption processes for multicomponent mixtures of polymers under flow conditions to understand and explain the behavior of a mixture of watersoluble associative polymers onto the silica and polysty-
(trelad)2total[U1(t), U2(t)]
(trelad)1total (U1 ≡ U2 ) 0)
(28b)
(28c)
(trelad)2total (U1 ) U2 ) 0) (trelad)1total (U1 ) U2 ) 1) (trelad)2total (U1 ) U2 ) 1)
(trelad)itotal (U1 ) U2 ) 0) ) (trelkin)oi + (treldif)oi (29a) i ) 1, 2
(27)
R ) Q/RT
(trelad)1total[U1(t), U2(t)]
(trelkin)oi )
Γoi ad
Ki coi(1 - θoΣ/2)
(treldif)oi )
exp
(
)
γi θ 2 8 oΣ
(29b)
Γoi coiKi(θΣ ) θoΣ/2)
(trelad)itotal(U1 ) U2 ) 1) ) (trelkin)∞i + (treldif)∞i (30a) i ) 1, 2 (treldif)∞i ) (trelkin)∞i )
[
Γoi λ˜ coiKi(θΣ ) θoΣ)
Γoi ∞
ad
λ˜ Γ coiKi (1 - θoΣ)
] [( ) ] exp
γi θ 2 2 oΣ
(30b)
(30c)
where (trelad)oitotal, (trelkin)oi, and (treldif)oi are the relaxation times for the ith species due to the adsorption kinetics and diffusive mass transfer simultaneously, due to the adsorption kinetics, and also due to the diffusive mass transfer, respectively, for short times for a two-component mixture; (trelad)∞itotal, (trelkin)∞i, and (treldif)∞i are the relaxation times for the ith species due to the adsorption kinetics and diffusive mass transfer simultaneously, due to the
Adsorption Kinetics of Mixtures on Planar Surfaces
Langmuir, Vol. 15, No. 8, 1999 2893
Table 3. Characteristics of the Adsorption Processes of Water-Soluble Nonionic Polymers onto Planar Polystyrene and SiO2 Substrates in a Flow Cella MWpol (kg/mol)
Kadco [10-2(1/s)]
D(y)(θf0) [10-7(cm2/s)]
R
Q (kJ/mol)
Adsorption of Associative Polymer in a Flow Cell onto SiO2 12 36 3.1 2.6 6.3 62 64 1.7 3.2 7.8 120 80 1.2 3.8 9.2 Adsorption of Associative Polymer in a Flow Cell onto Polystyrene 12 42 2.7 2.5 6.1 62 70 1.5 3.1 7.5 120 90 1.0 3.5 8.5 a MW ad pol ) molecular weight of polymers; K co(co ) 400 ppm) ) relative adsorption rate constant, D(y)(θf0) ) diffusion coefficient 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 ) 25 s-1; L ) 2.2 cm.
adsorption kinetics, and also due to the diffusive mass transfer, respectively, for long times for a two-component mixture; and λ˜ is the value which is found by using eq 17c. The values, o and 1, characterizing the relative rate of adsorption for short and long times, respectively, were calculated by using eqs 28a through 30c and data represented in Table 1 through 3; these values are listed in Table 4. As shown in Table 4, the values of o and 1 decrease with increasing molecular weights of polymers in a mixture since the adsorption process of the polymer mixture is controlled by the adsorption kinetics and diffusive mass transfer simultaneously for low molecular weights of polymers in a mixture and by the diffusive mass transfer for high and low molecular weights of polymers in a mixture.20-22 To estimate both the role of rates of exchange kinetics and interactions between polymer species on a planar interface for the kinetic-convective-diffusive-controlled adsorption eq 28a for the two-component mixture, it is convenient to analyze in the phase planes [U1(t), U2(t)]. In this case, as follows from eqs 14 through 15b, the second weakly adsorbed component in a two-component mixture is replaced by the first strongly component in a mixture; therefore, the second component in a mixture reaches a maximum value, U2*, when the relative concentration of the first strongly adsorbed component is to be U1* < 1. The coefficients, So and S1, and the values, U1* and U2*, were calculated by using eqs 15a and 15b, and data are represented in Table 4. The values U1* and U2* are listed in Table 4. The dependence of the relative amount of polymer adsorbed for the second weakly adsorbed polymer in a mixture, U2(t), versus the relative amount of polymer adsorbed for the first strongly adsorbed polymer in a mixture, U1(t), was calculated by using eq 14 and data represented in Tables 1 through 4. Figures 10 and 11 show the dependence of the relative amount of polymer adsorbed for the second weakly adsorbed polymer in a mixture, U2(t), versus the relative amount of polymer adsorbed for the first strongly adsorbed polymer in a mixture, U1(t), for the mixture of the individual watersoluble nonionic associative polymer (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol and 62 000 and 120 000 g/mol) from aqueous solution onto the silica and polystyrene film substrates in a flow cell, respectively, over a wide range of times for polymer concentrations ranging from 4 to 400 ppm. The rate of adsorption for the second weakly adsorbed component in a mixture is greater than the one for the first strongly adsorbed component in a mixture (o >1, 1 > 1); therefore, molecules of the second
Figure 6. Amount of polymer adsorbed, Γk(t), versus time for the adsorption process for a mixture of individual water-soluble nonionic associative polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol) on a SiO2 at a wall shear rate, γs, of 25 s-1, and L ) 2.2 cm, and polymer concentrations of (A) 4, (B) 40, and (C) 400 ppm, respectively; the amounts of individual polymer adsorbed, Γ1(t) and Γ2(t), were calculated from eqs 19a and 19b.
