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Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Istituto di Scienze e Tecnologie Molecolari
ISTM-CNR, Università degli Studi di Padova
e-mail: [email protected]
Silvia Gross
La chimica moderna e la sua comunicazione
Dipartimento di Scienze Chimiche
Università degli Studi di Padova
e-mail: [email protected]
http://www.chimica.unipd.it/silvia.gross/
Silvia Gross
Colloid Chemistry
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Colloid stability and DLVO theory
Understanding and controlling stability and structure (= spatial
organisation of the colloidal particles) of dispersion:
- Nature and origin of interparticles interactions (attractive, repulsive)
- Derivation of classical theory on colloid stability DLVO
- Effect of solution chemistry (A, y0, electrolyte nature and concentration) on
interparticles interaction energies (see vdW and electrostatic interactions):
- critical coagulation concentration (of electrolyte)
- Schulze-Hardy rule (effect of electrolyte valence)
- Effect of interparticles interactions on coagulation in dilute dispersions and on
structure of dispersion
- Quantitative determination of stability of a dispersion against coagulation (stability
ratio, W) (Smoluchowski and Fuchs theories of colloid stability)
- Develop quantitative relationship between stability and interparticle forces
- Theory of rapid and slow coagulation
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Concept of colloidal stability
Colloidal stability relates to the physical state of the system: it is
stable if it remains well dispersed
Colloidally stable means that the particles do not aggregate at a
significant rate: the precise connotation depends on the type of
aggregation under consideration.
http://old.iupac.org/reports/2001/colloid_2001/manual_of_s_and_t/no
de35.html
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Collod stability
Stability: ability of a dispersion to resist coagulation
Aggregation is the process or the result of the formation of aggregates.
When a sol is colloidally unstable (i.e. the rate of aggregation is not
negligible) the formation of aggregates is called coagulation or
flocculation.
coagulation, implying the formation of compact aggregates, leading to
the macroscopic separation of a coagulum;
flocculation, implying the formation of a loose or open network which may
or may not separate macroscopically. In many contexts the loose
structure formed in this way is called a floc.
http://old.iupac.org/goldbook/C01119.pdf
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Stability of colloidal dispersion
van der Walls: attractive forces and typically high intensity at short distances
→ if not counterbalanced: unstable dispersion and coagulation
Protection against vdW attraction provided through two main mechanisms:
1. Polymer-induced (steric) stability: suitable polymer/surfactant
adsorbed on particle surface; may also provide repulsive force, partially
due to pure steric effects (polymers chains overlap).
2. Electrostatic stability: electrostatic repulsive force deriving from
particles becoming charged (overlap of the electrical double layer):
understanding how it depends from experimental parameters
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Stability of colloidal dispersion
1. Polymer-induced (steric) stability: suitable polymer/surfactant
adsorbed on particle surface; may also provide repulsive force, partially
due to pure steric effects (polymers chains overlap).
- interaction polymer-dispersing medium
- configurational freedom of polymers
- analytical methods needed to be developed
- thermodynamic picture of steric stabilisation only accounting of:
- excluded volume of polymer chains
- elastic contributions determining interactions between polymer layers
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Stability of colloidal dispersion:
kinetics or thermodynamics?
barrier along the path
to coagulation
coagulation
Kinetic stability: consequence of a
energy barrier against collision and
eventually coagulation (favoured,
thermodynamically, by reduction of
DG)
But: Interaction energy barrier >> kBT
Thermodynamic stability
Example: related to the possibility of
the particle to assembly to give
ordered networks or association
colloids
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Stability of colloidal dispersion:
kinetics or thermodynamics?
Thermodynamics: what the equilibrium state will be
Kinetics: if that equilibrium state will be reached and how fast
Examples:
- Formation of micelles from surfactants at given concentration:
thermodynamics driven
- Lyophobic colloids (clays, silica dispersion) thermodynamically
unstable, kinetically stable (if potential and/or surface charges are
enough large)
→ see DLVO: classical theory of electrostatic stability of colloids
→ in colloid science many dynamic processes and variables
(buoyancy, flow, diffusion, sedimentation)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Colloid stability and DLVO theory
Quantitative relationship between interparticle forces and colloid stability:
Understanding relationship between interparticle forces and microstructure* of
dispersion (average distance, size and size distribution etc.)
