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interaction parameters like electrostatic interactions, coordinate bonds, and hydro-
gen bonding between the polymers (Li et al. 2020). For small hydrophobic
molecules, water tends to reorganize itself around the hydrophobic solute without
losing its hydrogen bond, which attributes to its low solubility (Schmid 2001) while
in the presen ce of a bulky hydrophobic moiety, it becomes challenging to maintain
the hydrogen bond interactions geometrically (Krieg et al.
2016). Techniques like
Monte Carlo simulations are used to study the relationship between the hydrogen
bond dynamics of the solvent and the formation of micelles from amphiphiles.
Heinzelmann et al. performed Monte Carlo simulations using a solution of
amphiphiles in water to determine hydrogen bonding in water and micellar dynamics
between a non-micellized and a micellized state. The results depicted a gradual
decrease in the number of hydrogen bonds with the growth of mi
celles, while the
number of hydrogen bonds increased with an increase in the exposed surface area of
the solute (Heinzelmann et al. 2012). Hu et al. depicted that the complexation of
polydimethylsiloxane-graft-poly(ethylene oxide) (PDMS-g-PEO) and poly(acrylic
acid)-block-polyacrylonitrile (PAA-b-PAN) led to the formation of micelles. This
was due to the hydrogen bonding between the ether oxygen and the carboxyl groups
of PEO and PAA (Hu et al. 2010).
58 A. Sikder et al.
3.3 Theories of Micelle Formation
The hydrophobic effect is responsible for amphiphilic molecules propensity to form
micelles in an aqueous solution, as discussed in the previous section. The stoichio-
metric constraint that prevents the demixing of polar and hydrophobic groups is the
considerable gap between the micellar assembly and macroscopic phase separation
(Boles et al. 2016). This section discusses the different mechanisms governing the
formation of micelles. The structure-property relationship and compu ter simulation
methods utilized in the dynamics of micelle formation are also covered.
3.3.1 Law of Mass Action
Mass action law applies to the micellar system which is considered a homogenous
one-phase system (Chen et al. 2005). In this method, aggregation of surfactants is
regarded as a reversible chemical reaction and is represented by the following
equation:
X
i
nB $ M
wherein M represents micelle, n and B are the aggregation numbers.
The above
equation is a fast reaction because of the short half-lives of surfactant
and micelles (Rusanov 1993). Amphiphiles tend to form aggregates of various sizes
and shapes. In this law, the aggregation number portrays the role of the
stoichiometric coefficient. The aggregation number is directly proportional to the
surfactant concentration; however, an insignificant change in aggregation number is
observed in the CMC region (Maibaum et al.
2004).
3 Dynamics of Micelle Formation 59
3.3.2 Flory-Huggin’s Interaction Parameters
Paul Flory and Maurice Huggins introduced Flory- Huggin’s interaction parameters
to study interaction effects in their lattice model of polymeric solutions (Russell et al.
2014). It is a dimensionless quantity and is represented by χ. This parameter is
employed to demonstrate thermodynamic properties like swelling ability and solu-
bility equilibrium of amphiphilic micelles. The interaction parameter also postulates
the amount of inconsistency between the subsequent amphiphilic blocks. The degree
of microphase segregation is governed by the segregation product represented as χN.
This parameter takes into consideration the enthalpy of the solvent and polymer
mixture and is expressed as
χ = δ
1
- δ
2
ðÞ
2
V
RT
here, V is the molar volume of the solvent, δ
1
and δ
2
are solubility parameters of the
solvent and polymer.
According to this theory, two moieties depicting χ < 0.5 is said to be soluble,
whereas χ > 0.5 is designated as phase separation. A group of scientists determined
the solubility of eleven drugs in micelles and found that the solubility decreased with
an increase in χ (Forrest et al.
2006).
3.3.3 Molecular Dynamics
Molecular dynamics (MD) is an emerging tool for investigating atomistic and
mesoscale dynamics (Lau et al. 2018). MD was used for the first time to determine
the self-assembling nature of surfactants. The observations depicted that the
surfactants formed micelles in an aqueous medium, whereas reverse micelles were
obtained in a lipophilic medium (Klein and Shinoda 2008; Seif and Montazeri 2022).
The MD simulations compute and compare numerous equilibrium properties of the
surfactants starting from their target phase with the experimentally available raw
data (Farafonov et al.
2020). MD can also interpret the dynamic properties like
stability parameters and rheology of the formed micelles (Liu et al. 2019). The more
prominent role of MD simulations is to determine the aggregation of surfactant
molecules into micelles by investigating their surface structure and hydrophobic
nature (Jafari et al. 2018). The results obtained from MD simulations depicted that
the increase in the number of amphiphilic groups led to an increase in the aggrega-
tion of micelles (Crespo et al.
