Diffusion Fundamentals (E-Journal)

Druzhinin, A. V.

Podgornyy, D. A.

Akinin, A. B.

Bykov, A. S.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Influence of heat treatment on magnetic properties of Cu-Sn-Co-based 
materials produced by powder metallurgy 
 
A.V. Druzhinin1*, D.A. Podgornyy1, A.B. Akinin2, A.S. Bykov1 
 
1National University of Science and Technology MISiS , Moscow, Russian Federation 
2Research Institute of Goznak, Moscow, Russian Federation  
*email: druzhininsas@mail.ru 
 
In the foundation of this work is the issue for production of non-magnetic materials with 
predetermined weak magnetic properties. Preferable solution is to dope these materials with 
ferromagnetic impurities in low concentrations. One of the efficient ways to obtain such materials is 
the use of powder metallurgy which gives an opportunity to produce complex powder compositions 
with variety of components in precisely specified concentrations. 
As an research object powder compositions of systems Cu-Sn (8 wt. % Sn) and Cu-Sn-P (8 wt. % Sn; 
0,5 wt. % P) with the addition of 1,5 wt. % Co were chosen. After mixing and pressing all samples 
were heat treated at 800 ˚C with different cooling rates. 
Noticeable solubility of components in Cu-Co system is observed at high temperatures [1]. This fact 
makes it possible to form specified magnetic properties only by formation of metastable solid solution 
with the further annealing in order to exclude Co particles from Cu (reduce Cu concentration in Co 
solid solution) and change the quantity of ferromagnetic domains in material [2]. Experimental results 
in this work with fast cooled Cu-Sn-Co system samples showed the same except there was no 
solubility of Co in Cu-Sn matrix. Anyway Co-Cu solid solution formation reduces sample 
magnetization in comparison with the pressed one. Formation of Cu-Sn-Co solid solution was 
discovered only after Cu3Sn chemical compound started to grow due to lower cooling rates. In the 
second case saturation magnetization became lower than for fast cooled sample because diffusion of 
Co into Cu-Sn solid solution was appeared and consequently the amount of ferromagnetic Co 
decreased. 
In the fast cooled samples of Cu-Sn-P-Co system besides the partially dissolution of Cu in Co 
diffusion of phosphorus in Cu-Sn matrix with further formation of Co2P chemical compound was 
occurred. Presence of cobalt phosphide decreased magnetic properties drastically because of 
paramagnetism of Co2P [3]. In slow cooled samples additional phase of eutectic copper phosphide 
Cu3P was discovered and saturation magnetization rose. Investigation of samples with various cooling 
rates proved that with the formation of tin-rich phase caused the dissolution of Co2P with simultaneous 
diffusion of phosphorus to the boundaries of Cu3Sn phase and formation of Cu3P.  
 
References 
[1] T. Nishizawa, K. Ishida: The Co-Cu (Cobalt-Copper) System. Bulletin of Alloy Phase 
Diagrams 2, 161–165 (1984) 
[2] A. Garcia Prieto, M. L. Fdez-Gubieda, C. Meneghini, A. Garcia-Arribas: Correlating among 
the structural and magnetic properties of CoCu granular alloys. Journal of Applied Physics 10, 8596–8598 
(2002) 
[3] S. Fujii, S. Ishida, S. Asano: Structures and magnetic properties of Fe2P, Co2P and CoMnP. 
Journal of Physics F: Metal Physics 18, 971–980 (1988) 
 
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English

Influence of heat treatment on magnetic properties of Cu-Sn-Co-based materials produced by powder metallurgy

Kiryukhantsev-Korneev, Ph.V.

Kuptsov, K.A.

Sagalova, T.B.

Shvindina, N.V.