*structure= spatial organisation of collodal particles
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Interplay interparticle forces/structure
Microstructure & stability determine quality, processability and properties of
colloidal systems.
Effect of:
- Interparticle forces
- Concentration
on structure of dispersions
Analysed example: microstructure (local arrangement of particles) in a
monodisperse colloid as a function of particles concentration
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Correlation between
interparticle forces and
corresponding microstructure)
in monodisperse colloid
1. Overall interparticle forces
dominated by strong
repulsion (TD stability)
2. Overall interparticle forces
dominated by strong
attraction
3. Intermediate situation
Volume fraction= fraction of the volume
of the system occupied = n4/3pa3
2
1
3
Interplay interparticle forces/structure
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
1. Overall interparticle forces
dominated by strong
repulsion (1)
surface charge large
strongly overall repulsive forces
extended over large distances (k-1)
thermodynamic stable dispersion
particles can assembly in ordered
crystalline structures even at very
low volume fractions (f= nvp< 0.001)
type of crystalline structure:
concentration and k-1
large k-1: body centered cubic structure
small k-1: face centered cubic structure
2
1
3
Interplay interparticle forces/structure
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
atom packing/crystalline structures
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Overall interparticle forces
dominated by strong
attraction (2)
Dispersion becomes
thermodynamically and
kinetically unstable:
Aggregated interconnetcted
networks (gel) with loose
fractal structure with strong
vdW interparticle “bonds”2
1
3
Interplay interparticle forces/structure
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Gels
Gel
Non-fluid colloidal network or polymer network that is expanded throughout its whole volume by
a fluid.
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
1. Overall interparticle forces
dominated by strong
repulsion
2. Overall interparticle forces
dominated by strong
attraction
3. Intermediate situation
2
1
3
Interplay interparticle forces/structure
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
1. Overall interparticle forces
dominated by strong
repulsion
2. Overall interparticle forces
dominated by strong
attraction
3. Intermediate situation
2
1
3
Interplay interparticle forces/structure
Repulsion/attraction of comparable
magnitude, system microstucture more
sensitive to changes in relative
contributions
Aggregation in secondary/primary minima
Dense and more efficient packing are
possible (e.g. ceramics processing)
Solution chemistry and surface chemistry
allows tuning
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory (ca. 1940)
B.V. Derjaguin, L.D. Landau, Theory of Stability
of Strongly Charged Lyophobic Sols and
Adhesion of Strongly Charged Particles in
Solutions of Electrolytes, Acta Physicochim.
URSS 14 (1941) 633-662.
E.J.W. Verwey, J.Th.G. Overbeek, Theory of
Stability of Lyophobic Colloids, Elsevier,
Amsterdam, 1948.
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory (ca. 1940)
Assumption 1 (approximation → see further “non-DLVO” forces)
Colloidal stability determined by a balance between:
- electrostatic repulsion
- van der Walls attraction
further contributions (”non-DLVO” forces, e.g. depletion interactions) neglected
Assumption 2
Additivity of the two contributions
Assumption 3
Even distribution of particles and ions in the dispersion
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
0
B
B
exp( x)
[exp(ze / 2k T) 1]
[exp(ze / 2k T) 1]
k
DLVO theory
Poisson-Boltzmann without limitations of DH
approximation → development of Gouy-Chapman
theory for EDL: derivation of an equation
describing variation in potential with distance
from the surface for diffuse double layer without
the simplifying DH approximation of low potential:
1 2
Bne 0
2212t 64k Tn ex
A( )( d1
d)2
p
p
k k
Case: two interacting parallel planes at a distance d
Vnet (d) = VR(d) + VA(d)
n∞ = bulk ions concentration
repulsion potential assuming:
- interacting planes
- not DH approximation (Gouy-Chapman)
- superposition approximation
- Taylor series
(interacting planes, d)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory (interacting planes, d)
The secondary minimum could cause
flocculation/reversible coagulation
