Adsorption Kinetics of Polyelectrolytes on Planar Surfaces under Flow Conditions

Journal of Colloid and Interface Science 211, 336 –354 (1999) Article ID jcis.1998.6018, available online at http://www.idealibrary.com on

Adsorption Kinetics of Polyelectrolytes on Planar Surfaces under Flow Conditions Nadezhda L. Filippova Russian Branch RTD Corporation, Moscow 111538, Russia Received July 20, 1998; accepted November 30, 1998

tions, owing to the difficulty in measuring dynamic processes at the interface. However, kinetic studies yield important information on how adsorbed amounts of polyelectrolyte are affected by phenomena such as macromolecular rearrangement, desorption, exchange with bulk solution, micellization, and flow-induced conformational changes. The interactions between polyelectrolytes and substrate under flow conditions have been intensively studied because of their applicability to various industrial processes. Therefore, it is of importance to understand how the equilibrium state and the nonequilibrium process on a planar surface are affected by different parameters under flow conditions. The behavior of polymer adsorption and adsorption kinetics in the adsorbed layer have been studied extensively; however, there has not been a satisfactory theoretical explanation to quantitatively describe the equilibrium adsorption and adsorption kinetics of polyelectrolytes from a flow solution onto a planar surface. A quantitative interpretation of the interfacial behavior of polyelectrolytes allows us to use these polymers successfully in different areas. The behaviors of polyelectrolyte adsorption and adsorption kinetics in both the double and the adsorbed layers have been studied extensively (1–21); however, there has not been a satisfactory theoretical explanation to quantitatively describe the equilibrium adsorption and adsorption kinetics of polyelectrolytes onto a planar surface in the flow cell. A quantitative interpretation of the interfacial behavior of polyelectrolytes in a flow cell allows us to use successfully these polymers in different areas. To describe the adsorption process in a flow cell, a number of investigators (22–24) have applied the approach of Leveque (25) for steady-state heat/mass transfer in a flow cell. In our opinion this approach cannot be used to describe adsorption in a flow cell since it is a nonstationary process as was shown in paper (26). Despite the previous arguments, a number of investigators (22–24) have applied the Leveque approach (25) to describe the non-steady-state adsorption and desorption processes in a flow cell. As shown by Leveque (25) and other investigators (27, 28), the steady-state flux is given by

The focus of our work has been to develop a theory of adsorption kinetics for polyelectrolytes in a flow cell onto planar surfaces in the framework of the two-dimensional model and to study adsorption processes of polyelectrolytes on a planar surface by ellipsometry. We have studied the adsorption kinetics of watersoluble cationic poly(vinylamine) hydrochloride homopolymer from aqueous solution onto both silicon wafers and polystyrene films by ellipsometry. Equations were derived to calculate (a) the equilibrium adsorption, (b) the thickness of the adsorbed layer, (c) the activation energy of adsorption for water-soluble polyelectrolytes, (d) the rate constant for the water-soluble polyelectrolytes, (e) the effective coefficients of diffusion in the adsorbed layer, and (f ) the time needed to attain the equilibrium state for the adsorption of the water-soluble polyelectrolytes in a flow cell. © 1999 Academic Press

Key Words: kinetics; diffusion; cationic polyelectrolyte; flow cell; ellipsometry.

INTRODUCTION

The properties of polyelectrolyte chains adsorbed on solid surfaces are involved in many technological applications such as flocculation, stabilization of colloidal suspensions, adhesion, and so on. Polyelectrolyte adsorption is the subject of increasing attention owing to the technological need for macromolecules at surfaces. However, the origin of the phenomena observed remains partly obscure, indicating that the present treatment of polyelectrolyte behavior is incomplete and that some fundamentally new views are needed to gain a full understanding of the adsorption behavior of electrolytes. In recent years the study of polyelectrolytes has been stimulated by the use of newly available experimental techniques [i.e., nuclear magnetic resonance spectroscopy, dynamic light scattering, neutron scattering, hydrodynamic methods, fluorescence excited by evanescent waves, radiotracer techniques, total internal reflection fluorescence, dynamic scanning angle reflectometry, ellipsometry, etc.] and by the introduction of new theoretical concepts which have been applied from other fields. Despite this activity, little work has focused on the kinetics of the polyelectrolyte adsorption under flow condi-

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336

337

ADSORPTION KINETICS OF POLYELECTROLYTES

uncharged macromolecules or low molecular weight electrolytes. The specific properties of polyelectrolyte solutions, particularly in the double and adsorbed layers, are directly affected by electrostatic effects arising from the charge in the double and adsorbed layers and on the interface. Mass and Charge Transport in the Bulk and the Double Layer FIG. 1. Flow geometry in a rectangular flow cell.

D

S D

­c~L, 0! D 2g s 5 b oc o ­y L

1/3

,

b o < 0.54,

[1]

where c( x, y, t) is the polyelectrolyte concentration, x and y are the coordinates in a rectangular cell shown in Fig. 1, c o is the polyelectrolyte concentration in the bulk, gs is the wall shear rate, D is the diffusion coefficient, and L is the distance in the direction ( x) from the entrance of the flow channel to the detection point (i.e., the point where adsorption is measured). A number of investigators (22–24) used the Leveque approach to describe the non-steady-state adsorption process and derived the following equation dG~L, 0, t! ­c~L, 0! 5D . dt ­y

[2]

Equation [2] indicates that (a) the rate of adsorption is constant and is independent of time and depends only on the normal flux [2D­c(L, 0)/­ y] while the axial flux [2D­c(L, 0)/­ x] is ignored, and (b) the normal flux is independent of time. Therefore, the motivation of the present research is (a) to develop a theory for both the diffusion-convective-controlled adsorption and the kinetic-convective-diffusive-controlled adsorption taking into account the nonequilibrium conditions in the adsorbed layer and (b) to derive equations to calculate the parameters characterizing the equilibrium states and the adsorption kinetics for polyelectrolytes which obey both the kinetic-convective-controlled adsorption and the convective-diffusive-controlled adsorption processes. Our experimental study of the behavior of water-soluble polyelectrolytes in the adsorbed layer utilizes a flow cell in combination with an ellipsometric technique which allows us to test the behavior of water-soluble polyelectrolytes on planar interfaces. Theory of Kinetic-Diffusive-Convective Adsorption for Polyelectrolytes in a Flow Cell The behavior of polyelectrolyte molecules in both the bulk and the double layer differs considerably from that of either

For the plane Poiseuille flow in a rectangular channel, the axial velocity, V x ( x, y, t), has a parabolic profile across the thickness when a full flow is developed (27, 28), as shown in Fig. 1. The two-dimensional transient convective-diffusion equation governing transport of polyelectrolyte molecules in a rectangular channel of the flow cell is given by ­c i~ x, y, t! ­j ~i x!~ x, y, t! ­j ~i y!~ x, y, t! 1 1 5 0, ­t ­x ­y 1 # i # r.

[3a]

In the general case, transport of each charged species is described by the two-dimensional Nernst-Planck diffusionmigration equation (5, 22) j ~i x! 5 V x~ x, y, t!c i~ x, y, t! 2 D ~i x! 3

F

­c i~ x, y, t! ­ w ~ x, y, t! 1 z ic i~ x, y, t! ­x ­x

< V x~ y!c i~ x, y, t! j~i y! 5 2D~i y!

G [3b]

F

G

­w~x, y, t! ­ci ~x, y, t! 1 zi ci ~x, y, t! ­y ­y

[3c]

F c ~ x, y, t! RT

[3d]

S D

[3e]

w ~ x, y, t! 5

V x~ y! 5 g s y 1 2

gs 5 V x~0! 5 V x~b! 5 0, ~V x! aver 5

E

y b

6Q b 2d

[3f]

SD

[3g]

Vx

b 3 5 V o 5 ~V x! aver 2 2

b

V x~ y!

d y g sb 5 , b 6

[3h]

0

where c i ( x, y, t), z i , D (i x) , D (i y) , j (i x) , and j (i y) are the concentration, charge number (valence, diffusion coefficient in the directions ( x) and ( y), and the flux in the directions ( x) and ( y) of the i-th species, respectively, and where i 5 1 is the

338

NADEZHDA L. FILIPPOVA

polyelectrolyte, i 5 2, 3, . . . , n are the counterions, and co-ions, n is the number of ions in the solution, t is time, x is a distance parallel to the interface (the flow direction), y is a distance normal to the interface, F is the Faraday constant, R is the ideal gas constant, T is absolute temperature, c ( x, y, t) and w ( x, y, t) are the dimensional and dimensionless total electrical potential, respectively, V x ( y) is the axial linear velocity, V o is the maximum linear velocity, (V x ) aver is the average linear velocity, gs is the wall shear rate, Q is the volumetric flow of polymer solution through the rectangular channel, and b and d are the thickness and width, respectively, of the flow cell. The origin ( x 5 0) is the entrance to the flow cell, as shown in Fig. 1. In Eq. [3b] the diffusion in the direction flow is ignored since V x ( y)c i ( x, y, t) @ D (i x) [­c i ( x, y, t)/­ x 1 z i c i ( x, y, t)­ w ( x, y, t)/­ x]. The system of Eqs. [3a] through [3h] is to be added to the initial Eq. [4a], boundary conditions Eqs. [4b] and [4c], and Eq. [4d] to describe the adsorption kinetics, at t 5 0,

c i 5 0 for all x, y . 0

[4a]

c i 5 c oi for all y, t

[4b]

at x 5 0, b at y 5 , 2

ci 5

H

0 for t , t o, t o 5 x/V o c oi for t $ t o

J

[4c]

­G~x, t! 5 Rad i @c1 ~x, b/2, t!, c2 ~x, b/2, t!, . . . , cn ~x, b/2, t!; ­t G1 ~x, t!, G2 ~x, t!, . . . , Gn ~x, t!# 2 Rdes i @G1 ~x, t!, G2 ~x, t!, . . . , Gn ~x, t!#

