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노창현, 최호석

충남대학교 화학공학부

Influence of a Preadsorbed Colloid Particle on the Flux Distribution of Depositing Colloid Particles under Gravity Field

Chang-Hyun Rho, Ho-Suk Choi

School of Chemical Engineering

Chungnam National University

220 Gung-Dong, Yousung-Ku, Daejeon City, 305-764, Korea

Introduction

The adsorption of colloid particles such as latexes, proteins, bacteria and enzymes has played an important role in many different industrial fields, which include chromatographic separation, filtration, water cleansing and biofouling. Despite the importance, the theoretical analysis of these processes are still at the beginning stage. So far, only Langmuir theory have been applied to all kinds of adsorption processes. However, in case that the size of solute particles is much larger than atomic spacing of binding sites on solid substrate, each solute particle randomly covers many active sites on the substrate and then has many bonds with substrate. Subsequently, the surface blocking effect of preadsorbed particle is very important and has to be taken into account. The adsorption of colloid particles is usually regarded as a random and (partially) irreversible process. Due to these special features, the classical equilibrium thermodynamic approaches can not be applied to this adsorption process.

Recently, the random sequential adsorption (RSA) process has been applied to explain the above two special features. In this process, particles are adsorbed randomly and sequentially to the plain substrate and fixed at the initially adsorbed place. Although the RSA model successfully describe the random, irreversible and blocking nature of the adsorption process, it does not properly incorporate the transport mechanism of the particles from bulk to surface. In a real system, the density difference between particles and solvent will be somewhere between the conditions of two extremes: a diffusion controlled process and a gravity controlled process[1,2]. In this study, the primary objective is to develop a more efficient way of simulation to generate the configuration of particle deposits. The secondary objective is to investigate the influence of gravity relative to diffusion on the particle flux distribution near a preadsorbed particle. However, we do not take into account the effect of hydrodynamic interactions and colloidal forces between particles or particle and surface.

Formulation of the transport equation

If there is a density difference between particle and solvent, one should take into account the effect of gravity. In this case, the governing steady state diffusion equation becomes, with the assumption of constant diffusivity and no inter-paritcle interactions,

[eqn: ∇^2 c - ∇·( Ｊ over {kT} c ) = 0,]                         (1)

where c is the concentration of the particles in the solution phase, F the force vector acting on a particle, k the Boltzmann constant, and T is the temperature of system. If we use the diameter of a particle, d, as a length scale and denote by c0 the particle concentration in the bulk phase, the above equation can be non-dimensionalized:

[eqn: ∇^2 Psi - ∇·({Ｊd over kT} Psi)=0,]                       (2)

where [eqn: Psi] is the normalized concentration distribution of particles in solution phase:

[eqn: Psi = c over c_0.]                               (3)

In a spherical coordinate system with azimuthal symmetry, the governing steady state diffusion equation takes the form

[eqn: {partial^2 Psi} over {partial r^2} + ({2 over r} + N_G sin theta ){partial Psi} over {partial r} + {1 over r^2} {partial^2 Psi} over {partial theta^2} - ({tan theta}over{r^2} - N_G {cos theta}over{r}) {partial Psi} over {partial theta} = 0,]   (4)

where [eqn: N_G (= {F_g d} over {kT})] is the dimensionless gravity number. It has an adsorbing boundary on the surface(Fig. 1),

(B.C.1)	[eqn: Psi = 0,~~~~at~y=0,]	(5)

a reflecting boundary at the exclusion surface of the preadsorbed disk:

(B.C.2)[eqn: {partial Psi} over {partial r} + N_G sin theta Psi=0,~~~~~~at~r=1,]              (6)

and a symmetric boundary at the center line through the preadsorbed disk:

(B.C.3)[eqn: {partial Psi} over {partial theta}=0,~~~~~at~theta=0.]                          (7)

We chose the last boundary condition, at a distance, [eqn: r_0 > > 1], as the solution of undisturbed diffusion equation:

[eqn: {partial^2 Psi}over{partial y^2} + N_G {partial Psi}over{partial y}=0]                             (8)

with its own boundary conditions:

[eqn: Psi=0,~at~y=0,~ and~Psi=1~at~y rightarrow infty,]                  (9)

based upon the assumption that the probability distribution of particles far from a preadsorbed particle is the same as the probability distribution of incoming particles on the empty surface. The solution of Eq.(8) is

[eqn: Psi = 1-exp(-N_G~ y).]                         (10)

So, the flux of incoming particles far away from a preadsorbed particle onto the surface is proportional to the gravity number.

[eqn: lim_{r rightarrow infty} {partial Psi(0)}over{partial y}=N_G.]                            (11)

Finally, the last boundary condition becomes

(B.C.4)[eqn: Psi=Psi_0=1-exp(-N_G ~r_0~sin theta)~~~~~at~r=r_0]             (12)

where we have chosen [eqn: r_0] as an arbitrary distance from the center of a preadsorbed particle. Once we have a set of convergent solutions, we can calculate the flux of particles at the interface by differentiating the probability function.

[eqn: J(r)□_{y=0} = -D({partial c} over {partial y})_{y=0} = -Dc_0 ({partial Psi} over {partial y})_{y=0}]	(13)

Result and Discussion

The normalized concentration distributions of new diffusing particles at steady state are displayed in Fig.2. At small [eqn: N_G], since the drifting of particles is not so significant compared to the thermal fluctuation of particles, the concentration profiles of particles does not seem to change dramatically with respect to geometry. In the range [eqn: 2 < N_G < 5],  however, the concentration profile changes dramatically in the vicinity of the preadsorbed particle. For [eqn: N_G geq 1], an accumulation of particles is observed on top of the preadsorbed particle. In comparison with the concentration profile in 1+1 D system[3], the effective range of thermal disturbance is reduced over the entire range of [eqn: N_G]. This phenomena is clear in Fig. 3 which shows the flux of particles onto the surface. Flux distributions(Fig. 3) at different [eqn: rm N_G] are obtained by the normalization of numerical values with following closure.

[eqn: 2 int_1^inf {J_1 (r)} over J_0 r dr = 1.]	(14)

Finally, we suggest a model equation which fits very well with the numerical solution of flux distribution of depositing particles over wide range of [eqn: rm N_G].

[eqn: J_1 over J_0 = alpha exp[-beta (ln x)^gamma ]]	(15)

References

[1] B. Senger, F. J. Bafaluy, P. Schaaf, A. Schmitt, and J.-C. Voegel, Proc. Nat. Acad. Sci., 89, 9449 (1992).

[2] B. Senger, R. Ezzeddine, F. J. Bafaluy, P. Schaaf, F. J. G. Cuisinier, and J.-C. Voegel, J. Theor. Biol., 163, 457 (1993).

[3] H. S. Choi and J. Talbot, Korean J. of Chem. Eng., 14, 117 (1997).

Fig 1. System boundary represented  Fig 2. Probability distribution of             cylindrical coordinate system.       finding an incoming particle                                                        in 2+1 D system. Lines                                                     represent contour plots of                                                   probability distribution of                                                    particles.(a)NG=0.1,(b)NG=1.0,

(c)NG=2.0,(d)NG=5.0.

(a)                                 (b)

Fig 3. Normalized flux distribution calculated at different valves of NG

(a) small NG (b) large NG