The primary minimum is the reason for
coagulation in most cases
1 2 2t
212ne B 0
A64k Tn exp( ) ( )d
12d
k p
k
Height of potential barrier determined by VR
Potential barrier can be lowered by:
- increasing electrolyte concentration (decrease
of k1)
- decreasing surface potential2
0
Unstability if barrier < several kBT
VR(d) VA(d)
Shape of curves: physical & geometrical
factors
12
1 0
2 2
r
i i
i
kT
e z n
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
General conclusions:
1. The higher potential at surface (and through the EDL, inner limit of the
diffuse part), the higher the repulsion
2. The lower the concentration of electrolytes, the longer the distance
from the surface before the repulsion drops significantly (k-1)
3. The larger the Hamaker constant, the larger the attraction between
macroscopic bodies
1 2 2t
212ne B 0
A64k Tn exp( ) ( )d
12d
k p
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
General conclusions:
1. The higher potential at surface (and through the EDL), the higher the
repulsion
2. The lower the concentration of electrolytes, the longer the distance
from the surface before the repulsion drops significantly (k-1)
3. The larger the Hamaker constant, the larger the attraction between
macroscopic bodies
1 21t
22ne B
2
0
A64k Tn exp( ) ( )d
12d
k p
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: effect of the potential
0 (case flat blocks): Gouy-Chapman
0
B0
B
0
0
0
0[exp(ze / 2k T) 1]
[exp(ze / 2k T) 1]
1
for large values of 𝛹0
Potential of the inner limit of the diffuse part of EDL
(not the «surface»)
𝛹0 𝛹0→ sensitivity to the value of decreses as increases
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: effect of the potential
0 (case flat blocks)
y0
Adjustable through the concentration of
potential determining ions
Value affected by specific adsorbtion
phenomena
More correct to refer to “inner limit of the
diffuse double layer” instead of “on the
surface”
→ zeta potential is the lower limit for y0
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: effect of the potential
0 (case flat blocks)
Plot of net /kBT
versus d for flat blocks
Different values of 0
Electrolyte z:z = 1:1, 0.093 M
Constant value of k = 109 m-1
Constant value of A= 2*10-19 J
0
1 2212net B
2 A64k Tn exp( d) ( )d
12
k k p
y0
increase in barrier = repulsion
potential can be tuned by p.d.i.
Large values of 0 0 1
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Inert and potential determining ions
potential-determining (p.d.) ions
Species which by virtue of their electron distribution between the solid and liquid phase (or by
their equilibrium with electrons in the solid) determine the difference in potential between
these phases. This definition requires that adsorbed p.d. ions are part of the adsorbent and
belong to the category of surface ions (http://goldbook.iupac.org/P04776.html)
Es: Ag+ and I- ions in AgI sols, H+ in oxides (potential pH-determined)
inert (indifferent) ions
They do not change the charge density at the surface of the particles, but may influence the
interfacial potential difference by virtue of their local distribution (e.g. NaNO3)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: surface charge density
DLVO potentials at constant salt concentration with varying surface charge density
Picture:
Courtesy Prof. Peter Lang,
Forschungszentrum Jülich,
Germany
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
General conclusions:
1. The higher potential at surface (and through the EDL), the higher the
repulsion
2. The lower the concentration of electrolytes, the longer the distance
from the surface before the repulsion drops significantly (k-1)
3. The larger the Hamaker constant, the larger the attraction between
macroscopic bodies
1 2 2t
212ne B 0
A64k Tn exp( ) ( )d
12d
k p
k
(interacting planes)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: electrolyte
concentration
1/ 2
1 0 r
i
B
2 2
A i
i
k T
1000e N z M
k
most «tunable» parameter from the experimental point of view
charge/valence of the electrolyte
concentration of the electrolyte
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: electrolyte
concentration
DLVO potentials at constant
surface charge density with
varying electrolyte concentration
Picture:
Courtesy Prof. Peter Lang,
Forschungszentrum Jülich,
Germany
1/ 2
1 0 r B
2 2
A i i
i
1
1/ 2
k T
1000e N z M
1
zM
k
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: electrolyte
concentration
1/ 2
1 0 r
