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Modelling of the aggregation process using the citrate synthesis of gold as case study

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Abstract

This work presents modelling for the aggregation process of metal nanoparticles following a theory proposed in literature. In this theory, metal atoms aggregate into particles of bigger size due to Van der Waal’s forces of attraction. Then, owing to the electrostatic forces of repulsion, the particles eventually stop aggregating and become stabilized. Based on this mechanistic description, we developed a model for the aggregation process. Because this process often occurs along with other processes, such as growth, we employed as case study the synthesis of gold nanoparticles by the citrate synthesis method for conditions where the aggregation process is decoupled from the growth process. Using this model, we calculated the seed particle sizes and compared them with the values previously reported in literature. Furthermore, we calculated the final particle sizes and compared them with experimental data. The results show excellent agreement.

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Abbreviations

Symbol :

Meaning Units

C :

Concentration mol/m3

C C :

Citrate mol/m3

CT, \( {C}_{Au{Cl_4}^{-}} \) :

Concentration of tetrachloroauric ion mol/m3

C Ct :

Concentration of all citrate species mol/m3

\( {C}_{H^{+}} \) :

Concentration of H+ ions mol/m3

\( {C}_{OH^{-}} \) :

Concentration of OH− ions mol/m3

C Au :

Concentration of gold in the particle phase mol/m3

C Pr1 :

Concentration of all other products from the reduction step, lumped together mol/m3

C Pr2 :

Concentration of all other products from the growth step, lumped together mol/m3

D s :

Seed correlation parameter −

D z :

Stability gradient correlation parameter −

e :

The charge on an electron, whose value C

E A :

Energy due to Van der Waal’s force of attraction J

E R :

Energy due to the charge repulsion J

E T :

the sum of the particles’ interaction energy due to Van der Waal’s force of attraction and that due the charge repulsion J

E agg :

the energy barrier to particle aggregation J

E max :

The maximum particles’ interactive energy attainable J

f(s):

number of particles per particle-length per total volume of fluid-particle mixture

1/(m3. m)

F z :

Stability gradient correlation parameter −

G s :

linear growth rate m/s

k n :

Rate constant for the nucleation step (m3)4/(mol5. s)

k h :

Rate constant for the growth step m/s

k r :

Rate constant for the reduction step [m3/mol]1.851/s

k p :

Rate constant for the passivation step m3/(mol. s)

k g :

Rate constant for the growth step m4/(mol. s)

K B :

Boltzmann constant J/K

m a :

Particles area shape factor −

m v :

Particles volume shape factor −

N av :

Avogadro ’ s number = 6.02e23−

p i :

The number concentration of an ion in the bulk of the solution 1/m3

R :

Universal gas constant 8.31J/(mol. K)

s :

Size m

s 0 :

Nucleus diameter (size of gold atom, i.e. 0.272 nm in the aggregation model)

m

s s :

Seed diameter m

s m :

Mean diameter with time m

s f :

Final mean diameter m

S :

Selectivity of the reduction step over the passivation step −

t :

Time s

t s :

Synthesis time s

T :

Temperature K

V :

volume of synthesis solution m3

W :

Stability factor −

\( {\overset{\sim }{\omega}}_A\left(\overline{s},\hat{s},t\right) \) :

Aggregation kernel m3/s

x :

Separation of particles m

y x :

relative mole fraction of CtH2− at the quasi-equilibrium pH −

y y :

relative mole fraction of CtH2− at the quasi-equilibrium pH −

Y i :

Molar mass of species i kg/mol

z i :

The charge on the ion C

Z :

The numerator of the stability gradient J

ϵ 0 :

The permittivity of free space F/m

ϵ c :

The dielectric constant of the solution −

κ :

the Debye-Huckel parameter 1/m

μ :

Fluid viscosity kg/(m. s)

ρ :

Molar density of gold mol/m3

ψ x :

Electrostatic charge potential V

τ :

Characteristic time s

τ p :

Reaction time for the passivation s

τ g :

Time for the growth step only in the citrate method s

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Acknowledgements

The authors are grateful to Dr. Luca Mazzei, Associate Professor in the Department of Chemical Engineering, University College London, and Dr. Michael Wulkow, managing director of Computing in Technology GmbH (CiT), the company that developed the numerical code Parsival, for their support and stimulating discussions. Emmanuel holds both Luca and Michael in very high esteem. Emmanuel Agunloye would also like to thank the Nigerian government for funding his PhD programme via the Petroleum Technology Development Fund and the National University Commission.

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Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, University of Lagos, Akoka, Lagos, Nigeria

    Emmanuel Agunloye & Mohammed Usman

Authors
  1. Emmanuel Agunloye
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  2. Mohammed Usman
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Correspondence to Emmanuel Agunloye.

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The authors declare that they have no conflict of interest.

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Research highlights

• New mathematical modelling for the aggregation of metal nanoparticles is presented.

• The modelling is based on the theory proposed by Polte (2015).

• This theory accounts for the interplay of attractive and repulsive forces among metal nanoparticles.

• The evolution of predicted nanoparticle size distributions closely agrees with those measured experimentally.

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Agunloye, E., Usman, M. Modelling of the aggregation process using the citrate synthesis of gold as case study. J Nanopart Res 22, 183 (2020). https://doi.org/10.1007/s11051-020-04871-1

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