[4d]

where c oi is the polyelectrolyte concentration of the i-th species in the bulk, c i ( x, b/ 2, t) is the surface polyelectrolyte concentration of the i-th species, G i ( x, t) is the amount of adsorbed polyelectrolyte of the i-th species at x 5 0, R ad i and R des are the rate of adsorption and desorption, respectively, of i the i-th species on a planar surface. Adsorption Kinetics on Planar Surfaces In the framework of the Arrehenius and Eyring approaches (5, 7, 12) and the Lucassen-Reynders convection (4, 6, 11, 13), the adsorption kinetics of polyelectrolyte onto a planar surface {Eq. [4d]} reduces to (29 –33)

S

D

­ u i~ x, t! b G 5 K ad , t @1 2 u i~ x, t!# i c i x, ­t 2 `

F

3 exp 2

gi 2 u ~ x, t! 2 z iw o~t! 2 S

H

2 K des i u i~ x, t!exp 2

u i~ x, t! 5

G i~ x, t! , G`

gi 5

~2DH! i RT

O u ~ x, t!,

[5b]

n

u S~ x, t! 5

i

[5c]

i51

des where K ad are the rate constants for the adsorption and i and K i desorption processes on a planar surface of the i-th species, u i ( x, t) is the surface coverage of the i-th species, u S ( x, t) is the total coverage, G` is the maximum amount of polyelectrolyte adsorbed, T is the absolute temperature, R is the gas constant, w o(t) is the surface electrical potential, and g i and (2DH) i are the interaction parameter and the activation energy of adsorption of the i-th species, respectively, characterizing the interaction between polyelectrolyte/interface, polyelectrolyte/polyelectrolyte, and polyelectrolyte/solvent.

Total Electrical Potential in the Bulk and in the Double Layer The total electrical potential, w ( x, y, t), depends on the total charge in the bulk (and/or in the double layer) and the total migration flux when the diffusion coefficients, D (i y) , of ions are different. The total electrical potential in the bulk (and/or in the double layer) is an unknown electrical potential which may be found from the charge-balance equation in the bulk (and/or in the double layer) (29 –33) ­ w ~ x, y, t! ­y 5

n ~ e /F!­E r~ y!~ x, y, t!/­t 2 ¥ i51 D ~i y!z i­c i~ x, y, t!/­ y , @6a# n ¥ k51 D ~k y!z 2k c k~ x, y, t!

where e is the dielectric constant and EW r ( x, y, t) is the vector electrical field due to the total electrical charge which is given by the Poisson equation (5, 12)

W W r~ x, y, t! 5 ¹E

S DO F e

n

z ic i~ x, y, t!,

[6b]

i51

W is the two-dimensional differential operator nabla where ¹ which is given as

G

W ; Wi ¹

gi gi @1 2 u S~ x, t!# 2 1 2 2

J

[5a]

­ ­ 1 Wj . ­x ­y

[6c]

Now we compare the suggested model in the form of Eqs. [3c] and [6a] and the models considered in the literature. In the papers of Murphy et al. (17) and MacLeod and Radke (20), the

339

ADSORPTION KINETICS OF POLYELECTROLYTES

diffusion-migration flux, j i ( x, t), in a non-flow cell (the onedimensional model) of the i-th species the Nernst-Planck equation was used in the following form

F

j i~ x, t! 5 2D i

G

­c i~ x, t! ­ w r~ x, t! 1 c iz i , ­x ­x

F c r~ x, t! , RT

w r~ x, t! 5

O ~D

n21

! ~ x, t!

eff ik

k51

­c k~ x, y, t! ­x ­ w eff~ x, y, t! 2 D ic iz i ­x

~Deff!ik 5 Di @Eik 2 dik ~x, y, t!#,

1 # i, k # n 2 1

d ii 5 z 2i ~D ~i y! 2 D ~ny!!c i~ x, y, t!/W~ x, y, t! d ik 5 z iz k~D

O ~D

~ y! k

2 D !c i~ x, y, t!/W~ x, y, t!

[8b]

F

[10]

O n

F2 e RT

c oi@exp~2z iw o! 2 1# 5

i51

O @z G # . n

2

k

ok

[11]

k51

Next, we consider the behavior of adsorption isotherms for polyelectrolytes for the two cases: the first case when the Debye length, k21, is greater than the Bjerrum length, lB, i.e.,

k 21 @ 1, lB

[12a]

where k21 is the Debye length [see Eq. (16a)] and lB is the Bjerrum length {see Eq. [16b]}, and the second case when the Debye length, k21, is less than the Bjerrum length, lB, i.e.,

j5

k 21 , 1. lB

[12b]

Now, we consider the case of Eq. [12a]. From Eq. [11] for a symmetrical polyelectrolyte (i.e., n 5 2, z 1 5 1, z 2 5 21) and u o1 @ u o2 , one finds

z 2 D ~ny!z n! z kc k~ x, y, t!

e ~ y! ­E r~ x, y, t! D z F n n ­x

­ w ~ x, y, t! e ­E r~ x, y, t! ­ E r~ x, y, t! 5 2 D ~ny! ­x F ­t ­ x2

1 # k, i # n,

[9]

ad des where K (p) i (5K i /K i ) is the equilibrium constant for the i-th species and f i is a function describing the adsorption isotherm for the i-th species. The surface equilibrium potential, wo, may be found from the following equation (29 –33)

[8d]

~ y! k k

2

G k~c oi! 5 f k~c o1, c o2, . . . , c on!,

G

[8a]

k51

eff

F

ui gi exp z iw o 2 ~1 2 2 u S! 1 2 uS 2

[8c]

~ y! n

1

K (p) i c oi 5

j5

n21

W~ x, y, t! 5

For the diffusive-convective-controlled adsorption process, des when the constant rate of adsorption, K ad i and desorption K i ad des is infinite (K i , K i 3 `) Eqs. [5a] and [5b] reduce to the equations for the adsorption isotherms as (29 –33)

[7]

where w r ( x, t) is the dimensionless electrical potential resulting from the electrical charge, r ( x, t), in the double layer. As shown in papers (29 –33), Eq. [7] may be used only to describe the steady-state adsorption process, the whole (b) non-steadystate adsorption process must be described using Eqs. [3c] and [6a], since the electrical potential, w r ( x, t), due to the charge, r ( x, t), and the total electrical potential, w ( x, t), in the double layer are different. Now we consider how to find the flux j i ( x, y, t). From Eqs. [3c] and [6a], the flux j i ( x, y, t) for each species i (1 # i # n 2 1) is given by

j ~i y!~ x, y, t! 5 2

Adsorption Isotherms for Polyelectrolytes on Planar Surfaces

GY

W~ x, y, t!,

2g 1sinh~ w o / 2! 5 u 1

[8e] g1 5

[8f]

where E ik is the unit matrix, (D eff) ik ( x, y, t) are the effective diffusion (i 5 k) and the effective mutual diffusion (i Þ k) coefficients, and w eff( x, y, t) is the effective electrical potential, which, according to Eqs. [6b] and [8f], depends only on the charge in the double layer.

S

D

l~1! eRT G2o1 D l(1) D 5 (1) , 2F2~G`!2co1 lad

[13a]

1/ 2

, l(1) ad 5

S D Go1 co1

[13b]

where l(1) D is the Debye length for the symmetrical polyelectrolyte. For the nonsymmetrical polyelectrolyte (i.e., n 5 2, z 1 @ 1, z 2 5 21) and u o1 @ u o2, from Eq. [11] we write 4z 1g˜ 21@exp~ w o! 2 1# 5 u 21 g˜1 5

S

D

l˜ (1) eRT G2o1 D l˜ (1) D 5 (1) , 2 2 lad 2z1 F ~G` !2 co1

1/ 2

,

l(1) ad 5

[14a]

S D

Go1 , co1

[14b]

340

NADEZHDA L. FILIPPOVA

where l˜ (1) D is the Debye length for the nonsymmetrical polyelectrolyte. From Eqs. [9], [11], [14a], and [14b] for the symmetrical (z1 5 1) and nonsymmetrical (z1 $ 2) polyelectrolyte (n 5 2 and z2 5 21) and uo1 @ uo2, the adsorption isotherms for polyelectrolytes are described, respectively, as b *1C 21 5

S

D

u 31 exp~ g 1q S!, 1 2 uS z 1 5 1, b *1 5 b 1r 1, 1! b˜ *1C ~11z 5 1

F

2 e RTc o1 r1 5 ~FG `! 2

1! u ~112z 1 exp~ g 1q S!, 1 2 uS

b˜ *1 5 b 1r˜ ~1z1!,

r˜ 1 5

z1 $ 1

2 e RTc o1 . z 1~FG `! 2

G

S

ee oMWpolRT n F 2 ¥ m51 z 2mc om

lB 5

Fe , 4 pee oRT

D

j5

[15a]

[15b]

[15c]

It should be noted that the adsorption isotherms for polyelectrolytes in the form of Eq. [15b] reduce to the adsorption isotherms of the Freundlich kind (1, 5) when z 1 $ 1. Next, we consider the case of Eq. [12b] when the Debye length, k21, is less than the Bjerrum length, lB, i.e., j , 1 (and/or j ! 1). To explain the behaviors of the adsorption isotherm polyelectrolyte we estimate the electrical interactions in the double and adsorption layers in the framework of a semiquantitatively point of view. The electrical interaction in the double layer in the presence of dissolved salt may be estimated by the Debye length. The Debye length is given by (5)

k 21 5

sion along the polyelectrolyte chain. For example, the Debye length, k21, of the water-soluble cationic hydrochloride homopolymer with a molecular weight of 142,000 g/mol from an aqueous solution at a polymer concentration of 500 mg/kg is found to be k21 5 0.21 nm. Therefore, it is reasonable to use the ratio of the Debye and Bjerrum lengths, j, as a criterion to estimate the screening effect in the double layer:

1/ 2

[16a]

