18.2 Polymeric Surface Modifi cation/
18.3.1 Colloidal Forces and Derjaguin–Landau–
Verwey–Overbeek Theory
Bare nanoparticles. For bare nanoparticles, aggregation and deposition in aqueous environments are generally modeled using Derjaguin–Landau–Verwey–
Overbeek ( DLVO) theory [ 31 , 33 ]. According to classical DLVO theory, the van der Waals forces ( V vdW ) are the primary attractive force, whereas repulsive forces are derived from the electrostatic double layer ( V ES ) ( Fig. 18.2(a) ). The V vdW attractive force between two spherical particles of radius R 1 and R 2 or a spherical particle of radius R and a fl at surface (a representative of a large collector grain relative to a nanoparticle) are given in Equations 18.1 and 18.2, respectively [ 34 ].
1 2 vdw
sphere– 1 2
sphere
( ) –
= 6
+ h R R
R V A
h R (18.1)
= ×
vdw sphere–
wall
( ) – 6 h A R
V h (18.2)
where A is the Hamaker constant, which is 1 × 10 –19 and 3.7 × 10 –20 N m for iron nanoparticles and titanium dioxide (anatase), respectively [ 35 , 36 ], and h (m) is separation distance between two interacting surfaces. The V vdW decays with increasing h and increases in magnitude and extent as the particle size
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increases. This attractive energy promotes aggregation and deposition.
Electrostatic repulsion between two identical particles of radius R and between a particle and a fl at surface are given in Equations 18.3 and 18.4, respectively [ 37 , 38 ].
2 –
ES 1
sphere–
sphere
2pe e0 z In[1 + ]
= r kh
V R e (18.3)
pe e0 z z z z
= – + 1+ 2 – 2
ES 1 2 – 2 2
sphere–
wall
[2 In(1 + ) ( ) In(1– )]
1–
kh
kh
r kh
V R e e
e (18.4)
where ε r is the relative dielectric constant of the liquid, and ε 0 is the permittivity of the vacuum. ζ 1 and ζ 2 are the zeta potentials of a particle and a collector, respectively. k is the inverse Debye length, which, for symmetrical ( z – z ) electrolytes, can be expressed as in Equation 18.5 [ 38 ]:
2Σ 2
0
k= e e i i
B
e n z
r k T
(18.5)
where e is the electron charge, n i is the number concentration of ion i in the bulk solution, and z i is the valence of ion i . The V ES decays exponentially with separation distance ( h ) and decreases in magnitude as the Debye length (1/ k ) decreases. In addition, V ES decreases as particle size decreases. This repulsive energy inhibits aggregation and deposition. Because of their small size, the energy barrier for nanoparticles to resist aggregation and deposition may be less than that of micron-sized particles of the same surface charge. The distance from the particle surface over which V ES acts to counter Van der Waal’s attractive forces V vdW is theoretically predicted to be similar to the Debye length (1/ k ). As shown in Equation 18.5, the Debye length decreases with increasing ionic strength ( n i ) and valency of ionic species present in the bulk solution. Therefore, the ionic strength and composition of groundwater will aff ect aggregation, deposition, and thus the mobility of nanoparticles [ 21 ].
Besides V vdW and V ES , other non-DLVO forces can aff ect aggregation and deposition in some circumstances. For example, NZVI particles that are single magnetic domain particle [ 13 , 32 ] have an intrinsic permanent magnetic dipole moment even in the absence of an applied magnetic fi eld [ 35 , 39 ]. When particle dipoles are oriented in a head-to-tail confi guration, the maximum magnetic attraction energy ( V M ) is expressed as Equation 18.6 [ 32 ]:
2 3 s M
9( 2)3
–8pm0
=
+ V M R
h R
(18.6)
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where m 0 is the permeability of the vacuum, and M s is the saturation magnetization of the particles. It should be noted that V M is a longer-range attractive interaction compared to V vdW and increases in magnitude with particle radius to the sixth power. Thus, the size of magnetic nanoparticles such as NZVI signifi cantly aff ects aggregation and deposition [ 22 ].
Classical DLVO theory assumes that the total interaction energy between a pair of particles (aggregation) or particle–fl at surface (deposition) is additive, that is, a combination of DLVO forces acting between two surfaces.
Figures 18.2(a) and 18.3 show a schematic diagram of total energy of interaction between a bare nanoparticle and a fl at surface. The sum of the interaction energy results in an attractive secondary minimum ( Fig. 18.3 ) where particles can agglomerate or deposit [ 22 , 40 ]. Typically, the magnitude of this attractive secondary minimum well is relatively small. Therefore, this aggregation or deposition mode is reversible if Brownian energy (1.5 k B T ) or external forces such as shear force due to fl uid fl ow is large enough to get particles out of this attractive energy well [ 22 , 40 , 41 ]. Natural geochemical conditions that decrease the magnitude of the energy barrier, such as high ionic strengths and multivalent ions in groundwater, can promote aggregation and deposition under a primary minimum that would be much less reversible.
Polymer-modifi ed nanoparticles. Polyelectrolyte surface modifi cation can improve the mobility of nanomaterials in porous media by introducing an additional electrosteric repulsion to counter V vdW , or magnetic attractive forces, that promote aggregation and deposition ( Fig. 18.2(b) ) [ 42–44 ]. Between two polymer-modifi ed surfaces, the electrosteric repulsion consists of: osmotic
Figure 18.2 Schematic representation of particle–particle interaction forces acting on (a) bare charged nanoparticles and (b) electrosterically stabilized polyelectrolyte-modifi ed nanoparticles including van der Waals attraction ( V vdW ), electrostatic double layer repulsion ( V ES ), magnetic attraction ( V M ), osmotic repulsion ( V osm ), and elastic-steric repulsion ( V elas ).