2020
). Luna et al. employed 1-alkyl-3-
methylimidazolium cations in an aqueous mixture of ionic liquids to determine the
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CMC and the formation of micelles using MD simulations. The simulation results
were consistent with those from the experimental evidence (Vicent-Luna et al.
2017).
60 A. Sikder et al.
3.3.4 Quantitative Structure-Property Relationship
The former discussed approach is inadeq uate for larger data sets and is a time-
consuming method. Simulations in MD encounter the necessity of enormous cost
and computational power (Liu et al. 2017). Hence, statistical methods like quantita-
tive structure-property relationship (QSPR) could be employed to estimate drug and
polymer compatibility in micelles (Upadhya et al. 2021). QSPR is extensively
utilized in toxicology and medicinal chemistry to determine the biocompatibility
and efficacy of small drugs (Cern et al. 2012). It involves the statistical analysis of
large data sets, forecasting drug solubilization in micelles, drug loading in micelles,
and cheminformatics study of micelles. Wu et al. utilized QSPR modeling and
genetic function approximation for determining the influence of polymer configura-
tion on the drug loading capacity of doxorubicin-loaded polymeric micelles. The
results depicted that QSPR modeling method was an optimized and good fitting
approach to predict drug loading capabilities in micelles (Wu et al. 2015).
3.4 Thermodynamics of Micelle Formation
Surfactants are composed of amphiphilic structures of hydrophilic and hydrophobic
groups. Hydrophobic chains repel the surrounding water molecules due to their
diminished affinity. This causes an increase in the surface free energy, which further
increases the Gibbs free energy at the micelle-water interface. Surfactants tend to
reduce their surface free energy to attain an equilibrium in two ways, i.e., by
concentrating at the surface via aligning the hydrophobic
chains toward the air,
known as adsorption (Reeve et al.
2021).
The other scenario includes the aggregation
of surfactants with hydrophobic chains oriented toward the interior while hydro-
philic chains face the exterior with a continuous aqueous phase. This process is
known as micellization (Jover et al. 2021). Micelle formation is governe d by a string
of factors like temperature, CMC, ionizing groups, aggregation number, hydropho-
bic and hydrophilic chain length and volume, continuous phase viscosity, etc.
(Sikder et al.
2022).
Standard thermodynamic parameters could help in understand-
ing the dynamics of the process undergone by surfactant molecules, i.e., either
adsorption or micelle formation. Generally, for ionized surfactants, the degree of
dissociation of the counter ion plays an essential role in governing micelle dynamics.
Xi et al. employed the postulates of the density functional theory to study thermo-
dynamics, alteration in the microstructure, and solubilization of blo
ck co-polymeric
micelles. Inhomogeneous statistical associating fluid theory was used to contemplate
the structure formation of block copolymers in dilute and concentrated solutions.
They unveiled that hexagonal, lamellar, and cubic mesophasic micelles existed with
an increase in concentration compared to spherical micelles at relatively dilute
concentrations. Temperature negatively influenced CMC, while aggregation number
positively correlated with the length of the hydrophobe (Xi et al.
2019).
3 Dynamics of Micelle Formation 61
3.5 Techniques to Evaluate Micelle Formation Dynamics
3.5.1 Isothermal Titration Calorimetry
Isothermal titration calorimetry considers the change in heat flow at a constant
temperature which is correlated with the meas urement of assembly in micelles.
Kadam and colleagues visualized the exothermic heat of Tetronic
®
T904 produced
in a sigmoidal manner. Concentration below CMC was distinguished by solid
enthalpy released, resulting in the disassembly of micelles to the generation of
monomers. A decrease in the enthalpy chain with an increasing concentration near
CMC demonstrated micelles’ formation. However, a significant drawback of this
method is that the CMC at temperatures lower than 25 °C and higher than 45 °C
cannot be accurately estimated. At lower temperatures, feeble enthalpy effects are
observed. In contrast, at higher temperatures, an association of surfactant monomers
to micelles is much higher and corresponds to pseudo-CMC, indicating erroneous
thermodynamic results for micellar formation (Kadam et al. 2010; Tso et al. 2020;
Sarolia et al.
2022).
3.5.2 Förster Resonance Energy Transfer
Förster resonance energy transfer (FRET) includes the exchange of energy from an
excited molecule termed as a donor to another molecule termed as acceptor. The
advantage of this technique includes the measurement of interaction of nearby
dipoles depending on Förster distance. Bos and coworkers employed FRET to
devise a model to study the underlying macromolecular exchange rates in complex
coacervate micelles among oppositely charged ions. They showed that the exchange
rate was strongly influenced by the ionic strength of the aqueous phase. The
understanding of thermodynamics was enriched by the structural association and
dissociation of micellar assembly. FRET is useful for studying micellar dynamics at
lower concentrations compared to other techniques preferably with ionic interactions
governing expulsion and insertion characteristics (Bos et al. 2021).
