768
Langmuir 1996, 12, 768-773
Mathematical Models of Polymer Adsorption at a Porous Adsorbent Surface V. N. Kislenko,† Ad. A. Berlin,*,† M. Kawaguchi,‡ and T. Kato‡ Department of General Chemistry, Lviv Polytechnic State University, 12 Bandera Street, Lviv 290646, Ukraine, and Department of Chemistry for Materials, Faculty of Engineering, Mie University, 1515 Kamihama, Tsu, Mie 514, Japan Received June 16, 1995. In Final Form: October 2, 1995X Mathematical models describing kinetics of adsorption and competitive adsorption of polymers into porous media have been developed. The models predict the constant of adsorption equilibrium, the rate constants of adsorption, of desorption, and of substitution of small polymer chains by large ones, and the maximum amount of polymer that can be adsorbed on a unit area of the surface. Using experimental adsorption data of monodisperse polystyrenes and porous silica, we show that there is good agreement between the kinetic models and the experimental data.
Introduction Much attention has been paid to recent advancements in adsorption of polymer chains at solid surfaces from their solutions. New and advanced developments in experimental techniques and theoretical methods have been extensively applied to understanding polymer adsorption behavior.1-4 As a result, the plateau or equilibrium polymer adsorption phenomena on various surfaces is relatively well understood in comparison with polymer adsorption kinetics. In the past years research groups have reported kinetic studies of polymer adsorption using several different techniques.5-15 However, it is important to recognize the role of surface roughness in adsorption. Although many naturally occurring surfaces are rough over many length scales, few systematic experimental and theoretical studies on such a problem have been made. Very recently, Kawaguchi et al. have investigated the surface geometry effect on homodisperse polystyrene (PS) and binary mixtures of homodisperse PS in terms of kinetics and adsorption isotherms under theta and good solvent conditions.5,14,16 From kinetic studies, when a PS
chain easily penetrated into the pores of the silica surfaces, the time to attain an equilibrium state was less than 10 h, whereas for the larger PS that is forced to penetrate into the pores with much deformation, an adsorption equilibrium time of 35 h was attained. For adsorption of the binary mixture at higher initial concentration, the small PS was preferentially adsorbed over the large PS at the early adsorption stage. With an increase in adsorption time, the large PS adsorbed more than the small PS, and finally an equilibrium adsorption was attained within 35 h. In this paper, we present a model describing kinetics of adsorption and competitive adsorption of polymers at a porous adsorbent surface and compare the model to experimental data for the competitive adsorption of homodisperse PS and their binary mixtures on wellcharacterized porous silica surfaces under theta solvent conditions, such as in cyclohexane at 35 °C as a function of initially added PS concentration. The binary mixture consists of the small size PS, whose molecular size is much smaller than the pore size in the silica, and the large size PS, whose molecular size is about half of the pore size. Experimental Section
* To whom correspondence should be addressed. † Lviv Polytechnic State University. ‡ Mie University. X Abstract published in Advance ACS Abstracts, December 15, 1995. (1) Cohen Stuart, M. A.; Cosgrove, T.; Vincent, B. Adv. Colloid Interface Sci. 1986, 24, 143. (2) Fleer, G. J.; Soheutjens, J. M. H. M.; Cohen Stuart, M. A. Colloids Surf. 1988, 31, 1. (3) Kawaguchi, M.; Takahashi, A. Adv. Colloid Interface Sci. 1992, 37, 219. (4) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993 and therein. (5) Kawaguchi, M.; Anada, S.; Nishikawa, K.; Kurata, N. Macromolecules 1992, 25, 1588. (6) Fu, T. Z.; Stimming, U.; Durning, C. J. Macromolecules 1993, 26, 3271. (7) Jonson, H. E.; Douglas, J. F.; Granick, S. Phys. Rev. Lett. 1993, 70, 3267. (8) Tripp, C. T.; Hair, M. L. Langmuir 1993, 9, 3523. (9) Xu, H.; Schlenoff, J. B. Langmuir 1994, 10, 241. (10) Dijt, J. C.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1994, 27, 3207, 3219, and 3229. (11) Couzis, A.; Gulari, E. Macromolecules 1994, 27, 3580. (12) Schneider, H. M.; Granick, S.; Smith, S. Macromolecules 1994, 27, 4714 and 4721. (13) Huguenard, C.; Pefferkorn, E. Macromolecules 1994, 27, 5271. (14) Kawaguchi, M.; Sakata, Y.; Anada, S.; Kato, T.; Takahashi, A. Langmuir 1994, 10, 538. (15) Tiberg, F.; Jonsson, B.; Lindman, B. Langmuir 1994, 10, 3714. (16) Anada, S.; Kawaguchi, M. Macromolecules 1992, 25, 6824.