Bondarev, A.V.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Diffusion-barrier properties and thermal stability of TiAlSiCN, 
TiAlSiCN/SiBCN, and TiAlSiCN/AlOx films 
 
Ph.V. Kiryukhantsev-Korneev*, K.A. Kuptsov, T.B. Sagalova, N.V. Shvindina, A.V. Bondarev 
 
National University of Science and Technology “MISIS”, Moscow, Russia 
*kiruhancev-korneev@yandex.ru  
 
The Ti-Al-N protective coatings are widely used in practice (metal cutting and forming tools, 
roller bearings and other machine parts) because of their high mechanical and tribological properties, 
good thermal stability and excellent oxidation resistance. The Si incorporation into the Ti-Al-N 
coating resulted in the formation of nanocomposite structure with the (Ti,Al)N crystallites embedded 
in the amorphous SiNx matrix. Nanocomposite structure of the Ti-Al-Si-N coatings promoted to the 
increasing of the hardness and maximal usage temperature. C-doped Ti-Al-N coatings are 
characterized by improved tribological properties due to the solid lubricant effect of the amorphous 
carbon. Recently developed Ti-Al-Si-C-N coating on Al2O3 model substrate show high thermal 
stability of structure and extremely high hardness ~40 GPa up to 1300°C. But for the practical 
application it’s very important to have the high diffusion barrier properties of coatings. The aim of the 
present work is to investigate the diffusion barrier properties, thermal stability, and oxidation 
resistance of the Ti-Al-Si-C-N base coatings as well as multilayer (ML) TiAlSiCN/SiBCN and 
TiAlSiCN/AlOx coatings with additional layers which improve listed characteristics. 
TiAlSiCN coatings were deposited using magnetron sputtering of composite targets produced 
by self-propagating high-temperature synthesis. ML coatings were manufactured by step-by-step 
magnetron sputtering (TiAlSiCN layer) and ion sputtering (SiBCN or AlOx layers). NiCrAlW disks 
and alumina plates were used as the substrates. To reveal the diffusion barrier properties and oxidation 
resistance the as-deposited coatings were subjected to the air annealing at temperatures from 800 to 
1000°C. Additional thermo-cycling experiments for 20, 50, and 100 cycles (20-1000°C) were fulfilled. 
Thermal stability of coatings was estimated by vacuum annealing. The structure, chemical and phase 
composition of as-deposited and thermal-treated coatings were studied by means of glow discharge 
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optical emission spectroscopy, X-ray diffraction, transmission and scanning electron microscopy. The 
coatings were characterised in terms of their hardness, elastic modulus, and elastic recovery.  
Results obtained show that in the ML coatings the SiBCN and AlOx layers play role of the 
barriers against Ni diffusion from the substrate into the coating and thus significantly improve the 
thermal stability of whole ML coating. TiAlSiCN/SiBCN and TiAlSiCN/AlOx coatings on Al2O3 
substrates also demonstrate better oxidation resistance (critical temperature of 1100°C) and thermal 
stability (1400°C) than basic TiAlSiCN coating. Combination of high diffusion barrier properties, high 
oxidation resistance and mechanical properties makes TiAlSiCN/SiBCN and TiAlSiCN/AlOx coatings 
promising candidates for protective purposes to be used in different high-temperature applications. 
 
 
2


Diffusion-barrier properties and thermal stability of TiAlSiCN, TiAlSiCN/SiBCN, and TiAlSiCN/AlOx films

Laurenz, E.

Füldner, G.

Schnabel, L.

Schmitz, G.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Application of frequency response methods for measuring heat and mass 
transfer in sorption materials for heat transformation 
 
E.	Laurenz1,2*,	G.	Füldner1,	L.	Schnabel1,	G.	Schmitz2	
 
1Fraunhofer Institute for Solar Energy Systems ISE, Freiburg, Germany 
2Hamburg University of Technology, Hamburg, Germany 
*eric.laurenz@ise.fraunhofer.de 
 