3.5.3 Conductometry
The alteration in conduction of surfactant molecules and assembled micellar
structures could result in the quantification of thermodynamic parameters governing
micelle formation. Thoppil et al. evaluated Gibb’s free energy, enthalpy and entropy
of micelle formation using conductometric principles. In the pre-micellar region, the
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þðÞ
increase in conductivity was observed with increasing surfactant concentration
attributed to the increasing amount of free ions. During micelle formation, the
alteration in slope led to the reduction of ionic moieties. Further, the decrease in
the slope of the post-micellar zone results in the effective loss of conducting ions
owing to the presence of counter ions on the surface. This further results in size
increment and lower micellar mobility (Thoppil et al. 2020
). The calculation of
Gibb’s free energy, enthalpy and entropy of micellization was done using the
application of pseudo-phase order equation respectively given by,
62 A. Sikder et al.
ΔG
m
° = 1 þ βðÞRT ln X
CMC
ΔH
m
° =- 1 β RT
2
∂ ln X
CMC
=
∂T
ΔS
m
° =
ΔH
m
° - ΔG
m
°
=
T
wherein β—degree of association of counter
ion, ΔG
m
—change in Gibb’s free
energy, ΔH
m
—change in enthalpy, and ΔS
m
—change in entropy of micelle
formation.
3.5.4 Dissipative Particle Dynamics
Dissipative particle
dynamics (DPD) is a stochastic mesoscopic simulation process
that delineates clust
ers of fragments collectively moving in a Lagrangian manner
when subjected to random, dissipative forces and conservative forces (Barnes et al.
2019). DPD utilizes a group of subtle intermingling particles which are employed to
simulate a fluid medium. Individual bead will symbolize severa l atoms (Ramezani
and Shamsara 2016). The dynamics behind these beads is governed by Newton’s
second law and is given as
dr
i
dt
= v
i
, m
i
dv
i
dt
= f
i
herein, m
i
is the mass of the bead which is assumed to be unity, r
i
, v
i
, f
i
are position
vector, velocity, and total force on the particle i. Masses of all beads is fixed as
1 DPD unit.
The convent
ional DPD cannot work in certain cases like non-isothermal and
two-phase fluid systems due to the lack of necessary degrees of freedom (Dzwinel
and Yuen 2000). To overcome this problem, extensions of DPD could be used like
Smoothed DPD (Müller et al. 2015), Energy Conserving DPD (Li et al. 2014), Fluid
Particle Model (Dzwinel et al. 2002), and Many-Body DPD (Arienti et al. 2011).
Wang and his coworkers performed DPD simulations to achieve the delivery of
multiple cargos using a single micellar structure. The results revealed that different
payloads occupied their peculiar preferred locations on the basis of their structure
(Wang et al.
2022).
3 Dynamics of Micelle Formation 63
3.6 Conclusion
Micelles are self-assembling aggregates designed using amphiphilic copolymers.
These nanocarriers are gaining highlights among researchers because of their ability
to improve the bioavailability of hydrophobic drugs, imparting target specificity with
sustained release, improving therapeutic efficiency, responding to stimuli, and
reducing toxicity. In this chapter, we have conveyed the fundamental dynamics
behind the micelle formation and the underlying theories. The micelles which are
fabricated from ACs are the perfect nanocarriers for drug delivery in the manage-
ment and diagnosis of different disease conditions. The role of ACs, types of block
copolymers, and how the value of packing parameters will influence the morphology
of micelles are discu ssed. The effect of CMC and hydrogen bonding in micelle
formation along with its solubility and stability is also conveyed through this
chapter. Aggregation number determines the size and morphology of the micelles
and is explained by the mass action law. The solubility of amphiphilic polymers is
further explained by the Flory-Huggin’s interaction parameter by taking enthalpy of
solvent into consideration.
Micelles are believed to be in a dynamic equilibrium with frequently deforming
and reforming. The MD simulation studies could be a useful tool to study the
behavior of active pharmaceutical ingredients and payloads in a solvent medium.
However, the unmet needs associated with this technique include the inability to
determine the values for large data sets and is an expensive method. QSPR modeling
was introduced to compute larger data sets in lesser time and cost.
The evaluation methods including dynamic light scatteri ng, differential scanning
calorimetry, etc. will determine the morphology, size, and CMC of micelles. How-
ever, the detailed dynamics of micelle formation could be only determined via
molecular level simulations like DPD. The variations in heat flow, conducting
property of surfactants, and energy transfer between moieties should be studied to
determine the formation of micelles which is also deliberated in this chapter.
Acknowledgments The authors would like to acknowledge the research funding support by the
Department of Pharmaceuticals (DoP), Ministry of Chemicals and Fertilizers, Govt. of India to
National Institute of Pharmaceutical Education and Research (NIPER), Hyderabad, INDIA.
Declaration of Interest The authors declare that there is no conflict of interest.
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