0743-7463/96/2412-0768$12.00/0
Materials. Two polystyrenes with narrow molecular weight distributions, having Mw ) 96.4 × 103 (PS-10) and 355 × 103 (PS-40) were purchased from Tosoh Co. The polydispersities of PS-10 and PS-40 were determined to be 1.01 and 1.02, respectively. Cyclohexane and dioxane were spectrograde quality and were used without further purification. Tetrahydrofuran was reagent grade quality and was used without further purification. The adsorbent used was a porous microbead (100-200 mesh) silica gel (MB-800, Fuji-David Chemical Co., Kasugai, Japan). The surface area (S) and the average pore diameter (d) were determined from N2 adsorption and a mercury porosimeter, respectively. The latter method characterizes the pore size distribution. From the pore size distribution, we characterized the breadth and skewness of the distributions as D90 ) 75 nm and D10 ) 112 nm; i.e. 90% of the pore diameters are larger than the value of D90 and 10% are larger than D10. The values of S and d were determined to be 45 m2/g and 81.3 nm, respectively. The silica particles were purified by washing with hot carbon tetrachloride using a Soxhlet apparatus for 3 days. The purified silicas were dried in a desiccator under vacuum using an aspirator and further dried in a vacuum oven at 130-150 °C for several days. The silica particles were kept in the vacuum oven to prevent contamination at room temperature before use. Adsorption of PS. Individual PS and 1:1 (w/w) mixtures of PS-10 and PS-40 were dissolved in cyclohexane to desired concentrations. A 0.25 g portion of the MB-800 silica was
© 1996 American Chemical Society
Models of Polymer Adsorption
Langmuir, Vol. 12, No. 3, 1996 769
transferred to a 50-mL flask and then mixed with 10 mL of the solvent. The sample was placed in an air incubator for 24 h at 35 °C to allow the solvent to fully penetrate into the pores to exchange with air. It was then mixed with 10 mL of PS solution. The mixture in the glass flask was mechanically shaken at a constant speed, usually 100 rpm in a Yamato BT-23 water incubator attached to a shaker for a fixed time interval to determine the amount of PS adsorbed at the silica particles. The temperature of water in the incubator was controlled to 35 ( 0.1 °C. The amount of PS adsorbed at the silica surfaces was determined from the difference in the concentrations between the dosage (c0) and the supernatant (c) and from the added silica amount. The value of c was determined as follows: After evaporation of the solvent, the residue was dried in a vacuum oven at room temperature for 24 h and then dissolved in a fixed amount of dioxane, and finally c was measured by using an Ohstuka Denshi System 77 UV spectrometer. The intensities of the dioxane solutions were measured at λ ) 266 nm, where the extinction coefficient was 76.5 L mol-1 cm-1. There was one reason why we took such a complex procedure to determine c, there is the possibility that a PS cyclohexane solution would be turbid at ambient temperature since its theta point is 35 °C. Determination of the composition of the supernatant was carried out using gel permeation chromatography as follows: After evaporation of the solvent and drying the residue as described above, the residue was dissolved in tetrahydrofuran, and finally the tetrahydrofuran solution was analyzed by gel permeation chromatography to determine the adsorbed amounts of the respective PS samples.