Periodically operated heat transformers use the difference in vapor pressure of the pure 
refrigerant (here water) and the refrigerant bound to a sorption material. They allow the 
construction of efficient thermally driven heat pumps, chillers or compact heat storages [1]. 
The principal component of such equipment is the sorption heat exchanger (SHX) which is in 
contact with the sorption material on one side. It is operated alternatingly between sorption 
and desorption of the refrigerant at low pressure/low temperature and at high pressure/high 
temperature respectively. The SHX acts as thermally driven compressor. In combination with 
an evaporator and a condenser, heat pumps and chillers are realized. Next to conventional 
sorption materials like zeolites and silica gels, hygroscopic salts have a high potential for the 
use in SHX [2, 3]. 
Mass transfer (mostly diffusion) of the refrigerant and the transfer of heat of sorption 
in the SHX are important transport mechanisms that often limit the overall performance. 
Design and optimization depend on a thorough understanding of these mechanisms. 
Challenges are their typically non-linear nature and the strong coupling between heat and 
mass transfer. 
One approach to identify the transport coefficients is the volume swing frequency 
response (FR) method [4]. The experiment depicted in Figure 21 has been set up. A sample of 
sorption material (e.g. a cutout from a SHX) is placed in a small measuring chamber with a 
pure refrigerant atmosphere. After a preconditioning phase, the chamber volume is excited 
sinusoidally so that the pressure and sample surface temperature response with sinusoidal 
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1
 
 
signals. The amplitude ratios and phase shifts between excitation and response are measured 
for different frequencies. By comparison to a heat and mass transfer model the transport 
coefficients (e.g. diffusion coefficients) can be identified. 
First FR measurements were conducted on zeolite Y samples with water as refrigerant. 
Results are presented and compared to results from pressure jump and temperature jump 
experiments [5]. 
 
Figure 21: Schematic of the new volume swing frequency response apparatus 
 
References 
[1] Y.D. Tu, R.Z. Wang, T.S. Ge, X. Zheng, Scientific Reports 7 (2017) 40437.  
[2] O. Hallström, G. Füldner, H.-J. Spahn, L. Schnabel, F. Salg, Energy Procedia 48 
(2014) 67–76. 
[3] Y. Aristov, G. Restuccia, G. Cacciola, V. Parmon, Applied Thermal Engineering 
22 (2002) 191–204. 
[4] L.M. Sun, F. Meunier, P. Grenier, D.M. Ruthven, Chemical Engineering Science 
49 (1994) 373–381. 
[5] A. Frazzica, G. Füldner, A. Sapienza, A. Freni, L. Schnabel, Applied Thermal 
Engineering 73 (2014) 1022–1031. 
2


Application of frequency response methods for measuring heat and mass transfer in sorption materials for heat transformation

Dvoyashkin, M.

Zaheer, M. A.

Wilde, N.

Haase, J.

Gläser, R.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Transport of methyl Oleate in hierarchically structured titaniumsilicalite-1 
catalysts probed by means of diffusion NMR  
 
M. Dvoyashkin1, M. A. Zaheer1,*, N. Wilde1, J. Haase2, R. Gläser1 
 
1Institute of Chemical Technology, Universität Leipzig, 04103 Leipzig, Germany. 
2Institute of Experimental Physics II, Universität Leipzig, 04103 Leipzig, Germany. 
*ayman.zaheer@uni-leipzig.de 
 
 
Diffusion of long-chain hydrocarbons in confined environments remains in the focus of modern 
science. It is relevant, e.g., in the heterogeneously catalyzed oil cracking, transesterification of oils for 
biodiesel production or in biodiesel epoxidation. Among the major challenges associated with the 
utilization of nanoporous catalysts in these type of conversions are diffusion limitations. The latter 
occur under conditions when the reaction time scale is notably smaller than the time needed for 
delivery of reactants to the catalytic sites and/or product removal. The diffusion process in such case 
becomes dominating and determines the overall reaction rate. Thus, obtaining quantitative information 
on the transport characteristics of long-chain hydrocarbons confined to nanopores is vital.  
In the present contribution, we demonstrate the applicability of the pulsed field gradient (PFG) NMR 
for probing diffusion of methyl oleate (C19H36O2) adsorbed into the mesopore system of post-
synthetically treated titanium silicalite-1 (TS-1) at room temperature. This catalyst is known to be 
highly selective in the liquid-phase oxidation of a wide range of hydrocarbons using aqueous H2O2 as 
the sole oxidant [1]. It is suggested that the introduction of larger (meso)pores might facilitate 
diffusion of methyl oleate into the crystals of hierarchically structured TS-1. In this context, the main 
focus of the present contribution is placed on a proof-of-principle demonstration of the direct 
assessment of self-diffusion coefficients of the long-chain methyl oleate adsorbed within TS-1 
possessing the mesopores by means of PFG NMR [2].  
 