[16b]

where k21(nm) is the Debye length, lB(nm) is the Bjerrum length, c om(mg/kg), (MWpol)m(g/mol), and z m are the concentration in the bulk, molecular weight, and the charge (valence) of the i-th species, respectively, n is the number of species in a solution, e is the electronic charge, R is the gas constant, eo is the permittivity of vacuum, e is the dielectric constant of a solution, F is the Faraday constant, and T is the absolute temperature. For an aqueous solution (e 5 78.3 and T 5 2938K), the Bjerrum length is equal to 0.72 nm. The energy of the electrical interaction [e2/(4peeolB)] is equal to the thermal energy (kT) when the distance, L, is equal to the Bjerrum length, lB. Therefore, the Bjerrum length is the appropriate length scale to estimate the electrostatic repul-

k 21 . LB

[17]

In fact, for j , 1 the Debye screening effect is smaller than the electrical interaction of the polyelectrolyte chains; therefore, the behaviors of the polyelectrolyte chains (conformation, conformational changes, and so on) resemble those of neutral chains. For j . 1, the electrostatic interaction is dominant and the behavior of the polyelectrolyte depends on the strong intra- and intermolecular interactions in the double layer. For example, for the water-soluble cationic hydrochloride homopolymer from an aqueous solution with a molecular weight of 142,000 g/mol at a polymer concentration of 4.4 mg/kg is found to be j 5 1. Thus, the Debye screening effect is increasingly important when the polyelectrolyte concentration increases to more than 4.4 mg/kg. From the previous analyses it follows that for the watersoluble cationic hydrochloride homopolymer with a molecular weight of 12,000, 76,000, and 142,000 g/mol from an aqueous solution the Debye screening effect is increasingly important since x , 1 {see Eq. [17]}; therefore, the behaviors of the polyelectrolyte chains for these polyelectrolyte in the vicinity of the adsorbed layer resemble those neutral chains. As result, for polyelectrolyte, when Debye length, k21, is less than the Bjerrum length, lB; i.e., x , 1 (and/or x ! 1), Eqs. [5a] and [9] reduce to

FSD

G J

­ u i~ x, t! gi 2 5 K *adic i~ x, 0, t!@1 2 u S~ x, t!#exp 2 u ~ x, t! ­t 2 S

HSD

2 K *desi u i~ x, t!exp 2

gi gi @1 2 u S~ x, t!# 2 1 2 2

[18]

S D

Gi Gi K* c 5 ` exp g i ` , G 2 GS G (p) i i

K* 5 (p) i

K *adi K *desi

O G, n

,

GS 5

i

i51

[19] where K *adi , K *des i are the effective rate constants for adsorption and desorption processes, respectively, on a planar surface

341

ADSORPTION KINETICS OF POLYELECTROLYTES

under flow conditions for the i-th species, K *(p)i is the effective equilibrium constant for the i-th species of polyelectrolyte, and c i ( x, 0, t) is the surface concentration for the i-th species of polyelectrolyte. The previous equations may be applied to describe the adsorption kinetics and the adsorption equilibrium for polyelectrolyte when Debye length, k21, is less than the Bjerrum length, lB; i.e., x , 1 (and/or x ! 1).

transfer and Gi(t) and Ci(L, 0, t) are the relative adsorption and the relative surface concentration, respectively, for the i-th species. It is reasonable to rewrite Eq. [18] in the following form

F G

~t kin! i dG i~t! 5 C i~L, 0, t! 2 f 21 i ~G 1, G 2, . . . , G n!, dt H ad i ~G i! 1#i#n

Adsorption Kinetics for Polyelectrolytes on Planar Surfaces under Flow Conditions The adsorption process takes place in the thin adsorbed layer near the adsorbing flow channel wall. Therefore, according to the general theory of boundary layers (27, 28, 34), we may use the following approximation for the linear velocity in a flow channel, V x~ y! < g s y,

0 # y # `.

~K dif! i 5 b

F

~D eff! 2iig s L

G

b < 0.65

dG i~t! 5 ~K eff dif! i@1 2 C i~L, 0, t!#, dt Gi ~t! 5

Gi ~L, 0, t! , Goi

Ci ~L, 0, t! 5

~K eff dif! i 5 ~K dif! i

c oi , G oi

1#i#n ci ~L, 0, t! coi

S

H ad i ~G i! 5 ~1 2 u S!exp 2

OdG, m

dm 5

m

m51

H des i ~G i! 5

[23b]

D

n

F

u om [23c] u oS

G

gi gi G iu o1 exp 2 ~1 2 u S! 2 1 , bi 2 2

Od n

b i 5 K *(p)i c oi

[23d]

H des i ~G i! ~G 1, G 2, . . . , G n! ; ad , H i ~G i!

[23e]

m

5 1,

m51

OG, n

GS 5

k

u o1 5

k51

f [21a]

G oi ` K ad i G c oi

~t kin! i 5

gi 2 u , 2 S uS 5

G o1 , G` 21 i

O GG n

u oS 5

ok `

,

k51

[22b]

where (t kin) i is the relaxation time for the i-th species due to the adsorption kinetics onto a planar surface for short time when G i 3 0 ( u i 3 0), f 21 i (G 1 , G 2 , . . . , G n ) is a reciprocal function describing the adsorption isotherms for the i-th species, and b i is the dimensionless equilibrium constant for the i-th species. At equilibrium [dG i (t 5 `)/dt 5 0, C i H iad(G i ) 5 des H i (G i )], from Eqs. [23a] through [23e] one writes that

[22c]

f 21 i ~G 1, G 2, . . . , G n! 5

1/3

,

~t kin! i , H ad i ~G i!

[20]

The simplified kinetic-diffusive-convective model in the form of the system of Eqs. [3a], [5a], [8a], [9], [18], and [20] may be applied to describe the adsorption and desorption processes in a flow cell. In the general case, the solutions of the system of Eqs. [3a], [8a], [9], [18], and [20] may be found using the two-dimensional Laplace transforms with respect to the axial coordinate x and the time t. However, it is useful to apply Nernst’s approach (12) to derive the equation in a simple analytical form. In the framework of this approach, the polyelectrolyte adsorption process due to the diffusive-convective mass transfer in the flow cell for intermediate and long times is described by (29 –33) dG i~L, t! 5 ~K dif! i@c oi 2 c i~L, 0, t!# dt

~t kin! i~G i! 5

[23a]

[21b]

[22a]

where (Kdif)i and (Keff dif)i are the rate constant and the effective rate constant, respectively, for the adsorption kinetics for the i-th species due to the diffusive-convective mass

G iu oi exp~ g iu S!, b i~1 2 u S!

Od G . n

uS 5

m

k51

m

[24]

342

NADEZHDA L. FILIPPOVA

Eliminating the relative surface concentration C i (L, 0, t) from the system of Eqs. [22a] and [23a] one writes

Equation [25a] may be applied to describe the adsorption process under flow conditions. Eq. [25a] may be rewritten as

dG i~t! ~t S! i~G i! 5 1 2 f 21 i ~G 1, G 2, . . . , G n! dt

[25a]

1 2 f 21 dG 1~t! 1 ~G 1! 5 R 1~G 1! 5 , dt t S~G 1!

~tkin!i ~Gi ! ~tdif!i ~Gi !

[25b]

~tS !i ~Gi ! 5 ~tdif!i ~Gi !@1 1 x~Gi !#, x~Gi ! 5

F

~D eff! ii~G i! ~t dif! i~G i! 5 ~t dif! i ~D oeff! ii

G

~tdif!i 5

F

G

t 2 t o 5 U 1~G 1!,

[25c]

L o 2 ~Deff!ii gs

D

1/3

,

~t S~G 1!! 1 < ~t dif! 1 1 ~t kin! 1.

[25d]

dx . R 1~ x!

[26b]

The integral U1(G1) may be found numerically by using a computer. However, the approximate solution of Eq. [26a] may be found in the analytical form by using asymptotic solutions for short and long times. For short times (t 3 0) and long times (t 3 `) the solutions of Eq. [26a] are given, respectively, by t 2 t o < ~t dif 1 t kin!G 1~t!,

F

t < 2 t `dif 1

~t 2 t o 3 0!

G

ln@1 2 G 1~t!# t kin , a` H ad ~G 5 1! 1 1

a` 5

~t 3 `!

df 21 u o1 1 ~G 1 5 1! 511 1 u o1g 1. dG 1 1 2 u oS

[27a] [27b]

[27c]

From the previous equations we write the following solution of Eq. [26a] which describes the adsorption process over a wide range of times

F

t 2 t o < t dif 1 t kin 2

G

t `dif t kin 2 G ~t! ad a ` a `H 1 ~G 1 5 1! 1

F

2 t `dif 1

G S

t kin ln@1 2 G 1~t!# a` H ad ~G 5 1! 1 1

H ad 1 ~G 1 5 1! 5 ~1 2 u oS!exp 2

D

g1 2 u . 2 oS

[28a]

[28b]

The above theory for the kinetic-convective-diffusive-controlled adsorption of polyelectrolyte under flow condition was developed in order to understand and explain the adsorption behavior of polyelectrolyte on planar surfaces. Now we consider the adsorption process described by Eq. [26a] for the following parameters

G S 3 1,

S

G1

[26a]

For long times (t 3 `) from Eq. [23c], it follows that

H ad 1 ~G 1 5 1! 5 ~1 2 u oS!exp 2

E

o

where (t S ) i (G i ) is the total relaxation time for the i-th species due to the diffusive-convective mass transfer and the adsorption kinetics onto a planar surface simultaneously, x (G i ) is a ratio characterizing which of the mechanisms for the i-th species is governed by the adsorption process, (t dif) i (G i ) is the relaxation time for the i-th species due to the diffusiveconvective mass transfer, and (D oeff) ii is the effective coefficient of diffusion for small surface coverage, u i 3 0. The ratio, x (G i ), may be used as the criteria in order to estimate which of the mechanisms for the i-th species is governed by the adsorption process. The adsorption process for the i-th species is governed by (a) the adsorption kinetics onto a planar surface for x (G i ) @ 1, (b) by the diffusive-convective mass transfer for x (G i ) ! 1, and (c) by the diffusive-convective mass transfer and the adsorption kinetics onto a planar surface simultaneously for x (G i ) ' 1. For short times (t 3 0) since u 1 3 0, H ad 1 (G 1 ) ' 1, from Eqs. [23b], [25c], and [25c], it follows that the algebraic additivity of relaxation times takes place; i.e.,

G 1 3 1,

U 1~G 1! 5

,

S DS

Goi Goi 5 1.54 coi ~Kdif!i ~ui 3 0! coi

whose solution is

2/3

~D oeff! ii 5 ~D eff! ii~G i 3 0, u i 3 0!