component in a mixture occupy the adsorption center on a planar surface of SiO2 and polystyrene film substrates under flow conditions for short times. For long times molecules of the first strongly adsorbed polymer component in a mixture replace molecules of the second weakly adsorbed component in a mixture. As a result, the amount of polymer adsorbed for the second weakly adsorbed component in a mixture reaches a maximum value which is greater than the equilibrium amount of polymer adsorbed for the second weakly adsorbed component in a mixture (U2* > 1), as shown in Figures 10 and 11 and Table 4. For mixture A of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol), the ratio of molecular weights of components in a
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Figure 7. Amount of polymer adsorbed, Γk(t), versus time for the adsorption process for a mixture of individual water-soluble nonionic associative polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol) on a SiO2 at a wall shear rate, γs, of 25 s-1, and L ) 2.2 cm, and polymer concentrations of (A) 4, (B) 40, and (C) 400 ppm, respectively; the amounts of individual polymer adsorbed, Γ1(t) and Γ2(t), were calculated from eqs 19a and 19b.
Figure 8. Amount of polymer adsorbed, Γk(t), versus time for the adsorption process for a mixture of individual water-soluble nonionic associative polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol) on a polystyrene film substrate at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of (A) 4, (B) 40, and (C) 400 ppm, respectively; the amounts of individual polymer adsorbed, Γ1(t) and Γ2(t), were calculated from eqs 19a and 19b.
mixture is to be 62/12 ) 5.17, and for mixture B of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol), the ratio of molecular weights of components in a mixture is to be 120/62 ) 2. The exchange effect between polymer molecules for the weakly and strongly adsorbed polymer in a mixture is proportional to the ratio of molecular weights of components in a mixture; therefore, the amount of adsorbed for the first strongly adsorbed component in a mixture onto SiO2 and polystyrene film substrates, respectively, for mixture A is greater than the one for mixture B, as shown in Figures 10 and 11 and in Table 4. It is of interest to compare the nature of exchange for the equilibrium and nonequilibrium cases. From eq 7b
and data represented in Figure 3, it follows that in the equilibrium state the dependence of U2eq(U1eq) versus U1eq is a monotone increasing function taking into account only interactions between polymer molecules and an interface. The nonequilibrium kinetic-convective-diffusive-controlled adsorption process is controlled both by the diffusive mass transfer and the un-steady-state processes onto the adsorbed layer (rearrangement, replacement, conformation, and so on) depending on interactions between polymer molecules and an interface. Therefore, the dependence of U2[U1(t)] versus U1(t) is a non-monotone function which depends on the rate of adsorption of each individual component in a mixture and interactions between polymer molecules and an interface. The interactions between polymer molecules and the interface on a polystyrene film
Adsorption Kinetics of Mixtures on Planar Surfaces
Langmuir, Vol. 15, No. 8, 1999 2895 Table 4. Relative Rate of Adsorption for the Two-Component Mixtures of Water-Soluble Nonionic Polymers onto Planar SiO2 and Polystyrene Substrates in Flow Cella MWpol co (kg/mol) (ppm) 12/62 62/120
4 40 400 4 40 400
SiO2
polystyrene film
o
1
U 1*
U 2*
θoΣ
o
1
θoΣ
U 1*
U 2*
2.0 2.4 2.6 1.8 1.9 2.1
2.0 2.2 1.8 1.8 1.7 1.6
0.75 0.73 0.71 0.82 0.8 0.77
1.19 1.23 1.33 1.12 1.16 1.17
0.48 0.68 0.86 0.52 0.71 0.87
2.1 2.4 2.7 1.9 2.0 2.3
2.1 2.0 1.8 1.9 1.8 1.7
0.46 0.67 0.86 0.51 0.69 0.86
0.78 0.77 0.76 0.78 0.77 0.76
1.21 1.29 1.39 1.17 1.21 1.25
a MW pol ) molecular weight of the two-component mixture of polymers with molecular weights of 12 000 and 62 000 g/mol (12/ 62) and 62 000 and 120 000 g/mol (62/120), respectively; o and 1 ) relative rate of adsorption for the two-component mixture for the low (Ui f 0) and high (Ui f 1) relative amount of polymer adsorbed, respectively; U1* ) the relative amount of polymer adsorbed for the first strongly adsorbed component of a two-component mixture when the second weakly adsorbed component reaches a maximum value of U2*; θoΣ ) total surface coverage.