Theoretical Part Mathematical models of adsorption kinetics of polymers elaborated earlier describe adsorption under dynamic conditions,17-22 in particular polymer adsorption in stirred suspensions on a surface of unporous particles.19-22 If suspension contains porous particles, polymer adsorption predominantly proceeds on the internal surface. Therefore adsorption is diffusion controlled, even under stirring of suspension. Polymer desorption from a solid surface is kinetically blocked due to the high activation energy required to peel all trains off the surface at once.23 However irreversibility of the polymer adsorption24 is apparently deceptive because segment-by-segment substitution of the adsorbed polymer with other polymer from solution can occur, with lesser activation energy connected with such a sequential process.23 The change of adsorbed amount of homodisperse polymer in suspension of porous particles can be described in the general case by eq 1.
dca/dt ) va - vd
(1)
where ca is mass of adsorbed polymer in unit volume of suspension, t is time, and va and vd are the rates of polymer adsorption and desorption, respectively. The rate of adsorption must be proportional to the concentration of unadsorbed polymer in solution, c, and (17) Peterson, C.; Kwei, T. K. J. Phys. Chem. 1961, 65, 1330. (18) Torocheshnikov, N.; Keltsev, N.; Shumyatskiy, Yu. In Kinetics and Dynamics of Physical Adsorption; Dubinin, I., Rodushkevich, L., Eds.; Nauka: Moscow, 1973; p 110. (19) Gregory, J. J. Colloid Interface Sci. 1988, 31, 231. (20) Kislenko, V. N.; Berlin, Ad. A.; Moldovanov, M. A. J. Colloid Interface Sci. 1993, 156, 508. (21) Kislenko, V. N.; Berlin, Ad A., Moldovanov, M. A. Colloid J. (Russia) 1991, 53, 480. (22) Berlin, Ad. A.; Minko, S. S.; Kislenko, V. N.; Luzinov, L. A.; Moldovanov, M. A. Colloid J. (Russia) 1994, 56, 264. (23) Frantz, P.;p Granick, S. Phys. Rev. Lett. 1991, 66, 899. (24) Fleer, G.; Lyklema, J. In Adsorption from Solution at the Solid/ Liquid Interface; Parfit, G. D., Rochester, C. H., Eds.; Academic Press: London, 1983; p 153.
to the fraction of bare sites of adsorption on the surface, R.
va ) K1Rc
(2)
where K1 is the rate constant of adsorption. The rate of polymer desorption must be proportional to mass of adsorbed polymer in unit volume of suspension, ca.
vd ) K2ca
(3)
where K2 is the rate constant of desorption. It is corroborated by linearity of half-logarithmic anamorphosis of kinetic curves for desorption of polystyrene adsorbed on a surface of oxidized silicon.23 During adsorption of polydisperse polymers on a nonporous surface, long chains adsorb preferentially.25 At a high ratio of adsorption area to solution volume S/V, both long and small chains can adsorb. However at low S/V, long chains adsorb predominantly. As a result, surface fractionation takes place during adsorption of a polydisperse sample.25 In contrast, for adsorption of polydisperse polymer samples or of mixtures of homodisperse samples on a porous adsorbent, small chains are preferentially adsorbed over long chains at the early stages of adsorption.14,16 This is due to the fact that it is more difficult for the long chains to penetrate into the pores of the sorbent, especially under good solvent conditions. However with an increase in adsorption time, long chains were adsorbed more than the small ones. In these cases kinetic curves for small polymers pass through the maximum at high initial polymer concentration in solution (see Figure 6). Taking account of the above, kinetic equations should be completed with the substitution rate, vs, of the small chains by the long ones for adsorption of polydisperse polymer samples or of mixtures of homodisperse samples into porous adsorbent, especially at high initial polymer concentration in solution and (or) at low S/V ratio. Then, for adsorption of a binary mixture of homodisperse polymers, a system of two kinetic equations (4 and 5) should be used instead of eq 1.