[26a]

D

g1 2 u ! 1. 2 oS

[26b]

Thus, for long times (t 3 `) the role of the kinetic-controlled step increases for the kinetic-convective-diffusive-controlled adsorption process under flow conditions.

n 5 2,

u o1 @ u o2,

x5

t kin t dif

[29a]

g 1 5 5,

b 1 5 0.18,

u o1 5 0.1

[29b]

g 1 5 5,

b 1 5 810,

u o1 5 0.9.

[29c]

ADSORPTION KINETICS OF POLYELECTROLYTES

343

FIG. 2. (A) Relative polyelectrolyte adsorption, G 1 ( t ), and (B) the relaxation function, F[G 1 ( t )], versus relative time, t, for different isotherms for t kin 5 t dif by using Eqs. [28a] through [29c] (solid curves) and by using numerical integrations of Eqs. [26a] and [29a] through [29c] (dotted curves).

The relaxation times tkin and tdif characterizing the rate of adsorption process are proportional to the mass resistance due to the adsorption kinetics on a planar surface and the diffusive-convective mass transfer, respectively. Therefore, the kinetic-diffusiveconvective-controlled adsorption process obeying arbitrary adsorption isotherms is governed by both the adsorption kinetics on a planar surface and the diffusive-convective mass transfer simultaneously, i.e., when x ' 1(tdif ' tkin). From Eqs. [23b] and [25d], it follows that tkin/tdif ; g1/3 s . Therefore, the role of the diffusiveconvective mass transfer resistance under flow conditions decreases with increasing the wall shear rate gs. Figure 2A shows the time-dependent relative adsorption, G1(t), versus relative time, t 5 t/tdif, for the kinetic-diffusive-convective-controlled adsorption process under flow conditions (x 5 tkin/tdif 5 1) obeying different adsorption isotherms: from uo1 5 0.1 (the quasilinear adsorption isotherm, b1 ! 1) to uo1 5 0.9 (almost attaining the rectangular adsorption isotherm, b1 @ 1). From Eq. [27b], it follows that to describe the adsorption process for long times, it is reasonable to use the relaxation function F[G1(t)] in the following form F@G 1~t!# 5 2ln@1 2 G 1~t!#, G 1~t! 5

G 1~t! , G o1

G o1 5 G 1~t 5 `!.

[30]

For long times Eqs. [27b] and [30] reduce to F@G 1~t!# < S˜ `relt, ~t`rel!dif 5

t`dif , a`

1 ` ` ` ˜S `rel 5 ~t rel! S 5 ~t rel! dif 1 ~t rel! kin

[31a]

~t`rel!kin tkin , x`rel 5 ` a` H1 ~G1 5 1! ~trel!dif

[31b]

1 1 x `rel 1 , ` 5 a` S rel

[31c]

~t`rel!kin 5

F@G 1~ t !# 5 S `relt ,

t5

t t

` dif

,

` where S˜ ` rel, S rel are the slope of the straight line of the relaxation function F[G 1 (t)] versus time, t, and the relaxation function F[G 1 ( t )] versus relative time, t, respectively, (t ` rel) S is the relaxation time characterizing the rate of the kineticconvective-diffusive-controlled adsorption process under flow conditions on a planar surface. Figure 2B shows the timedependent relation function, F[G 1 ( t )], versus relative time, t 5 t/t dif, for the kinetic-convective-diffusive-controlled adsorption process under flow conditions ( x 5 t kin/t dif 5 1) obeying different adsorption isotherms: from u o1 5 0.1 (the quasilinear adsorption isotherm, b 1 ! 1) to u o1 5 0.9 (almost attaining the rectangular adsorption isotherm, b 1 @ 1). The slope, S ` rel, was calculated by using Eqs. [27c], [28b], [29a]–[29c], and [31c]. From Fig. 2B and Eq. [31c], it follows that for intermediate and long times the time-dependent relaxation function, F[G 1 ( t )], versus relative time, t, is a linear function. Figure 3A shows the time-dependent relative adsorption, G 1 ( t ), versus relative time, t 5 t/t dif, for the diffusivecontrolled adsorption process under flow conditions ( x 5 t kin/t dif 5 0.01) obeying different adsorption isotherms: from u o1 5 0.1 (the quasilinear adsorption isotherm) to u o1 5 0.9 (almost attaining the rectangular adsorption isotherm). Figure 3B shows the time-dependent relation function, F[G 1 ( t )], versus relative time, t 5 t/t dif, for the diffusive-controlled adsorption process under flow conditions ( x 5 t kin/t dif 5 0.01) obeying different adsorption isotherms. From Fig. 3B and Eqs. [31c], it follows that for long times the time-dependent relaxation function, F[G 1 ( t )], versus relative time, t, is a linear function. Eqs. [26a] and [29a] through [29c] were integrated numerically by using a computer. From Figs. 2A and 3A it follows that Eq. [28a] may be used to described over a wide range of times the kinetic-diffusive-convective-controlled process obeying arbitrary adsorption isotherms. It is of interest to compare the suggested adsorption model

344

NADEZHDA L. FILIPPOVA

FIG. 3. (A) Relative polyelectrolyte adsorption, G 1 ( t ), and (B) the relaxation function, F[G 1 ( t )], versus relative time, t, for different isotherms for t kin 5 0.01t dif by using Eqs. [28a] through [29c] (solid curves) and by using numerical integrations of Eqs. [26a] and [29a] through [29c] (dotted curves).

and the adsorption model based on the Leveque approach which is used by a number of investigators (22–24) to describe the adsorption process over a wide range of time. According to the Leveque approach (25) and Eq. [1], the non-steady-state adsorption process in a flow cell is described by

S D

dG 1~L, t! D 2g s 5 b oc o1 dt L

1/3

,

b o < 0.54.

[32]

From the previous equation it follows that (a) the rate of the adsorption process does not depend on the convective transfer, (b) the rate of adsorption is constant, and (c) the rate does not depend on the surface concentration, c 1 (L, 0, t). However, from our previous analysis and the experimental data (22–24), it follows that (a) the rate of adsorption process depends on the convective transfer, (b) the rate of adsorption is variable, and (c) it depends on the surface concentration, c(L, 0, t). Figure 4 shows the time dependence of the relative polyelectrolyte

adsorption on planar surfaces for the typical experimental data (22–24) (solid line) and the theoretical model based on the Leveque approach (circle). According to the suggested model and Eqs. [21a] and [21b], the rate of adsorption varies with the gradient concentration [c o1 2 c 1 (L, 0, t)]. As follows from the experimental data shown in Fig. 2A, the rate of polyelectrolyte adsorption decreases when the adsorption process approaches the equilibrium state, since [c o1 2 c 1 (L, 0, t 3 `)] 3 0. It is impossible to describe this fact in the framework of the model of Eq. [32]. The previous analysis leads to the following important conclusions. The adsorption model of Eq. [32] based on the Leveque approach cannot be applied to describe the non-steady-state polyelectrolyte adsorption process in a flow cell since this model (a) is based on Eq. [2] and (b) ignores convective mass transfer. Thus, to describe the non-steady-state polyelectrolyte adsorption process in a flow cell over a wide range of time, the suggested adsorption model of [18], [23a], and [25a] may be used. EXPERIMENTAL METHODS AND INSTRUMENTATION

FIG. 4. Relative polyelectrolyte adsorption, G 1 (t), versus time, t, for typical experimental data (22–24) (solid line) and for the theoretical model based on Eq. [30].

Silicon wafers. The silicon wafers were purchased from IBM Company. Polystyrene films. A solution of polystyrene (PS) (with a molecular weight of 190,000 g/mol) in toluene (0.01% by wt) was spin-coated on 2.5 cm polished silicon wafers; the thickness of SiO2 was 23 nm; the resulting thickness of the polystyrene film was 15 nm. Polystyrene films were dried in a vacuum oven at 70°C. Polymers. The model water-soluble associative polymer used in the kinetic study was a cationic poly(vinylamine) hydrochloride homopolymer with a molecular weight of 142,000 g/mol (M W /M n 5 1.15) and a structure of [OCH2OCHONH•2HCl]n.