- - - - -
+ +
+ +
+ +
+ +
+ +
+ + + + + +
- - -+ - -
- - -
- - - - --
-- - -
-- --
- - -- -
- --
- - -
- - -- --- - - - - - -
-
- -
- - -
-
- - -- - -
- - - - -
- -- - - - - - - -- -
- -- - -- ---- -
- - - Collision due to diffusion,
sedimentation, interception. and hydrodynamic effect
Collision due to diffusion, sedimentation, interception. and
hydrodynamic effect
Repulsion due to VES
Attraction due to VvdW+vM Attraction due to VvdW+vM
Repulsion due to VES+Vosm+velas h
(a) (b)
d h
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repulsion ( V osm ) and elastic–steric repulsion ( V elas ). Overlap of the polyelectrolyte layers on two approaching surfaces increases the local polymer segment concentration and thus increases the local osmotic pressure in the overlap region ( V osm ). Any compression of the adsorbed polyelectrolyte layers below the thickness of the unperturbed layer ( d ) leads to a loss of entropy and gives rise to the elastic repulsion ( V elas ) [ 42 , 44 ]. The range and magnitude of the electrosteric repulsion between a pair of particles or a particle and a collector depends on the surface concentration of adsorbed polyelectrolyte, and extension and charge density of the adsorbed polyelectrolyte layer. The conformation of the adsorbed polymer can also contribute to the magnitude of the electrosteric repulsion [ 42 ]. However, the range of electrosteric repulsion is solely controlled by the adsorbed layer thickness. As shown in Fig. 18.3 ( V T XDLVO with steric), for separation distances beyond 2 d , V vdW attractive forces are dominant and result in the net attractive potential. However, when the polyelectrolyte- modifi ed particles approach one another to a distance less than 2 d , there is a large energy barrier due to the adsorbed layer. In fact, the polymer coatings make it very diffi cult for these particles to aggregate in a primary minimum, so all agglomeration or deposition must occur under a shallower secondary minimum. This has very important consequences regarding the dispersion sta- bility and mobility of nanoparticles in the subsurface because agglomeration and attachment under a secondary minimum are subjected to disaggregation and
Figure 18.3 Hypothetical interaction energy profi les showing various components of the particle–collector/particle–particle interaction energy profi le including total interaction energy with and without the electrosteric repulsive forces aff orded by adsorbed or grafted polyelectrolytes ( V osm and V elas ).
Velas
VES V/kBT
Vosm Energy
barrier
VTXDLVO with steric (elastic and osmotic)
VTXDLVO without steric
VvdW
2d Primary minimum
Secondary minimum
Separation distance (h)
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detachment due to shear force and Brownian energy. For this reason, aggregation and mobility of surface modifi ed nanoparticles should be largely controlled by the adsorbed polyelectrolyte layer properties. A recent study [ 22 ] correlated the adsorbed polyelectrolyte layer properties ( d estimated using Ohshima’s soft particle analysis, to be discussed next) and the adsorbed polyelectrolyte mass ( Γ ) on NZVI to their dispersion stabilities ( Fig. 18.4(a) ). The higher the adsorbed polyelectrolyte mass and the more extended the layer thickness, the larger the magnitude and extent of electrosteric repulsion and the smaller the secondary minimum well for agglomeration. For similar reason, considering the particle–collector interaction under the infl uence of adsorbed polymer layers, collector ripening, and pore plugging are not expected for the transport of colloidally stable polyelectrolyte modifi ed nanoparticles in saturated porous media. Figure 18.4(b) illustrates the enhanced mobility of NZVI particles modifi ed by various surface modifi ers compared to bare NZVI particles. The infl uence of electrosteric stabilization on the enhancement of dispersion sta- bility and nanoparticle transport is also generally observed as summarized in Table 18.1 .
Another unique feature of electrosteric stabilization with implications on the subsurface transport of nanoparticles is the ability to maintain the strong repulsive forces at high ionic strength and in the presence of multivalent ionic species. Aggregation and deposition of electrostatically stabilized nanoparticles are sensitive to ionic strength and divalent cations due to charge neutralization and shrinking of the Debye length. Electrosterically stabilized nanoparticles are much less sensitive to ionic strengths and types of ionic species [ 21 , 42 ] ( Fig. 18.4(b) ).
Figure 18.4 (a) Correlation between the colloidally stable fraction (wt %) of nanoparticles and the measured surface excess ( Γ , mg/m 2 ) and layer thickness ( d , nm) of each adsorbed polyelectrolyte [ 22 ]. (b) Eff ect of ionic strength on the elution of bare and modifi ed reactive nanoscale iron particle (RNIP) through a 12.5-cm column with water-saturated silica sand having a porosity of 0.33. Particle mass concentration is 3 g/L. All samples also contained 1 mM NaHCO 3 to control pH at 7.4. The approach velocity was 9.5 m/d [ 15 ].
bare Bare
Stable fraction (wt%) after 400 mm d(nm)
T(mg/m 2) PAP10K
PSS70K
PAP2.5
CMC90K CMC700K
% Elution
0
0.0 0
20 40 0 10 20 30 40 50
0.5 1.0 1.5 2.0 2.5 20 40 60 80 100 120
(b) (a)
PAP PMAA48-PMMA17-PSS650 1 mM
50 mM 100 mM
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