dcal/dt ) val - vdl - vs
(4)
dcah/dt ) vah - vdh + vs
(5)
where subscripts l and h concern to low and high molecular mass sample of polymer in a binary mixture of homodisperse polymers. Under adsorption of polydisperse samples or of mixtures of several homodisperse samples into porous adsorbent, a set of kinetic equations analogous to eqs 4 and 5 should be used. It is natural to assume that the substitution rate should be proportional to mass of adsorbed low molecular mass polymer in unit volume of suspension, cal, and to the concentration of unadsorbed high molecular mass polymer in solution, ch
vs ) Kscalch
(6)
where Ks is the rate constant of substitution. Results and Discussion 1. Adsorption of Homodisperse Polymers. The fraction of bare sites of adsorption can be described as (25) Fleer, G. J.; Sheutjens, J. M. H. M. In Coagulation and Flocculation; Bobias, B., Ed.; Marcel Dekker: New York, 1993; p 209.
770
Langmuir, Vol. 12, No. 3, 1996
R ) 1 - A/Am
Kislenko et al.
(7)
where A and Am are the adsorbed amount and the maximum amount of polymer that can be adsorbed on a unit area of the surface of a porous adsorbent.
ca ) A(S/V)
(8)
c ) c0 - ca ) c0 - A(S/V)
(9)
where c0 is the initial concentration of polymer in solution. Taking account of expressions 7-9, one can derive a kinetic equation (10) from eqs 1-3.
dA/dt ) K1(V/S)(1 - A/Am)(c0 - AS/V) - K2A
(10)
In the equilibrium state, in the plateau sections of kinetic curves (see Figures 1 and 2) dA/dt ) 0. Then expression 11 can be written from eq 10
(c0 - ApS/V)/Ap ) (K2/K1)(S/V) + (1/Am)(c0 - ApS/V) (11) where Ap is adsorbed amount of polymer on unit area of the surface in the plateau section of a kinetic curve. As Figure 3 demonstrates, the experimental data on polystyrene adsorption at a porous silica surface converted according to eq 11 are situated on a straight line for adsorption of PS-10 and PS-40 as well as for their binary mixtures (the correlation coefficient is 0.990 that prevails over the critical value 0.765 for the 0.01 significance level26). It allows us to calculate values of K2/K1 ) 0.09 ( 0.03, of the constant of adsorption equilibrium K ) K1/ K2 ) 12 ( 4 and of Am ) (2.7 ( 0.3) × 10-3 g/m2. The ratio of the rate constants K2/K1 and Am are almost independent of molar mass of PS. Perhaps it is due to a compensation effect: long chains adsorb better but penetrate into pores of adsorbent more slowly, and on the contrary small chains adsorb less strongly, however they easily penetrate into the pores. Integration of eq 10 at the initial condition A(t)0) ) 0 leads to the expression
ln(F/Fo) ) K1(E/Am)t
(12)
where
F ) |(2A + E1 - E)/(2A + E1 + E)| Fo ) |(E1 - E)/(E1 + E)| E ) (E12 - 4E2)1/2 E1 ) -(c0V/S + Am + AmK2/K1) E2 ) c0AmV/S As can be seen in Figure 4, the experimental data for the initial stages of kinetic curves on PS adsorption into a porous silica for various initial polymer concentrations lie on straight lines when plotted according to eq 12: correlation coefficients 0.969, 0.983, and 0.960 prevail over the critical values 0.449, 0.449, and 0.661, respectively,25 for the 0.01 significance level. It is straightforward to find values of the rate constant of PS adsorption from the tangent of the slope angle of the straight lines as well as (26) Mueller, P. H.; Neumann, P.; Storm, R. Tafeln der mathematischen Statistik; VEB Fachbuchverlag: Leipzig, 1979.