345

ADSORPTION KINETICS OF POLYELECTROLYTES

Flow cell. The fused quartz trapezoid flow cuvette (Hellma Cells, Germany) has a volume of about 1.3 ml; its fused quartz windows were placed perpendicular to the incident and reflection beams (angle of incidence of 70°). The flow cell was used to measure the values of the refractive index of the silicon wafers and polystyrene film substrates in DDI (distilled-deionized) water and the ellipsometric values for polyelectrolyte solutions during the adsorption processes at room temperature. We have used the ellipsometer to measure the thickness and amount of polyelectrolyte adsorbed on a planar surfaces comprised of the silicon wafers and polystyrene film substrates. Ellipsometric analysis was carried out by a Shimadzu ellipsometer, Model EP-10 (Shimadzu Manufacturing Co. Ltd.). An incident light of 546.1 nm wavelength was applied to the sample at a incident angle of 70° as shown in Fig. 1. The thickness of SiO2 for the polished silicon wafers (IBM Company) was 123 nm. The bare surface of silicon wafers were placed into a fused quartz trapezoid flow cuvette. The flow cuvette has a volume 1.3 ml, its fused quartz windows were placed perpendicular to the incident and reflection beams (angle of incident 5 70°). The water-soluble polyelectrolytes used in the adsorption study are cationic poly(vinylamine) hydrochloride homopolymer with a molecular weight of 142,000 g/mol (MW/Mn 5 1.15). Fresh polyelectrolyte solutions were prepared for each run. The silicon wafers were cleaned after the adsorption runs were complete when the equilibrium state is reached. We have used the ellipsometer to measure the thickness and amount of polyelectrolyte adsorbed on a planar surface comprised of silicon wafers. The optical system used consists of the following layers: bulk silicon (Si) with a complex refractive index, n *3 5 n 3 2 ik 3 , a layer of silica (SiO2) with a refractive index n 2 and a thickness d 2 , the adsorbed layer with a refractive index n 1 5 n ad.layer and a thickness d 1 5 d ad.layer, and a surrounding solution with a refractive index n o. The thickness, d ad.layer, and the refractive index, n ad.layer, of the polyelectrolyte adsorption layer were found simultaneously by using the following equation (32, 33), 2 2 n 2solsin2q sol# 1/ 2, Re~ d ! 5 ~2 p / l !d ad.layer@n ad.layer

Im~ d ! 5 0, [33]

where Re(d) and Im(d) are the real and imaginary parts of the phase shift, respectively, l is the wavelength of light, d ad.layer is the thickness of the adsorbed polyelectrolyte layer, and n ad.layer and n sol are the refractive index of the adsorbed polyelectrolyte layer and the polyelectrolyte solution, respectively. The values of d ad.layer and n ad.layer from Eqs. [33] are calculated by using ellipsometric experimental data (D and c). The

amount of polyelectrolyte adsorbed on a planar surface is given by (32, 33) G 5 d ad.layerX ad.layerr pol X ad.layer 5 R ad.layer 5

[34a]

6n ad.layer~n ad.layer 2 n sol! 2 ~R pol 2 R sol!~n ad.layer 1 2! 2

2 21 n ad.layer , 2 n ad.layer 1 2

R sol 5

n 2sol 2 1 , n 2sol 1 2

[34b]

[34c]

where G is the amount of polyelectrolyte adsorbed (in mg per m2 of surface), X ad.layer is the weight fraction of polyelectrolyte in the adsorbed layer, and rpol is the polyelectrolyte density (g/cm3). Equations [33] through [34c] are useful for calculating the weight fraction and amount of polyelectrolyte adsorbed in the adsorbed layer. RESULTS AND DISCUSSION

The above theory for the equilibrium and the kineticconvective-diffusion-controlled adsorption process of polyelectrolytes in the flow cell was developed in order to understand and explain (a) the behavior of polyelectrolytes in the double and adsorbed layers on planar surfaces of various substrates [i.e., SiO2 and the polystyrene film], (b) the behavior of water-soluble polyelectrolytes in the double and adsorbed layers on various planar surfaces, and (c) the difference between the behavior of water-soluble polyelectrolytes in the double and adsorbed layers on planar surfaces in a flow cell and a nonflow cell. The adsorption of watersoluble polyelectrolytes from aqueous solution onto SiO2 and the polystyrene film [with a polystyrene molecular weight of 190,000 g/mol] was studied in the flow cell for polyelectrolytes in a concentration range of 2 ppm (mg/kg) to 500 ppm by ellipsometry at room temperature. First, we studied the equilibrium adsorption of the watersoluble polyelectrolytes in a flow cell from aqueous solution onto SiO2 and onto polystyrene films by ellipsometry. Figure 5 shows the thickness of the adsorbed layer which was calculated from the experimental ellipsometric data using Eqs. [33] through [34c]. The weight fraction, X ad.layer, and the amount of adsorbed polyelectrolytes, G, were calculated by using Eqs. [34a] and [34b]. The weight fractions of water-soluble polyelectrolyte in the adsorbed layer on SiO2 and polystyrene films under flow conditions were found to be approximately 0.16; i.e., the adsorbed layer consists of 16 wt% water-soluble polymer and 84 wt% water. Figure 6 shows the adsorption isotherms of the water-soluble polyelectrolyte onto SiO2 and polystyrene films. The values of G max, d ad.layer, and X ad.layer are listed in Table 1. To explain the behaviors of the adsorption isotherm polyelectrolyte we estimate the electrical interactions in the

346

NADEZHDA L. FILIPPOVA

FIG. 5. The thickness of the adsorbed layer, d ad.layer, versus the polyelectrolyte concentration, c o1, for the water-soluble polyelectrolyte with a molecular weight of 12, 76, and 142 kg/mol on (A) a SiO2 substrate and (B) on a polystyrene film (PS) at a wall shear rate, gs, of 27 s21 and L 5 2.2 cm.

double and adsorption layers in the framework of a semiquantitatively point of view. The electrical interaction in the double layer in the presence of dissolved salt may be estimated by the Debye length, k21, and the Bjerrum length, lB, since the Bjerrum length is the appropriate length scale to estimate the electrostatic repulsion along the chain of polyelectrolyte. Therefore, it is reasonable to use the ratio of the Debye and Bjerrum lengths, j, as a criterion to estimate the screening effect in the double layer {see Eqs. [12a] and [12b]}. In fact, for j , 1 the Debye screening effect is smaller than the electrical interaction of the polyelectrolyte chains; therefore, the behaviors of the polyelectrolyte chains (conformation, conformational changes, and so on) resemble those of quasineutral chains. For j . 1, the electrostatic interaction is dominant and the behavior of the polyelectrolyte depends on the strong intra- and intermolecular interactions in the double layer. For the water-soluble cationic hydrochloride homopolymer from an aqueous solution with

a molecular weight of 12,000 g/mol at a polymer concentration of 53.4 mg/kg, with a molecular weight of 76,000 g/mol at a polymer concentration of 8.3 mg/kg, and with a molecular weight 142,000 g/mol at a polymer concentration of 4.4 mg/kg, j 5 1. Thus, the Debye screening effect is increasingly important when the polyelectrolyte concentration increases to more than about (co1)B [the Debye length equals to the Bjerrum length when co1 5 (co1)B]. From the previous analyses it follows that for the water-soluble cationic hydrochloride homopolymer with a molecular weight of 12,000, 76,000, and 142,000 g/mol from an aqueous solution the Debye screening effect is increasingly important since j , 1 {see Eq. [12b]}; therefore, the behaviors of the polyelectrolyte chains for these polyelectrolyte in the vicinity of the adsorbed layer resemble those quasineutral chains. As result, Eq. [9] reduces to Eq. [19]. To estimate the parameters characterizing the adsorption isotherm for individual polyelectrolyte species (n 5 1), when

FIG. 6. The amount of polyelectrolyte adsorbed, G 1 (c o1), versus the polyelectrolyte concentration, c o1, for the water-soluble polyelectrolyte with a molecular weight of 12, 76, and 142 kg/mol on (A) a SiO2 substrate and (B) on a polystyrene film (PS) at a wall shear rate, gs, of 27 s21 and L 5 2.2 cm.

347

ADSORPTION KINETICS OF POLYELECTROLYTES

TABLE 1 Characteristics of the Adsorbed Layers of Water-Soluble Polyelectrolyte onto Planar Polystyrene and SiO2 Substrates in a Flow Cell G` (mg/m2)

MWpol (kg/mol)

K *(p)1 (1/ppm)

sm (nm2)

d *ad.layer (nm)

X pol

xArch

g1

(2DH) 1 (kJ/mol)

3.6 1.9 1.8

2.2 3.2 4.3

5.4 7.8 10.5

2.4 1.8 1.7

2.5 3.9 4.6

6.1 9.5 11.2

Adsorption of associative polyelectrolyte in a flow cell onto SiO2 12 76 142

0.34 0.84 1.12

36 9.1 1.9

59 150 210

2.4 6.0 9.0

0.14 0.14 0.14

Adsorption of associative polyelectrolyte in a flow cell onto polystyrene 12 76 142

0.48 1.1 1.42

18 3.7 1.4

42 115 166

3.0 6.8 8.8

0.16 0.16 0.16

Note. MWpol 5 molecular weight of polymers; G` 5 amount of polymer adsorbed per unit area; K *(p)1 5 equilibrium constant characterizing the adsorption isotherm; sm 5 area occupied by one polymer molecule; d *ad.layer 5 thickness of the adsorbed layer corresponding the adsorption isotherm plateau; X pol 5 weight fraction of polymers in the adsorbed layer corresponding to the adsorption isotherm plateau; xArch 5 parameter characterizing the architecture of the adsorbed layer; g1 5 parameter characterizing the interaction in the adsorbed layer; (2DH) 1 5 activation energy of adsorption; gs 5 27 s21; and L 5 2.2 cm.

G 1 @ G 2 (G S ' G 1 ), it is reasonable to rewrite Eq. [19] in a more convincing forms

SD

ln

S D

c1 G1 5 2ln~K *(p)1 ! 1 g 1 ` 2 ln~G ` 2 G 1! G1 G ln

SD

I os 5 2ln~K (p)1 G `!, * Ss 5

c1 5 I os 1 S os G 1, G1 S os 5

G1 3 0

[35a]

[35b]

1 1 g1 , G`

F S DG

­ c1 ln ­G 1 G1

,

S os 5 S s~G 1 3 0!,

[35c]

where K *(p)1 is the effective equilibrium constant, G` is the maximum amount of polymer adsorbed, g1 is the parameter characterizing the interaction between polyelectrolyte/ interface, polyelectrolyte/polyelectrolyte, and polyelectrolyte/ solvent, I os and S os are the intercept and slope, respectively, of the straight line of ln(c 1 /G 1 ) versus G1 for low and intermediate amount of polyelectrolyte adsorbed (G 1 3 0). Figure 7 shows the function of ln(c 1 /G 1 ) versus the amount of polyelectrolyte adsorbed, G1, which was calculated from experimental data represented in Fig. 6 for polyelectrolyte adsorbed onto SiO2 and polystyrene films, respectively. From data represented in Fig. 7 it follows that the function of ln(c 1 /G 1 ) versus the amount of polyelectrolyte adsorbed, G1, is the straight line for the low and intermediate amount of polyelec

FIG. 7. The function of ln(c 1 /G 1 ) versus amount of polyelectrolyte adsorbed, G1, for the water-soluble polyelectrolyte with a molecular weight of 12, 76, and 142 kg/mol on (A) a SiO2 substrate and (B) on a polystyrene film (PS) at a wall shear rate, gs, of 27 s21 and L 5 2.2 cm.