i
B
2 2
A i
i
k T
1000e N z M
k
we concentrate on M
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: effect of the electrolyte
(case spheres of equal r = 100 nm)
Constant value of 0 = 25.7 mV
Radius (equal) of spheres = 100 nm
Constant value of A= 2*10-19 J
Different values of k (concentration
10-3 to 10-2)
1/ 2
1 0 r B
2 2
A i i
i
1
1/ 2
k T
1000e N z M
1
zM
k
k
k
k1
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
General conclusions:
1. The higher potential at surface (and through the EDL), the higher the
repulsion
2. The lower the concentration of electrolytes, the longer the distance
from the surface before the repulsion drops significantly (k-1)
3. The larger the Hamaker constant, the larger the attraction between
macroscopic bodies
1 2 212net B 0
2A64k Tn exp( ) ( )
12d d
k p
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: effect of Hamaker
constant
Plot of net /kBT
versus d for flat blocks
Constant value of 0 = 103 mV
Constant value of k = 109 m-1 = k-1 = 1 nm
Different values of A 212
1 2 2t
212ne B 0
A64k Tn exp( ) ( )d
12d
k p
k
Energy barrier decreases, secondary minimum depth
increases with increasing A
Lowest degree control (chemical nature of the
phases)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: actual values
Contributions to the total interaction energy between two spheres
Picture:
Courtesy Prof. Peter Lang,
Forschungszentrum Jülich,
Germany
kBT scala
kBT at 298 K = 4.11 *10-21 J
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Critical Coagulation Concentration
Experimental starting point:
addition of an indifferent (i.e. not potential determining)
electrolyte to a lyophobic colloid can cause coagulation
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory
General conclusions:
1. The higher potential at surface (and through the EDL, inner limit of the
diffuse part), the higher the repulsion
2. The lower the concentration of electrolytes, the longer the distance
from the surface before the repulsion drops significantly (k-1)
3. The larger the Hamaker constant, the larger the attraction between
macroscopic bodies
1 2 2t
212ne B 0
A64k Tn exp( ) ( )d
12d
k p
k
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Critical Coagulation Concentration
Sharply defined concentration of the electrolyte
needed to induce coagulation (settling out of the dispersed
phase)
Critical coagulation concentration (CCC)
Experimental determination:
1. test tubes of the colloidal suspension with increasing
electrolyte concentration
2. addition of water and electrolyte (dilution being constant)
3. mixing/waiting an arbitrary but consistent lenght of time
4. visual/microscopic inspection of the tube (settling out of
the coagulum)
the highest concentration that leaves the colloid unchanged
and the lowest determining coagulation brackets the CCC
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Critical Coagulation Concentration
Actual concentration of electrolyte at CCC affected by:
a) time allowed to elapse after electrolyte addition
b) polydispersity of the sample
c) potential at the surface
d) the value of Hamaker constant
e) the valence of the ions of the electrolyte
Typically (a)-(d) remain constant, so CCC can be used for a quantitative
measure of the effect of the valence of the ions (Schulze-Hardy rule)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Critical Coagulation Concentration
Actual values of concentrations: array of different (and partially unknown) parameters
Relative values of concentrations: valence of the counterions: generalisation possible: SH rule
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Schulze-Hardy rule (1900)
Generalization about the effect of electrolyte
It is the valence of the ion of opposite charge with respect to the surface of
the colloid which has the prevailing effect on colloid stability (regadless the
nature of the ion with the same charge as the surfcae).
Definition of Schulze-Hardy Rule by IUPAC
The generalization that the critical coagulation concentration for a typical lyophobic sol is extremely
sensitive to the valence of the counter-ions (high valence gives a low critical coagulation
concentration).
http://goldbook.iupac.org/S05501.html
H. O. Schulze (1853–1892), Sir W. B. Hardy (1864–1934)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Critical Coagulation Concentration
stable
coagulate
can we translate it into a qualitative
relationship?
value of k for which the barrier becomes 0?