Figure 1. Change of the adsorbed amount of PS-10 at a porous silica, A, with time, t, for various concentrations, c0: c0 ) 2.0 (1), 1.0 (2) and 0.5 g/L (3). Experimental data are marked with points; curves are calculated according to eq 13.
Figure 2. Change of the adsorbed amount of PS-40 at a porous silica, A, with time, t, for various concentrations, c0: c0 ) 2.0 (1), 1.0 (2), and 0.5 g/L (3). Experimental data are marked with points; curves are calculated according to eq 13.
values of the rate constant of PS desorption taking the value of K2/K1 into account: K1 ) 0.39 ( 0.04 1/h, K2 ) 33 ( 15 1/h for PS-10 and K1 ) 0.086 ( 0.005 1/h, K2 ) 7.4 ( 2.9 1/h for PS-40. The found values of the rate constants of adsorption and desorption of PS into a porous silica allow calculation of kinetic curves A vs time by eq 13 obtained from eq 12.
A ) 0.5[E - E1 + (E + E1)Fo × exp(K1Et/Am)]/[1 - Fo exp(K1Et/Am)] (13) Curves in Figures 1 and 2 testify about satisfactory agreement of calculated kinetic curves with experimental data and about adequacy of the suggested mathematical model (13) to the experiment on homodisperse polystyrene adsorption into a porous silica. 2. Competitive Adsorption of Binary Mixtures of Homodisperse Polymers. However, a mathematical model (13) cannot describe kinetic data on PS adsorption from binary mixtures of homodisperse polymers. Binary mixture kinetics can be approximated by the mathematical model obtained from eqs 4 and 5 taking account eqs 2, 3,
Models of Polymer Adsorption
Langmuir, Vol. 12, No. 3, 1996 771
Figure 3. Relationship between the equilibrium adsorbed amount of PS-10, PS-40, and their mixture, Ap, and polymer concentration in solution in the plateau section of kinetic curves, cp, when plotted according to eq 11.
Figure 4. Change of the adsorbed amount of PS-40 (1), of PS-10 (3), and of their mixture (2), A, with time, t, when plotted according to eq 12.
Figure 5. Adsorbed amounts, A, of a mixture of PS-10 and PS-40 at a porous silica as a function of adsorption time, t, at c0 ) 1.0 g/L: 1, total adsorbed amount; 2, adsorbed amount of PS-40; 3, adsorbed amount of PS-10. Experimental data are marked with points; curve 1 is calculated according to eq 22, curve 2 according to eq 31, and curve 3 according to formula 32.
dAh/dt ) F1 + KsF2
(14)
dAl/dt ) F3 - KsF2
(15)
Figure 6. Adsorbed amounts, A, of a mixture of PS-10 and PS-40 at a porous silica as a function of adsorption time, t, at c0 ) 4.0 g/L: 1, total adsorbed amount; 2, adsorbed amount of PS-40; 3, adsorbed amount of PS-10. Experimental data are marked with points; curve 1 is calculated according to eq 22, curve 2 according to eq 31, and curve 3 according to formula 32.
where Ah and Al are adsorbed amounts of high and low molecular mass samples of polymer in a binary mixture on unit area of the surface
F4 ) (1 - A/Am)(V/S)[(K1h + K1l)c0/2 - K1l(S/V)A + (K1l - K1h)(S/V)Ah] - [K2lA + (K2h - K2l)Ah] (21)
and 6-9
F1 ) K1h(V/S) (1 - A/Am)(c0/2 - AhS/V) - K2hAh F2 ) A1(c0/2 - AhS/V) F3 ) K1l(V/S) (1 - A/Am) (c0/2 - AlS/V) - K2lAl
(16)
Integration of eq 19 at the initial condition A(t)0) ) 0 leads to expression (22).