348

NADEZHDA L. FILIPPOVA

FIG. 8. The function of (c 1 /G 1 ) versus polyelectrolyte concentration, c 1 , for the water-soluble polyelectrolyte with a molecular weight of 12, 76, and 142 kg/mol on (A) a SiO2 substrate and (B) on a polystyrene film (PS) at a wall shear rate, gs, of 27 s21 and L 5 2.2 cm.

trolyte adsorbed onto SiO2 and polystyrene films, respectively. The values of I os and S os were found from experimental data represented in Fig. 7. The equilibrium constant, K *(p)1 , and the value of S os 5 (1 1 g 1 )/G ` were calculated by using Eq. [35c]. To estimate the maximum amount of polymer adsorbed, G`, for individual polyelectrolyte species (n 5 1), when G 1 @ G 2 (G S ' G 1 ), it is reasonable to rewrite Eq. [19] in more convincing forms

S D

[36a]

c 3 c o1

[36b]

G1 c1 c1 1 5 ` 1 (p) ` exp g 1 ` G1 G G K *1 G

SD

c1 5 I `s 1 S `s c 1, G1

S`s 5

1 , G`

Ss 5

g1 5

SD

­ c1 , ­c1 G1

S`s 5 Ss~c1 3 co1!

~2DH! 1 S os 5 ` 2 1, RT Ss

[36c]

[36d]

where K *(p)1 is the effective equilibrium constant, c o1 is the bulk polyelectrolyte concentration, (2DH) 1 is the activation energy characterizing the interaction between polyelectrolyte/ interface, polyelectrolyte/polyelectrolyte, and polyelectrolyte/ ` solvent, I ` s and S s are the intercept and slope, respectively, of the straight line of (c 1 /G 1 ) versus c 1 for intermediate and high polyelectrolyte concentrations (c 1 3 c o1). Figure 8 shows the function of (c 1 /G 1 ) versus the polyelectrolyte concentration, c 1 , which was calculated from experimental data represented in Fig. 6 for polyelectrolyte adsorbed onto SiO2 and polystyrene films. From data represented in Fig. 8 it follows that the function of (c 1 /G 1 ) versus the polyelectrolyte concentration, c 1 , is the straight line for the intermediate and high polyelectrolyte concentration for adsorption onto SiO2 and polystyrene

films. The value of S ` s was found from experimental data represented in Fig. 8. The maximum amount of polyelectrolyte adsorbed, G`, was calculated by using Eq. [35c]. The parameter, g1, and the activation energy of adsorption, (2DH) 1 , were calculated by using Eqs. [35c] and [36d]. The values, G`, g, and (2DH) 1 , characterizing the adsorption isotherms onto SiO2 and polystyrene films are listed in Table 1. The adsorption of polyelectrolyte molecules, in many respects, is different from the adsorption of nonionic polymer molecules from solution. The adsorbed polyelectrolyte molecule may also remain in a conformation closer to that of the molecule in solution. Therefore, the thickness of the adsorbed layer depends on the number of attachments per molecule and the distribution of these attachments. The difference in the adsorbance with solution concentration results from the differences in the conformation of the adsorbed molecule. The higher the solution concentration, the fewer the number of attachments per molecule, until a limiting adsorbance is attained. As result for the adsorption onto SiO2, (a) the maximum amount of polyelectrolyte adsorbed, G`, increases from 0.34 to 1.12 mg/m2 (about 3.29 times) and (b) the activation energy of adsorption, (2DH)1, increases from 5.4 to 10.5 kJ/mol (about 1.94 times) onto SiO2 when a polymer molecular weight increases from 12,000 to 142,000 g/mol (11.8 times), as shown in Table 1. The parameter characterizing the adsorption onto SiO2 and polystyrene films for water-soluble associative polymer with different molecular weight may be estimated by using data represented in Table 1 and Eqs. [34a], [34b], [35c], and [36d]. The surface area occupied on the SiO2 and polystyrene film substrates by one polyelectrolyte molecule, sm, in the adsorbed state corresponding to the plateau is given by

s m~nm2! 5

MWpol , G maxN A

[37]

where MWpol is the polyelectrolyte molecular weight, Gmax is

349

ADSORPTION KINETICS OF POLYELECTROLYTES

FIG. 9. The relative amount of polyelectrolyte adsorbed, G 1 (t)/G o1, versus time, (t 2 t o), for the water-soluble polyelectrolyte with a molecular weight of 12, 76, and 142 kg/mol on SiO2 and polystyrene films substrates at a wall shear rate, gs, of 27 s21, L 5 2.2 cm, and at different polymer concentrations: (A) 400 ppm and (B) 4 and 40 ppm, respectively.

the amount of the adsorbed polyelectrolyte on a planar surface corresponding to the isotherm plateau, and N A is Avogadro’s number. The value of sm for the polyelectrolyte calculated from the experimental data in Fig. 6 using Eq. [37] is represented in Table 1. The length, L sur, characterizing the surface area on a planar surface occupied by one polyelectrolyte molecule, is estimated as

L sur <

S D 4sm p

1/ 2

.

[38]

The architecture of the adsorbed layer may be characterized using the ratio, xArch, which is equal to L sur x Arch 5 * , d ad.layer

[39]

where d ad.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, xArch, since the architecture of the adsorbed layer is like that of a polyelectrolyte brush when x Arch # 0.5 and is like that of a polyelectrolyte pancake when x Arch $ 2. From the experimental data presented in Figs. 5 and 6 and by using Eqs. [37] through [39] the ratio, xArch, was calculated and is shown in Table 1. We also studied the adsorption kinetics for the watersoluble polyelectrolyte with a molecular weight of 12,000, 76,000, and 142,000 g/mol on a planar surfaces of SiO2 and polystyrene films under flow conditions by ellipsometry.

The amount of polyelectrolyte adsorbed was calculated from the ellipsometric kinetic data by using Eqs. [34a] through [34c]. Figure 9 shows the time dependence of the relative amount of polyelectrolyte adsorbed, G1(t), over a wide range of times for a high polyelectrolyte concentration of 400 ppm and low and intermediate polyelectrolyte concentrations of 4 and 40 ppm, respectively, onto SiO2 and polystyrene films, respectively. According to the developed theory of the kinetic-convective-diffusive-controlled adsorption process for polyelectrolytes, it is reasonable to represent the timedependent adsorption, G1(t), in the form of the relative adsorption, G1(t), versus t for short times as shown in Fig. 9. There are the two ways to find the parameters characterizing the kinetic-convective-diffusive-controlled adsorption processes on a planar surface under flow conditions. The first way is based on the experimental kinetic ellipsometric data for intermediate times when the relative amount of polyelectrolyte adsorbed reaches G1 ' 41. The polyelectrolyte diffusion coefficient, (D eff) 11 ( u 1 ), which depends on the surface coverage, q1, is given by (12, 30 –33)

~Deff!11 ~u1 ! 5 ~Deff!11 ~u1 3 0!exp~2a1 q1 !,

a1 5

Q1 , RT

[40]

where (Deff)11(u1 3 0) is the polyelectrolyte diffusion coefficient when u1 3 0, and where a1 and Q1 are the parameter and activation energy of the diffusion process characterizing the interaction of the polyelectrolyte molecules in the adsorbed layers which may be found from the experimental kinetic data. From Eqs. [25a]–[25d] and [40] for intermediate times it

350

NADEZHDA L. FILIPPOVA

follows that the relative amount of polyelectrolyte adsorbed, G 1 (t), is given by G1~t! < Sorel~t 2 to!, t $ to ~intermediate times!

S D

S D

1 t kin g1 2 a1 exp u o1 1 t difexp u , o 5 32 6 o1 S rel 1 2 u o1/4

[41a]

1 G1 5 4

u o1 , K ad 1 c o1

t dif 5 1.54

S DF G o1 c o1

L ~D eff! 11~ u 1 3 0! g s

G

1/3

G o1 u o1 5 ` , G

,

I os 5

[41c]

S DF

G o1 S 5 1.54 c o1 o s

[42a]

S D G S D

g1 2 t kin exp u 1 2 u o1/4 32 o1 L ~D oeff! 11

a1 exp u , 6 o1

K ad 1 [10 (1/ppm z s)] 24

(D eff) 11 ( u 1 3 0) [1027(cm2/s)]

a1

Q1 (kJ/mol)

12 76 142

2.9 1.9 1.1

1.2 1.0 0.9

2.1 3.0 4.2

5.1 7.3 10.2

Adsorption of associative polyelectrolyte in a flow cell onto polystyrene

where Sorel is the slope of the straight line of the relative adsorption, G1(t) versus time for the intermediate times at G1 ' 41. Taking into account Eq. [41c], Eq. [41b] is rewritten as S os 1 o 5 I 1 s S orel g 1/3 s

MWpol (kg/mol)

Adsorption of associative polyelectrolyte in a flow cell onto SiO2

[41b] t kin 5

TABLE 2 Characteristics of the Adsorption Processes of Water-Soluble Polyelectrolytes onto Planar Polystyrene and SiO2 Substrates in Flow and Nonflow Cells

12 76 142

2.7 1.7 1.0

1.4 1.2 1.1

2.3 3.6 4.4

5.6 8.7 10.7

Note. K ad 1 5 adsorption rate constant for polyelectrolyte; (D eff) 11 ( u 1 3 0) 5 diffusion coefficient for polyelectrolyte in the bulk; a1 5 parameter characterizing the surface coverage dependence of the activation energy of diffusion in the double and adsorbed layers; and Q 1 5 activation energy of diffusion for polyelectrolyte in the double and adsorbed layers.

by the adsorption kinetics when the wall shear rate is infinite ( g s 3 `) since in this case Eq. [43] is valid

[42b] I os 5

1/3

1 , S orel

g s 3 `.