Figure 9. Amount of polymer adsorbed, Γk(t), versus time for the adsorption process for a mixture of individual water-soluble nonionic associative polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol) on a polystyrene film substrate at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of (A) 4, (B) 40, and (C) 400 ppm, respectively; the amounts of individual polymer adsorbed, Γ1(t) and Γ2(t), were calculated from eqs 19a and 19b.
substrate is greater than the one on SiO2 [γ(polystyrene film) > γ(SiO2)], as shown in Table 1. The exchange effect because of the un-steady-state processes onto the adsorbed layer increases with increasing (a) polymer concentrations in a mixture, (b) the interactions between polymer molecules and the interface, and (c) the ratio of molecular weights of components in a mixture. Therefore, the maximum value of the second weakly adsorbed component in a mixture, U2*, for mixture A of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol) is greater than the one for mixture B of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of
Figure 10. Relative amount of polymer adsorbed for the second weakly adsorbed component of a mixture, U2[U1(t)], versus the relative amount of polymer adsorbed for the first strongly adsorbed component in a mixture, U1(t), on SiO2 at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of 4, 40, and 400 ppm, calculated from eq 14, for a mixture of individual water-soluble nonionic associative polymers with a weight fraction of 0.5 for each individual component with molecular weights of (A) 12 000 and 62 000 g/mol and (B) 62 000 and 120 000 g/mol.
62 000 and 120 000 g/mol), as shown in Figures 10 and 11 and Table 4. The exchange effect for the kinetic-convective-diffusivecontrolled adsorption process may be estimated by using the activity coefficients βk[UΣ(t)]. Figures 12 and 13 show the activity coefficients βk[UΣ(t)], which were calculated by using eqs 18, 19a, and 19b and data represented in Figures 10 and 11. The behavior of the activity coefficients for the equilibrium, βkeq(UΣeq), and nonequilibrium state,
2896 Langmuir, Vol. 15, No. 8, 1999
Figure 11. Relative amount of polymer adsorbed for the second weakly adsorbed component in a mixture, U2[U1(t)], versus the relative amount of polymer adsorbed for the first strongly adsorbed component in a mixture, U1(t), on a polystyrene film substrate at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of 4, 40, and 400 ppm, calculated from eq 14, for a mixture of individual water-soluble nonionic associative polymers with weight fraction of 0.5 for each individual component with molecular weights of (A) 12 000 and 62 000 g/mol and (B) 62 000 and 120 000 g/mol.
βk[UΣ(t)], is different due to an essential role of the kinetic and mass transfer factors, as shown in Figures 3, 12, and 13. The activity coefficient for the first strongly adsorbed component of mixtures, β1[UΣ(t)], dramatically increases and the activity coefficient for the second weakly adsorbed component of mixtures, β2[UΣ(t)], dramatically decreases at saturation [UΣ(t) f 1]. Thus, the exchange processes between polymer molecules in a mixture dramatically increase at saturation. The rate of exchange processes increases with increasing the ratio of molecular weights of components in a mixture. Therefore, the rate of exchange processes for mixture A of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol) is greater than one for mixture B of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol), as shown in Figures 12 and 13. The above developed approach allows us to calculate the amount of polymer adsorbed for individual components of mixture from the experimental data for the total amount of polymer adsorbed for a mixture by using the values of the activity coefficient βk[UΣ(t)]. Figures 6 through 9 show the time dependence of the amount of polymer adsorbed for the individual components, Γ1(t) and Γ2(t), for a mixture of individual water-soluble nonionic associative polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 12 000 and 62 000 g/mol and
Filippova
Figure 12. Activity coefficients, βk[UΣ(t)], versus the relative total amount of polymer adsorbed, UΣ(t), on SiO2 at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of 4, 40, and 400 ppm, calculated from eqs 19a and 19b and data represented in Tables 1 through 4 for a mixture of individual water-soluble nonionic associative polymers with a weight fraction of 0.5 for each individual component with molecular weights of (A) 12 000 and 62 000 g/mol and (B) 62 000 and 120 000 g/mol.