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Schulze-Hardy rule (1900)
Derivation of SH rule
We assume: border stability/unstability at k values corresponding to energy
barrier = 0 (kBT units)
Maximum in the potential energy occurs at zero means
net = 0 at d = dm (location at the maximum of the potential)
and
dnet/dd = 0 at d = dm
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Schulze-Hardy rule (1900)
Derivation of SH rule
since holds: 1 2 2212net B 0
A64k Tn exp( d) ( )d
12
k k p
1 2 2212B 0 m m
A64k Tn exp( d ) ( )d
12
k k p
net = 0 at d = dm
dnet/dd = 0 at d = dm
2 3212B 0 m m
A64k Tn exp( d ) 2( )d
12
k p
conditions of zero barrier are satisfied for:
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Schulze-Hardy rule (1900)
Derivation of SH rule (looking for dependence of CCC from valence)
since holds: 1 2 2212net B 0
A64k Tn exp( d) ( )d
12
k k p
1 2 2212B 0 m m
A64k Tn exp( d ) ( )d
12
k k p
net = 0 at d = dm
dnet/dd = 0 at d = dm2 3212
B 0 m m
A64k Tn exp( d ) 2( )d
12
k p
1
m2 dk from which we obtain
we have obtained:
1
mdk case: spherical particles:
criterion for stability
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Schulze-Hardy rule (1900)
Derivation of SH rule: dependence of CCC from valence
since holds:
3n k
1
m2 dk
(neglected the dependence of on z) we obtain:
and
0
2 2
i i
i
0 r B
e z n
k T
k
1/2
3 3/ 2n z n
since holds:
and6
1c
z
2 3212B 0 m m
A64k Tn exp( d ) 2( )d
12
k p
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
Rules of thumb for the calculation of
k for different types of electrolytes.
The salt concentration, c has to be
introduced in the unit mol/L
2 3
6 6
1 1 1 1Na : Mg : Al CCC 1: : 1: :
2 3 64 729
Schulze-Hardy rule (1900)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
2 3
6 6
1 1 1 1Na : Mg : Al CCC 1: : 1: :
2 3 64 729
Schulze-Hardy rule (1900)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
“Inverse” Schulze-Hardy rule (2015)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
“Inverse” Schulze-Hardy rule (2015)
Schulze-Hardy Rule CCC ≈ 1/z6
Inverse Schulze-Hardy Rule CCC ≈ 1/z
EXPERIMENTAL
Polystyrene latex 300 nm diameter
Amidine and sulphate groups functionalisation
Dispersed in salt solutions 10 mg/L
Different multivalent electrolytes
Evolution followed by DLS
Typically positively charged particles coagulated by multivalent anions and viceversa
Multivalent ions are the co-ions (same sign as the colloidal particles)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
“Inverse” Schulze-Hardy rule (2015)
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
CCC: summary
- Ions opposite in charge with respect to the surface: predominant
role in EDL (basic statment of Schulze-Hardy rule, SHR)
- In the Gouy-Chapman theory, a Boltzmann factor introduced to
describe relative concentration of ions in EDL with respect to bulk
ni/ni∞ = exp (-zi·e·y /kBT)
Ions having the same charge as the surface
exponent is negative → (Coulombic repulsion of the ions from the surface)
→ ions with same charge as the surface present a lower concentration in
EDL (ni) than in the bulk (ni∞)
Oppositely charged ions: concentration increased in EDL
→ only counterions contribute to the diffuse double layer at CCC
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
CCC: summary
- The valence chiefly determine the CCC (Schulze-Hardy rule)
- Small difference in ions with same valency: to coagulate:
As2S3 sol: 0.058 M of Li+ and 0.051 M of Na+
AgI sol: 0.165 M of Li+ and 0.140 M of Na+
- Li+ less effective than Na+ in coagulation
Effectiveness to promote coaugulation
- Cs+ > Rb+ > NH4+ > K+ > Na+ > Li+
- F- > Cl-> Br-> NO3-> I- >SCN-
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
strength of specific adsorbtion
I- >SCN->Br->Cl->OH->F-
N(C2H5)4+ > N(CH3) 5
+ > Tl+> Cs+> Na+
Specific adsorption enhanced by larger size (= larger polarizability)
and lower hydration
→ The stronger the specific adsorption, the lower the effectiveness in
determining coagulation
Silvia Gross – Chimica dei Colloidi – Laurea Triennale in Chimica
DLVO theory: summary
additional approximations introduced, but:
- effective and versatile tool for understanding colloid stability
- quantitative theoretical analysis well matching with experimental
outputs
- threshold of stability in terms of
(a) valence and
(b) concentration of added indifferent electrolyte easily measured