[43]

[42c]

where I os and S os are the intercept and slope, respectively, of the o straight line of (1/S rel ) versus (gs)21/3. From Eqs. [42a] through [42c] it follows that the adsorption process is governed

FIG. 10. The reciprocal slope, (1/Sorel), versus (wall shear rate)21/3 for the water-soluble polyelectrolyte with a molecular weight of 142 kg/mol on a SiO2 substrate and on a polystyrene film at a polymer concentration of 40 ppm.

The slope, S orel, of the straight line of the relative adsorption G 1 (t) versus time was found from the experimental ellipsometric kinetic data for intermediate times represented in Figs. 9 and 10 for the adsorption of the water-soluble polyelectrolyte onto SiO2 and polystyrene film, respectively, for different values of wall shear rate, gs. Figure 10 shows the reciprocal slope, (1/S orel) as a function of (gs)21/3. From the experimental data represented in Fig. 7, it follows that a function of the reciprocal slope, (1/S orel), versus (gs)21/3 is linear for the adsorption of polyelectrolyte onto a SiO2 or polystyrene film. The intercept, I os , and the slope, S os , were found from the experimental kinetic ellipsometric data represented in the form of 1/S orel versus uo1 for polyelectrolytes with different molecular weights. The rate constant for the adsorption process on a planar surface, K ad 1 , and the effective diffusion coefficient, (D eff) 11 ( u 1 3 0), were then found from the experimental kinetic data for low polyelectrolyte concentrations (correspondingly, for low surface coverage, u 1 3 0) by using Eqs. [42b] and [42c]. These values are listed in Table 2. Next we consider the second way to estimate the parameters characterizing the kinetic-diffusive-convective-controlled adsorption process. The second way is based on use the experimental kinetic ellipsometric data for long times when the relative amount of polyelectrolyte adsorbed reaches G 1 3 1.

351

ADSORPTION KINETICS OF POLYELECTROLYTES

FIG. 11. The relaxation function, 2ln[1 2 G 1 (t)], versus time, (t 2 t o), for the water-soluble polyelectrolyte on a SiO2 substrate and on a polystyrene film at a wall shear rate, gs, of 27 s21, L 5 2.2 cm, and at different polymer concentrations: (A) 400 ppm and (B) 4 and 40 ppm, respectively.

According to Eqs. [25a], [28a], [28b], [30], [31a], and [31b] for long times the relaxation function, F[G 1 (t)] is given by F@G 1~t!# 5 2ln@1 2 G 1~t!# < S `relt,

G 1 3 1 ~long times! [44a]

S D S DF

S

D

tkin g1 2 2 1 5 exp u 1 tdifexp a1uo1 2 o1 3 S`rel 1 2 uo1 tkin 5

uo1 Go1 L , tdif 5 1.54 ad co1 ~Deff!11 ~u1 3 0!gs K1 co1

G

[44b]

1/3

.

[44c]

Figure 11 shows the relaxation function, F(t), versus time for adsorption of the water-soluble polyelectrolyte on SiO2 and polystyrene films under flow conditions. From the experimental ellipsometric kinetic data represented in Fig. 11, it follows that a function of the relaxation function F(t) versus time for long times is linear. Therefore, the slope, S ` rel, of the straight line of the relaxation function F[G 1 (t)] versus time was found from the experimental data is presented in Fig. 11. Taking into account Eqs. [44c], Eq. [44b] is rewritten as S `s 1 ` 5 I 1 s S `rel g 1/3 s I `s 5

S D G S D

t kin g1 2 exp u 1 2 u o1 2 o1

S DF

S`s 5 1.54

[45a]

Go1 L co1 ~Deff!11~u1 3 0!

1/3

exp

2 au , 3 1 o1

[45b]

[45c]

` where I ` s and S s are the intercept and slope, respectively, of the ` straight line of (1/S rel ) versus (gs)21/3. From Eqs. [45a] through [45c] it follows that the adsorption process is governed

by the adsorption kinetics when the wall shear rate is infinite ( g s 3 `) since in this case Eq. [46] is valid 1 , S `rel

I `s 5

g s 3 `.

[46]

` The intercept, I ` s , and the slope, S s , were found from the experimental kinetic ellipsometric data represented in the form of (1/S ` rel)( u o1) versus time for different polyelectrolyte concentrations, c o1, (correspondingly, for different the surface coverage, u o1) onto SiO2 and polystyrene film, respectively. It is reasonable to rewrite Eq. [45c] in the following convenient form

ln~S `s ! 5 I a 1 S au o1,

H S DF

I a 5 ln 1.54

G o1 c o1

Sa 5

L ~D eff! 11~ u 1 3 0! 2 a, 3 1

GJ 1/3

[47a]

[47b]

where I a and S a are the intercept and slope, respectively, of the straight line of ln(S ` s ) versus surface coverage, uo1. The function of ln[S ` s ( u o1)] was calculated from the experimental kinetic data [1/S ` rel( u o1)] for different polyelectrolyte concentrations (or different surface coverage, uo1,) by using Eq. [45c]. The parameter a1 and the activation energy of the diffusion, Q 1 , were then found as the slope, S a , of the straight line of ln[S ` s ( u o1)] versus (uo1) by using Eq. [47b]. These values are listed in Table 1. The polyelectrolyte diffusion coefficient, (D eff) 11 ( u ), in the double and adsorbed layers depends strongly on the surface coverage (and correspondingly, the polyelectrolyte concentra-

352

NADEZHDA L. FILIPPOVA

tion in the bulk). According to Eq. [50], for the polyelectrolyte adsorbed with a molecular weight of 142,000 g/mol onto SiO2 in a flow cell with a 1 5 4.2, one finds that (D eff) 11 (c 5 2 ppm)/(D eff) 11 (c 5 500 ppm) 5 (0.9 3 10 27 cm2/s)/ (0.13 3 10 28 cm2/s) 5 66.6 and for the polyelectrolyte adsorbed onto a polystyrene film with a 1 5 4.4, one finds that (D eff) 11 (c) (c 5 2 ppm)/(D eff) 11 (c) (c 5 500 ppm) 5 (1.1 3 10 27 cm2/s)/(0.16 3 10 28 cm2/s) 5 81.4. The polyelectrolyte diffusion coefficient, (D eff) 11 (c), in the double and adsorbed layers is decreased due to the interaction of the polyelectrolyte chains. These quantities may be estimated from the activation energy of diffusion: Q 1 5 10.2 kJ/mol for polyelectrolyte adsorbed onto SiO2 in a flow cell and Q 1 5 10.7 kJ/mol for polyelectrolyte adsorbed onto the polystyrene film. It should be noted that the activation energies of adsorption, (2DH) 1 5 10.5 kJ/mol for the polyelectrolyte adsorbed onto SiO2 and (2DH) 1 5 11.2 kJ/mol for polyelectrolyte adsorbed onto the polystyrene film are close to the activation energy of diffusion, Q 1 . These facts may be explained as follows: (a) the density of the double and adsorbed layers is decreased in the direction ( y) normal to the interface ( y 5 0), and (b) the architecture of the double and adsorbed layers is nonhomogeneous because the diffusive-convective mass transfer in the double and adsorbed layers is extensive. According to Eqs. [29a], [41b], [41c], [44b], and [44c], the ratios xorel, x` rel may be used as the criteria in order to estimate which mechanism is governed by the adsorption process on a planar surface under flow conditions for intermediate and long times, respectively, as

x orel 5 ~t orel! kin 5

~t orel! kin , ~t orel! dif

x `rel 5

S D

g1 2 t kin exp u , 1 2 u o1/4 32 o1

~t `rel! kin ~t `rel! dif

[48a]

S D a u 6 o1

~torel! dif 5 t dif exp

[48b] ~t `rel! kin 5

S D

g1 2 t kin exp u , 1 2 u o1 2 o1

S

~t`rel! dif 5 t dif exp

D

2 au . 3 1 o1 [48c]

The adsorption processes on a planar surface under flow conditions are governed (I) by the adsorption kinetics (AK) for x orel, x `rel $ 4, (II) by the adsorption kinetics and the diffusiveconvective mass transfer simultaneously (AK) 1 (DCMT) for x orel, x `rel . 1, and (III) by the diffusive-convective mass 1 transfer simultaneously (DCMT) for x orel, x ` rel # 4 ; i.e.,

x rel 5 ~ x orel, x `rel! 5

H

$ 4, ~AK! . 1, ~AK! 1 ~CD! # 1/4, ~CD!

J

.

[49]

TABLE 3 Mechanisms Controlling the Adsorption Process for Polyelectrolytes in a Flow Cell for Intermediate and Long Times Intermediate times C o1 (ppm)

xorel

Mechanisms

Long times

x`rel

Mechanisms

Adsorption of polyelectrolyte onto SiO2 in a flow cell 4 40 400

0.26 0.2 0.13

(AK) 1 (DCMT) (DCMT) (DCMT)

0.2 0.5 0.9

(DCMT) (AK) 1 (DCMT) (AK) 1 (DCMT)

Adsorption of polyelectrolyte onto polystyrene film in a flow cell 4 40 400

0.31 0.22 0.17

(AK) 1 (DCMT) (DCMT) (DCMT)

0.24 0.63 1.1

(DCMT) (AK) 1 (DCMT) (AK) 1 (DCMT)

Note. c o1 5 polyelectrolyte concentration in the bulk; (AK) 5 adsorption kinetics controlling the adsorption process on a planar surface; (AK) 1 (DCMT) 5 adsorption kinetics and the diffusive-convective mass transfer simultaneously controlling the adsorption process on a planar surface; (DCMT) 5 diffusive-convective mass transfer controlling the adsorption process on a planar surface under flow conditions.