62 000 and 120 000 g/mol, respectively) on a SiO2 and polystyrene film substrates over a wide range of polymer concentrations of 4, 40, and 400 ppm, respectively, which were calculated by using eqs 19a and 19b and data represented in Figures 12 and 13. From data represented in Figures 6 through 9, it follows that the second weakly adsorbed component in a mixture is replaced by the first strongly adsorbed component in a mixture; therefore, the amount of polymer adsorbed for the second weakly adsorbed component in a mixture reaches the maximum value of Γ2*. The times, t2*, when the second weakly adsorbed component in a mixture reaches the maximum value of Γ2*, are 86, 11, and 2.1 min for mixture B of individual water-soluble polymers (with a weight fraction of 0.5 for each individual component with molecular weights of 62 000 and 120 000 g/mol) for polymer concentrations of 4, 40, and 400 ppm, respectively, as shown in Figure 7. The value of time, t2*, decreases with increasing (a) polymer concentrations in a mixture, (b) the interactions between polymer molecules and the interface, and (c) the ratio of molecular weights of components in a mixture, as shown in Figures 6 through 9. From data represented in Figures 6 through 9, it follows that the classical adsorption kinetics occurs since the time dependence of the total amount of polymer adsorbed, ΓΣ(t), versus time is a monotone function since the rate of adsorption for the second weakly adsorbed component in a mixture is greater than that for the first strongly adsorbed component in a mixture (o > 1, 1 > 1) over a wide range of times.
Adsorption Kinetics of Mixtures on Planar Surfaces
Langmuir, Vol. 15, No. 8, 1999 2897
6 through 9 for the nonequilibrium state, it follows that the exchange behavior between polymer molecules in a mixture for the equilibrium and nonequilibrium states is quite different due to an essential role of the kinetic and mass transfer factors. Conclusions
Figure 13. Activity coefficients, βk[UΣ(t)], versus the relative total amount of polymer adsorbed, UΣ(t), on a polystyrene film substrate at a wall shear rate, γs, of 25 s-1, L ) 2.2 cm, and polymer concentrations of 4, 40, and 400 ppm, calculated from eqs 19a and 19b and data represented in Tables 1 through 4 for a mixture of individual water-soluble nonionic associative polymers with a weight fraction of 0.5 for each individual component with molecular weights of (A) 12 000 and 62 000 g/mol and (B) 62 000 and120 000 g/mol.
From data represented in Figures 1B, 1C, 2B, and 2C for the equilibrium state and data represented in Figure
We have developed new approaches that allow us (a) to calculate the amount of polymer adsorbed for individual species in a mixture from the ellipsometric experimental data for the total amount of polymer adsorbed for a mixture of water-soluble nonionic polymers in a flow cell on a planar silica (SiO2) substrate and a polystyrene film by using the activity coefficients for equilibrium state and (b) to calculate the amount of polymer adsorbed for individual species in a mixture from the ellipsometric experimental kinetic data for the total amount of polymer adsorbed for a mixture of water-soluble nonionic polymers in a flow cell on a planar silica substrate and a polystyrene film by using the activity coefficients for nonequilibrium state. The exchange effect because of the un-steady-state processes onto the adsorbed layer increases with increasing (a) polymer concentrations in a mixture, (b) the interactions between polymer molecules and an interface, and (c) the ratio of molecular weights of components in a mixture. The exchange effect for the kinetic-convectivediffusive-controlled adsorption process may be estimated by using the activity coefficients βk[UΣ(t)]. The behavior of the activity coefficients for the equilibrium, βkeq(UΣeq), and nonequilibrium state, βk[UΣ(t)], are quite different due to an essential role of the kinetic and mass transfer factors. It is shown that the classical adsorption kinetics occurs since the time dependence of the total amount of polymer adsorbed versus time is a monotone function over a wide range of polymer concentrations. LA980950M