(17) t) (18)
One can derive eq 19 by summation of eqs 14 and 15
dA/dt ) F4
(19)
A ) Ah + Al
(20)
where
∫0A(1/F4) dA
(22)
Numerical integration of the right-hand side of eq 22 was performed by the Romberg method. Equation 22 defines the relationship between total amount A of adsorbed polymer from a binary mixture and time. Curves 1 in Figures 5 and 6 demonstrate a satisfactory agreement between kinetic curves calculated in such a way and experimental data on PS adsorption from a mixture of PS-10 and PS-40 into a porous silica.
772
Langmuir, Vol. 12, No. 3, 1996
Kislenko et al.
Table 1. Initial Rates of Adsorption, va0, Rates of Adsorption and Desorption in Point of Tangent Crossing, vac and vdc, as Well as in Plateau Sections of Kinetic Curves, vap and vdp, in 10-4 g m-2 h-1, for Adsorption of Homodisperse PS and of Their Binary Mixtures into the Porous Silica c0, g/L 0.5 1.0 2.0 0.5 1.0 2.0 0.5 + 0.5 1.0 + 1.0 1.5 + 1.5 2.0 + 2.0 a
va0
vac
3.42 6.84 13.7 0.757 1.52 2.94 4.21 (4.18) 8.32 (8.36) 12.6 16.6 (16.6)
vdc
vap
vdp
1.46 2.59 3.93
PS-10 0.145 0.270 0.468
0.262 0.494 0.755
0.262 0.494 0.755
0.609 1.16 2.06
PS-40 0.010 0.020 0.036
0.058 0.109 0.166
0.058 0.109 0.166
Binary Mixtures of PS-10 and PS-40 (1:1 (w/w), Total Curves)a 1.99 (2.07) 0.147 (0.155) 0.303 (0.320) 3.36 (3.75) 0.259 (0.290) 0.405 (0.603) 4.23 0.353 0.412 4.66 (5.98) 0.414 (0.504) 0.429 (0.921)
0.302 (0.320) 0.405 (0.603) 0.414 0.421 (0.921)
Sums of rates for homodisperse PS-10 and PS-40 are presented in parentheses.
Calculation of a value of Ks is as follows. Equation 14 may be recast as
dAh/dt ) dAa/dt + dAs/dt
(23)
Ah ) Aa + As
(24)
dAa/dt ) F1
(25)
dAs/dt ) KsF2
(26)
where
Contribution of the augend, Aa, to Ah (see eq 24) may be estimated by integration of eq 25 at the initial condition Aa(t)0) ) 0. Herewith Ah is substituted by Aa in the expression for F1 (16). It leads to the equation
Figure 7. Plot of X vs time, t, according to eq 29.
Integration of eq 14 at the initial condition Ah(t)0) ) 0 leads to the relationship
Aa
t)
∫(1/F1) dAa
(27)
0
(28)
Next, eq 29 can be obtained by integration of eq 26 taking formula 24 into account at the initial condition As(t)0) ) 0.
X ) K st
(29)
where As
X)
∫(1/F2) dAs
∫[1/(F1 + KsF2)] dAh
(31)
0
Numerical integration of the right-hand side of eq 27 was realized by the Romberg method. It determines a relationship between Aa and time. Then, one may calculate values of As according to expression 24 as
As ) Ah - Aa
Ah
t)
(30)
0
Numerical integration of expression 30 was realized by the Romberg method. As can be seen in Figure 7, the experimental data lie on a straight line when plotted according to eq 29: correlation coefficient 0.918 prevails over the critical value 0.449 for the 0.01 significance level.26 One can find the value of Ks from the tangent of the slope angle of the straight line: Ks ) (2.8 ( 0.5) × 10-2 L/(g h).
Numerical integration of the right-hand side of eq 31 was performed by the Romberg method. It is straightforward to calculate values of Al for any value of time according to formula 32 following from eq 20.