By using the values G o1, g 1 , K ad 1 , (D eff) 11 ( u 1 3 0), a 1 , u o1, and Q 1 of from Tables 1 and 2, we calculated the relaxation criteria, xorel, x` rel for intermediate and long times for the watersoluble polyelectrolytes onto a planar surface SiO2 and polystyrene film by using Eqs. [48a] through [48c]; these values are shown in Table 3. Next we consider the mechanisms controlling the adsorption process over a wide range of times using the experimental data represented in Figs. 9 and 11 and Table 3. From the experimental data it follows that the adsorption of the water-soluble polyelectrolyte from aqueous solution onto SiO2 and polystyrene films under flow conditions is governed (a) by the adsorption kinetics for short times at low polyelectrolyte concentrations (4 ppm and/or less), (b) by the polyelectrolyte adsorption kinetics and the convective-diffusive mass transfer simultaneously for long times at intermediate and high polyelectrolyte concentrations (about of 40 ppm (and more), and (c) by the convective-diffusive mass transfer for intermediate times at intermediate and high polyelectrolyte concentrations (about of 40 ppm (and more). As shown in our papers (29 –33), the adsorption process for water-soluble polyelectrolyte onto polystyrene films under nonflow conditions is governed by (a) by the adsorption kinetics for short times or low polyelectrolyte concentrations (less than 40 ppm), (b) simultaneously by the kinetics of polyelectrolyte adsorption and the diffusion of polyelectrolyte molecules in the double and adsorbed layers for the intermediate times or midrange polyelectrolyte concentrations (greater than 4 ppm and less than 10 ppm), and (c) by the diffusion of polyelectrolyte molecules in the double and adsorbed layers for

ADSORPTION KINETICS OF POLYELECTROLYTES

TABLE 4 Characteristics of the Architecture of the Adsorbed Layer of Water-Soluble Polyelectrolytes onto Planar Polystyrene and SiO2 Substrates under Flow and Nonflow Conditions MWpol (kg/mol)

xArch

Architecture of the adsorbed layers

t ad G

Adsorption of polyelectrolyte onto SiO2 under flow conditions 142

2.42

Pancake

6.5 min

Adsorption of polyelectrolyte onto polystyrene under flow conditions 142

1.8

Pancake brush

10 min

Adsorption of polyelectrolyte onto SiO2 under nonflow conditions 142

2.06

Pancake

27 h

Adsorption of polyelectrolyte onto polystyrene under nonflow conditions 142

1.5

Pancake brush

86 h

Note. MWpol 5 molecular weight of polyelectrolytes; xArch 5 parameter characterizing the architecture of the adsorbed layer; t ad G 5 time needed to reach the quasiequilibrium state for the polyelectrolyte adsorbed layer (c o1 5 400 ppm).

long times or high polyelectrolyte concentrations (greater than 100 ppm). From the previous analysis and the values represented in Tables 1 through 3 it follows that the behaviors of the adsorption kinetics of polyelectrolytes in flow and nonflow cells are significantly different. Now we consider the architecture of the adsorbed layer and the time needed to reach the quasiequilibrium state, i.e., the state when the conformational changes in the adsorbed layer are almost completed. According to our previous papers (26, 29 –33), the quasiequilibrium state becomes realizable in finite times if we take the following range of quasiequilibrium adsorption values (1 2 z )G o1. From Eqs. [28a] and [44a] through [44c], the time, t Gad, when the polyelectrolyte adsorption, G 1 (t ad G ), reaches the quasiequilibrium state, (1 2 z )G o1, is given by o o t ad G < ~t rel! kin 1 ~t rel! dif 1

SD

1 1 . ` ln z S rel

[50]

From experimental data presented in Figs. 6, 9, and 10 and Eqs. [39] and [30] the architecture parameter, xArch, and the time, t ad G , needed to reach the quasiequilibrium state for the polymer adsorbed layer were found; these values are listed in Table 4. As shown in Table 4, the xArch ratio was found to be 2.42 for the polyelectrolyte adsorbed onto SiO2 and 1.8 for the polyelectrolyte adsorbed onto the polystyrene film, respectively. Thus, the architecture of the adsorbed layer of the watersoluble polyelectrolyte onto SiO2 and onto the polystyrene film

353

under flow conditions is like that of a polyelectrolyte pancake and a polyelectrolyte pancake brush, respectively. According to Eq. [50] and the experimental data presented in Figs. 9 and 10, the time, t Gad, when the adsorption reaches the quasiequilibrium state, is found to be approximately 6.5 min for the polyelectrolyte adsorbed onto SiO2 and 10 min for the adsorption onto the polystyrene film, respectively, which is also shown in Table 4. From Eq. [50] and the data presented in Table 3 it follows that the amount of polyelectrolyte adsorbed, G 1 (t), reaches the equilibrium state, Go1, under flow conditions faster than under nonflow conditions. It is of interest to compare adsorption of polyelectrolyte onto SiO2 and the polystyrene film under flow and nonflow conditions. The equilibrium adsorption and adsorption kinetics of the water-soluble polyelectrolyte with molecular weight of 142 kg/mol under flow conditions onto SiO2 and the polystyrene film by ellipsometry have been studied in our previous investigations (26, 29 –33); the results are summarized in Tables 1 through 4. As shown in our papers (26, 29 –33), the adsorption process in a nonflow cell is governed by the adsorption kinetics at low polyelectrolyte concentrations, by the adsorption kinetics and the diffusion in the adsorbed layer simultaneously for intermediate and long times at intermediate and high polyelectrolyte concentrations, and by the diffusion in the adsorbed layer for long times. The behaviors of the adsorption process under flow conditions and under nonflow conditions are qualitatively different. In fact, the adsorption process in a flow cell due to the intensive diffusive-convective mass transfer in the adsorbed layer with high polyelectrolyte concentrations is governed by the adsorption kinetics for long times. Therefore, the rate of adsorption for long times in the flow cell is greater than that in a nonflow cell. As a result, the time, t ad G , needed to reach the quasiequilibrium state for the adsorbed polyelectrolyte layer in a nonflow cell is greater than that found in a flow cell. ad As shown in Table 4, t ad G (nonflow cell)/t G (flow cell) 5 250 for ad the adsorption kinetics onto SiO2 and t G (nonflow cell)/t ad G (flow cell) 5 516 for the adsorption kinetics onto a polystyrene film. The architecture of the adsorbed layers in the flow and nonflow cells is also quite different. From the data presented in Table 1, the amount of polyelectrolyte adsorbed, Gmax, and the thickness of the adsorbed layer, d ad.layer, under flow conditions are less than those obtained under nonflow conditions; i.e., Gmax(nonflow cell)/Gmax(flow cell) 5 1.11 and d ad.layer(nonflow cell)/d ad.layer(flow cell) 5 1.11 for the polyelectrolyte adsorbed onto SiO2 and Gmax(nonflow cell)/Gmax(flow cell) 5 1.15 and d ad.layer(nonflow cell)/d ad.layer(flow cell) 5 1.12 for the polyelectrolyte adsorbed onto a polystyrene film. As a result, the architecture parameter, xArch, increases from 1.8 (nonflow cell) to 2.42 (flow cell) for polyelectrolyte adsorbed onto SiO2 and increases from 1.5 (nonflow cell) to 2.06 (flow cell) for the polyelectrolyte adsorbed onto the polystyrene film. However,

354

NADEZHDA L. FILIPPOVA

the architecture of the adsorbed layer for nonflow and under flow conditions is the same.

for the polyelectrolyte adsorption process under nonflow conditions.

CONCLUSIONS

REFERENCES

We have developed a theory which describes the adsorption kinetics of water-soluble polyelectrolyte on a planar surface in a flow cell using a new approach. We showed that the rate of the adsorption process depends on convective transfer, and that the rate of adsorption is variable and also depends on the surface polyelectrolyte concentration, c(L, 0, t). These are the new results, since the model equations presented in the literature are based on the assumptions that the rate of adsorption process does not depend on the convective transfer, and that the rate of adsorption is constant and does not depend on the surface concentration, c(L, 0, t). Equations were derived to calculate the parameters which characterize the polyelectrolyte adsorption processes in a flow cell. Using a theoretical approach and analysis of the ellipsometric experimental data, it is shown that the adsorption processes for water-soluble polyelectrolyte onto SiO2 and polystyrene film on a planar surface in a flow cell are governed (I) by the kinetics of polyelectrolyte adsorption for short times, (II) simultaneously by the kinetics of polyelectrolyte adsorption and the diffusive-convective mass transfer in the adsorbed layer, and/or (III) by the diffusive-convective mass transfer in the adsorbed layer for the intermediate and long times. Tables 1 through 4 presented the parameters characterizing (I) the adsorbed layers—(a) the adsorption, Gmax, of the water-soluble polyelectrolytes on planar polystyrene film and silicon surfaces, (b) the structure (i.e., the thickness, d ad.layer, the area occupied by one polyelectrolyte molecule, sm, and the weight fraction, X ad.layer) and architecture (the parameter, xArch) of the adsorbed layer, and (c) the activation energy of adsorption, (2DH) 1 —and (II) the adsorption process—i.e., (d) the adsorption rate constant, K ad 1 , (e) the polyelectrolyte diffusion coefficient, (D eff) 11 ( u 1 3 0), (f) the activation energy of polyelectrolyte diffusion in the adsorbed layers, Q 1 , and (g) the time, t ad G , needed to reach the quasiequilibrium state in the adsorbed layer. It is shown that because of the intensive convective-diffusive mass transfer in the adsorbed layer under flow conditions, the behavior of the adsorption processes for the polyelectrolyte under flow and nonflow conditions is significantly different. The architecture of the adsorbed layer for the water-soluble polyelectrolytes for nonflow and under flow conditions is the same. The rate constant of the adsorption process in the adsorbed layer under flow conditions is significantly greater than the rate constant in the adsorbed layer in a nonflow cell. Therefore, the equilibrium state is reached faster for the polyelectrolyte adsorption process under flow conditions than that

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