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Statistical thermodynamics of irreversible aggregation: the sol-gel transition.

Matsoukas T - Sci Rep (2015)

Bottom Line: Here we show that the connection to thermodynamic phase transition is rigorous.We develop the statistical thermodynamics of irreversible binary aggregation in discrete finite systems, obtain the partition function for arbitrary kernel, and show that the emergence of the gel cluster has all the hallmarks of a phase transition, including an unstable van der Waals loop.We demonstrate the theory by presenting the complete pre- and post-gel solution for aggregation with the product kernel.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802.

ABSTRACT
Binary aggregation is known to lead, under certain kinetic rules, to the coexistence of two populations, one consisting of finite-size clusters (sol), and one that contains a single cluster that carries a finite fraction of the total mass (giant component or gel). The sol-gel transition is commonly discussed as a phase transition by qualitative analogy to vapor condensation. Here we show that the connection to thermodynamic phase transition is rigorous. We develop the statistical thermodynamics of irreversible binary aggregation in discrete finite systems, obtain the partition function for arbitrary kernel, and show that the emergence of the gel cluster has all the hallmarks of a phase transition, including an unstable van der Waals loop. We demonstrate the theory by presenting the complete pre- and post-gel solution for aggregation with the product kernel.

No MeSH data available.


Related in: MedlinePlus

Cluster distributions in a population with M = 40 as a function of the number of clusters N.Shaded curve: mean distribution (exact calculation by direct enumeration of all distributions); vertical sticks: most probable distribution (exact calculation); symbols: Monte Carlo simulation (average of 5000 repetitions); dashed line: equation (19) (thermodynamic limit) with θ = 1 − N/M in the pre-gel region (N ≥ M/2), and θ = N/M in the post-gel region.
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f2: Cluster distributions in a population with M = 40 as a function of the number of clusters N.Shaded curve: mean distribution (exact calculation by direct enumeration of all distributions); vertical sticks: most probable distribution (exact calculation); symbols: Monte Carlo simulation (average of 5000 repetitions); dashed line: equation (19) (thermodynamic limit) with θ = 1 − N/M in the pre-gel region (N ≥ M/2), and θ = N/M in the post-gel region.

Mentions: These calculations are compared in Figure 2, which shows selected distributions ranging from N = 33 (early stage of mostly small clusters) to N = 6 (nearly fully gelled). Since the MPD is an actual member of the ensemble, it contains integer numbers of clusters. The mean distribution is a composite of the entire ensemble and is not restricted to integer values. The giant cluster forms at N* = 22 and its presence is seen very clearly in the MPD. The gel phase is less prominent in the mean distribution because its peak is smeared by lateral fluctuations. Not all distributions in the vicinity of the MPD contain a giant cluster; as a result, the gel fraction grows smoothy at the gel point. In the sol region 1 ≤ (M − N + 1)/2, the theoretical distribution from equation (19) and the mean distribution are in excellent agreement. The analytic result eventually breaks down when N → 1 (the thermodynamic limit is violated at this point), yet even with N as small as 6, agreement with theory remains acceptable. The mean distribution from MC is practically indistinguishable from that by the exact calculation. This confirms the validity of equation (14), which forms the basis of the exact calculation.


Statistical thermodynamics of irreversible aggregation: the sol-gel transition.

Matsoukas T - Sci Rep (2015)

Cluster distributions in a population with M = 40 as a function of the number of clusters N.Shaded curve: mean distribution (exact calculation by direct enumeration of all distributions); vertical sticks: most probable distribution (exact calculation); symbols: Monte Carlo simulation (average of 5000 repetitions); dashed line: equation (19) (thermodynamic limit) with θ = 1 − N/M in the pre-gel region (N ≥ M/2), and θ = N/M in the post-gel region.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4352918&req=5

f2: Cluster distributions in a population with M = 40 as a function of the number of clusters N.Shaded curve: mean distribution (exact calculation by direct enumeration of all distributions); vertical sticks: most probable distribution (exact calculation); symbols: Monte Carlo simulation (average of 5000 repetitions); dashed line: equation (19) (thermodynamic limit) with θ = 1 − N/M in the pre-gel region (N ≥ M/2), and θ = N/M in the post-gel region.
Mentions: These calculations are compared in Figure 2, which shows selected distributions ranging from N = 33 (early stage of mostly small clusters) to N = 6 (nearly fully gelled). Since the MPD is an actual member of the ensemble, it contains integer numbers of clusters. The mean distribution is a composite of the entire ensemble and is not restricted to integer values. The giant cluster forms at N* = 22 and its presence is seen very clearly in the MPD. The gel phase is less prominent in the mean distribution because its peak is smeared by lateral fluctuations. Not all distributions in the vicinity of the MPD contain a giant cluster; as a result, the gel fraction grows smoothy at the gel point. In the sol region 1 ≤ (M − N + 1)/2, the theoretical distribution from equation (19) and the mean distribution are in excellent agreement. The analytic result eventually breaks down when N → 1 (the thermodynamic limit is violated at this point), yet even with N as small as 6, agreement with theory remains acceptable. The mean distribution from MC is practically indistinguishable from that by the exact calculation. This confirms the validity of equation (14), which forms the basis of the exact calculation.

Bottom Line: Here we show that the connection to thermodynamic phase transition is rigorous.We develop the statistical thermodynamics of irreversible binary aggregation in discrete finite systems, obtain the partition function for arbitrary kernel, and show that the emergence of the gel cluster has all the hallmarks of a phase transition, including an unstable van der Waals loop.We demonstrate the theory by presenting the complete pre- and post-gel solution for aggregation with the product kernel.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802.

ABSTRACT
Binary aggregation is known to lead, under certain kinetic rules, to the coexistence of two populations, one consisting of finite-size clusters (sol), and one that contains a single cluster that carries a finite fraction of the total mass (giant component or gel). The sol-gel transition is commonly discussed as a phase transition by qualitative analogy to vapor condensation. Here we show that the connection to thermodynamic phase transition is rigorous. We develop the statistical thermodynamics of irreversible binary aggregation in discrete finite systems, obtain the partition function for arbitrary kernel, and show that the emergence of the gel cluster has all the hallmarks of a phase transition, including an unstable van der Waals loop. We demonstrate the theory by presenting the complete pre- and post-gel solution for aggregation with the product kernel.

No MeSH data available.


Related in: MedlinePlus