Al ) A - Ah
(32)
Equations 31 and 32 define the relationships between Ah and Al and time. Satisfactory agreement of kinetic curves 2 and 3 calculated in such a way and experimental data on adsorption of samples of homodisperse polystyrene from their binary mixtures into a porous silica are demonstrated in Figures 5 and 6. This testifies in favor of the correctness of the suggested kinetic model (14-19). Found values of the rate constants enable us to calculate the initial rates of adsorption, va0, the rates of adsorption and desorption in point C of crossing of tangents to initial and final sections of a kinetic curve, vac and vdc, where tc ) Ap/(K1c0V/S) (see Figure 1) as well as in the plateau section of kinetic curves, vap and vdp for adsorption of homodisperse PS and their mixtures into the porous silica b y means of a suggested kinetic model (Table 1). It allows us also to calculate these rates for each sample of PS in the binary mixture as well as the rates of substitution of PS-10 by PS-40, vsc (Table 2). As seen in Table 1, increase of c0 leads to rise in values of all rates for each homodisperse PS. Unlike this, increase of Mw leads to their drop for all values of c0 because
Models of Polymer Adsorption
Langmuir, Vol. 12, No. 3, 1996 773
Table 2. Initial Rates of Adsorption, va0, Rates of Adsorption, vac, of Desorption, vdc, and of Substitution, vsc, in Points of Tangent Crossing to Kinetic Curves for Adsorbed Amounts of PS-40 (see Figures 5 and 6), in 10-4 g m-2 h-1, for Adsorption of Binary Mixtures of Homodisperse PS into the Porous Silica c0, g/L
polymer
va0
vac
vdc
vsc
2.0
PS-40 PS-10 PS-40 PS-10 PS-40 PS-10
1.51 6.82 2.29 10.4 3.02 13.6
0.277 0.491 0.205 0.556 0.140 0.510
0.045 0.436 0.059 0.513 0.074 0.498
0.235
3.0 4.0
0.452 0.588
herewith polymer penetration into adsorbent pores and detachment of adsorbed polymer from the surface are hampered. The adsorption rate diminishes from va0 to vap, and the desorption rate grows from 0 to vdp when adsorption time increases. If vac is markedly larger than vdc, then the rate of adsorption and desorption have balanced in the plateau sections of kinetic curves at the equilibrium state. It is worth noting that only the initial adsorption rate for each binary mixture of PS is equal to sum of initial adsorption rates for homodisperse PS-10 and PS-40 for the same concentrations (Table 1, numbers in parentheses), i.e., adsorption of PS-10 and PS-40 from their mixture runs independent only in early stage of the process while the surface is still free. Nearly the same state remains until the end of the process for low initial polymer
concentration. In contrast, at higher polymer concentrations, the rates of adsorption or desorption from mixture during the process and especially in the equilibrium state are markedly smaller than corresponding sums of the rates for homodisperse PS. Herewith this distinction rises when c0 increases. Evidently, this fact is due to a competition between long and small chains at adsorption into a porous adsorbent that has to become stronger when the fraction of free surface decreases during the process at c0 ) constant or when initial polymer concentration grows at S/V ) constant. As Table 2 illustrates, the initial adsorption rates of PS-40 are significantly smaller than rates of PS-10 for all values of polymer concentration because penetration of long chains into pores of adsorbent is more difficult. Small chains are being desorbed from the adsorbent surface more easily as was described above. Therefore the desorption rates of PS-10 are much greater than those of PS-40 for all values of polymer concentration. It is interesting that values of vac are markedly greater than vdc for PS-40. In contrast, this distinction is very small for PS-10. The substitution rate rises strongly when polymer concentration increases because competition between long and small PS chains intensifies at S/V ) constant. In conclusion, suggested kinetic models describe well adsorption of homodisperse polystyrenes and their binary mixtures at a porous silica surface. LA950478U
