Article pubs.acs.org/JPCC
A Thermodynamic View of Agglomeration Stanislav V. Sokolov,†,‡ Enno Kaẗ elhön,†,‡ and Richard G. Compton*,† †
Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford University, Oxford OX1 3QZ, United Kingdom S Supporting Information *
ABSTRACT: We report an entirely new view of solutions containing agglomerated nanoparticles, colloids or other particulates. Assuming their stability (a previously solved problem), we use a maximum entropy of mixing approach to predict the distribution of monomers, dimers, trimers, etc., in a solution of minimally interacting particles. The predictions are close to experiment, which approximates a log-normal distribution. We further consider the perturbations when interparticle forces operate and predict the resulting agglomerate size distributions.
■
INTRODUCTION Agglomeration1the reversible clustering of particles in solutionlies at the heart of diverse fundamental chemical processes associated with nanoparticulate and colloidal systems as well as having enormous technological and environmental importance. Traditional methods have focused on the stability of the systems and considered the forces acting between the agglomerated particles. Most notably, DLVO theory2−7 is based on considering the net potential energy between particles as a result of van der Waals attractive forces and electrostatic repulsions and seeks to predict the conditions under which the systems are stable or unstable with respect to agglomeration. Additionally steric forces particularly between polymeric surfaces have been implicated for example, where particles are ‘capped’.8−10 However, such approaches while identifying ‘stability’, offer little if any, insights into the distributions of particles found in agglomerating systems. Alternative approaches such as potential of mean force (PMF) have also been successfully applied to a wide range of systems including graphene,11 nanotubes,12 and proteins13 in the modeling of the temporal evolution of systems of interacting particles. While PMF models provide highly accurate results, they feature an inherent complexity and can require exceedingly long computation times for large systems. In the present paper we take a de novo view of agglomeration, noting its inherent reversibility and consider the process in the light of the entropy of mixing of an agglomerating system as an initial and possibly primary consideration. In particular, we propose a theory which complements existing approaches and places the entropy of mixing of the system as a central concept. In this way, presuming stability of the system, we are able to generate predictions of particle size distributions, specifically the number of monomer, dimer, trimer, etc. units which are in remarkably good agreement with experiments, and further explore the role of the enthalpy of formation of agglomerates in this process, where the entropy of mixing based approach shows that some agglomeration is expected even in the presence of mildly repulsive forces between the particles. © XXXX American Chemical Society
Moreover experimental studies of agglomerating systems typically show n-mers with n < 6 suggesting that entropy of mixing effects must contribute to the observed distributions. There is a wide variety of techniques used for characterizing agglomeration or aggregation processes such as dynamic light scattering (DLS),14 nanoparticle tracking analysis (NTA),15 cryo-TEM,16 and nanoimpacts.17 For the purposes of the present investigation, NTA was used to characterize agglomeration processes due to its simplicity and high accuracy that is well-established for metallic nanoparticles. Further we note the prevalence of log-normal distributions encountered in the experimental literature not only regarding the particle size distributions but generally and throughout a diversity of sciences:18−21 from the abundance of species in ecosystems to the distribution of minerals in the planet Earth’s crust. We show how the entropy of mixing concept can be consistent with the approximate generation of such distributions.
■
THEORY
We consider a system of initially Ninit monomers that may reversibly agglomerate forming clusters containing up to imax individual monomers. We further assume that all agglomerate states are equivalent in terms of their energy (i.e., there is no energetically relevant interaction between the agglomerating monomers) and that all monomers and agglomerates freely diffuse in a solution. The most probable distribution of agglomerate states will hence be the distribution that features the highest entropy. For the given system we can find this state by maximizing the system’s entropy of mixing S, which can be calculated as the mixing entropy of i different ideal species: Received: August 13, 2015 Revised: September 26, 2015
A
DOI: 10.1021/acs.jpcc.5b07893 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C imax ⎛ ⎞ max⎜⎜S = −kBN ∑ xi ln(xi)⎟⎟ ⎝ ⎠ i=1
(1)
where: imax
N=
∑ Ni i=1
Figure 1. Two agglomeration states of a tetramer. On the left, a close packed structure is shown featuring the maximum number of possible contact points; on the right, the case of the minimum number of contact points forming a chain-like structure is sketched.
(2)
and N is the total number of entities in solution after the agglomeration process, Ni is the number of agglomerates comprising i individual monomers (i.e., i = 2 for a dimer, i = 3 for a trimer, etc.), and xi = Ni N−1 is the mole fraction of each type of agglomerate present in the solution such that ∑ixi = 1. For a finite number of particles the calculation can be done numerically by considering all possible configurations of the system, evaluating the associated entropy of each state, and selecting the state which leads to the highest value of the entropy. A constraint on this variational problem is given by the fact that the total number of monomers (that are either in a monomeric or an agglomerate state) in the system remains constant, which can be formulated mathematically as
Table 1. Number of Contact Points Per Agglomeration State for Two Different Agglomeration Models
imax
Ninit =
∑ iNi i=1
(3)
(4)
where H is the enthalpy and T is the temperature. The enthalpy change caused by the agglomeration process can be estimated if we regard the agglomeration as the formation of virtual ‘bonds’ between monomers. These ‘bonds’ are formed at contact points between two monomers and may feature either positive or negative enthalpies for the cases of repulsive or attractive interactions, respectively. For a dimer, one contact point is formed; for a trimer, depending on the agglomeration model, two or three contact points are formed, and so on. By defining a fixed energy value for the enthalpy per contact point and considering the total number of contact points present in each agglomeration state, the enthalpy of the system can be estimated: imax
H=
∑ NH i bni i=1
number of contact points in the chain model (strong repulsion)
number of contact points in the close packed model (weak repulsion)
monomer dimer trimer tetramer pentamer hexamer octamer decamer
0 1 2 3 4 5 7 9
0 1 3 6 9 12 18 24
likely to be chain like. In the here-discussed case of repulsive particle interactions, we consider two limiting cases in order to demonstrate the contrasting behavior. We investigate distributions for the case of a minimum number of contact points observed in chain-like structures and compare it to the case of the highest possible number of contact points that is observed in the case of close packing. These two extreme configurations are exemplarily sketched for a tetramer in Figure 1. Real systems will adopt an intermediate between the two limiting cases. In Table 1 the number of contact points is shown for either model and various agglomerate states. The presented model is solely based on the number of contact points between the monomers and does not take into account the shape effect. Scalability of the Results. The above-described theory uses the total number of initial monomers Ninit as a parameter for the determination of the energetically most favorable distribution of agglomerate states. This, however, is a value that is typically not exactly known in experiments, which may potentially impose a constraint on the applicability of this theory. In the following, we therefore show that the resulting distributions of agglomeration states do not depend on this parameter and that the obtained distributions are hence freely scalable, even though the overall Gibbs energy of the system is, of course, affected. To this end, we first show that the constraint on the variation problem given in eq 3 is independent of Ninit by rewriting it in terms of the molar fractions xi of each agglomeration state. If we divide eq 3 by Ninit, we obtain the expression
where Ninit is the total number of monomers in the system before the agglomeration process started. Perturbations Arising from Particle Interactions. For a nonideal system where interactions between individual species take place, the most probable distribution will no longer have the maximum entropy, but the minimum Gibbs energy G, which can be defined in accordance with the following equation:
G = H − TS
agglomerate state
(5)
where Hb is the enthalpy per contact point and ni is the number of contact points for a given agglomerate state, which depends on the respective agglomeration model. The determination of the ni is complicated as the observed behavior is strongly dependent on the strength of interparticle interaction. Particles experiencing strong attractive forces tend to form multiple contact points, which is energetically favorable and may result in closely packed structures if the change in enthalpy is large enough. Strongly repulsively interacting particles are less likely to form contact points due to the associated energetic costs and the resulting structures are more
imax
1=
∑ ixi i=1
(6)
As this expression is independent of Ninit, the constraint (eq 3) must be equally independent of Ninit, and hence applicable to systems of any size. In the next step, we determine the dependency of the total number N of all monomers and agglomerates in solution (i.e., the sum of all monomers, dimers, etc.) and their distribution Ni on Ninit. To this end, we consider a system of agglomerates that B
DOI: 10.1021/acs.jpcc.5b07893 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C was formed using a sufficiently large number of initial monomers, allowing us to treat N and Ni as real numbers rather than integers. Each microstate of a scalable system can then be expressed through a sequence of imax − 1 coefficients ci that each represent the mole fractions of monomers bound in a certain agglomerate state, which comprises i monomers per agglomerate. In contrast to the molar fractions xi that represent the fraction of all agglomerates N in the system that are bound in a certain agglomerate state i, the coefficients ci hence represent the fraction of initial monomers that are bound in a certain agglomerate state. Using this definition of ci, the distribution of the absolute numbers of agglomerates Ni found in solution can then be expressed through Ni = ciNinit
i < imax
for
Table 2. Effect of the Number of Considered Agglomeration States on the Computation Time for a Starting Population of 200 Monomers
imax − 1
∑
(7)
i=1
⎞ ci⎟⎟ ⎠
(8)
which are all proportional to Ninit and hence linealry scale with the number of initial monomers in the system. The total number of all monomers and agglomerates in solution N = ∑i Ni, which is given by the sum of all Ni, must hence be equally proportional to Ninit. Having shown that N and Ni both linearly scale with Ninit, we can conclude that the mole fractions xi = Ni/ N of the agglomerates must be independent of Ninit as well. In the last step, we introduce an expression for the Gibbs energy G per agglomerate by dividing Equation 4 by N: G′ =
G = ∑i Ni
imax
imax
∑ xiHbni − kBT ∑ xi ln(xi) i=1
i=1
average computation time (min)