References 
[1] N. Wilde, M. Pelz, S.G. Gebhardt, R. Gläser: Highly efficient nano-sized TS-1 with micro-/mesoporosity 
from desilication and recrystallization for the epoxidation of biodiesel with H2O2. Green Chem. 17, 
3378-3389 (2015). 
[2] J. Kärger, H. Pfeifer, W. Heink: Principles and application of self-diffusion measurements by nuclear 
magnetic resonance. Adv. Magn. Reson. 12, 2-89 (1988). 
 
 
Acknowledgment 
M.D. acknowledges financial support by a Max‐Buchner‐Forschungsstipendium #3427 (DECHEMA 
e.V.). 
 
 
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1


Transport of methyl Oleate in hierarchically structured titaniumsilicalite-1 catalysts probed by means of diffusion NMR

Tietz, Frank

Ma, Qianli

Guin, Marie

Naqash, Sahir

Tsai, Chih-Long

Guillon, Olivier

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Scandium-Substituted Na3Zr2(SiO4)2(PO4) as Superior Sodium-Ion
Conductors
Frank Tietz, Qianli Ma, Marie Guin, Sahir Naqash, Chih-Long Tsai, Olivier Guillon
1 Institute of Energy and Climate Research – IEK-1, Forschungszentrum Jülich GmbH, 52425
Jülich, Germany; 2 Jülich Aachen Research Alliance, Section JARA-Energy, 52425 Jülich,
Germany 3 Helmholtz-Institute Münster – Ionics in Energy Storage, Forschungszentrum Jülich
GmbH, Corrensstraße 46, 48149 Münster, Germany;
E-Mail: f.tietz@fz-juelich.de
Room-temperature Na-ion batteries (NIBs) show advantages of unlimited sources, low
price of starting materials and easy recycling [1], which make them promising candidates
for stationary energy storage applications. NIBs in all-solid-state design are regarded as
the batteries of the next generation due to their non-leaking, non-volatile, non-flammable
electrolyte and adaptability to temperature changes [2]. Trivalent cation-substituted
Na3Zr2(SiO4)2(PO4)3 in NASICON structure are regarded as promising electrolyte
materials for solid-state or high-temperature batteries because of their high Na+ ion
conductivity and reliable chemical and physical stability. In this study, scandium has been
chosen as the substitution element in the solid solution Na3+x+yScxZr2−x(SiO4)y(PO4)3−y
(NSZSP), because among all the trivalent cations, the ionic radius of Sc3+ (74.5 pm) is
the closest to that of Zr4+ (72.0 pm), indicating that the substitution only generates a de-
ficiency in positive charge, but does not cause distortions in the crystal structure. NSZSP
has received hardly any attention so far and only few reports are available on this kind of
materials [3, 4]. However, in the past the performance of NSZSP was compared at high
temperatures (≈ 300 ◦C). Only a few conductivities at room temperature were mentioned,
and they were rather moderate (maximum 5.0 · 10−4 S · cm−1). In this study, a solution-
assisted solid-state reaction method was applied by which a series of NSZSP nano-powders
with favorable microstructures were prepared. With increasing scandium substitution,
the conductivity of sintered NSZSP samples also increases. Na3.4Sc0.4Zr1.6(SiO4)2(PO4)
shows the highest conductivity of the system. Higher amounts of scandium decrease the
conductivity due to the decreasing number of available vacancies for Na+ ion transport.
The total ionic conductivity of Na3.4Sc0.4Zr1.6(SiO4)2(PO4) reaches 4.0 · 10
−3 S · cm−1
at 25 ◦C [5]. To our knowledge, it is not only the best value for all materials with
NASICON-type structure, but also the best reported value for all polycrystalline oxidic
Na+ ion conductors. It is even comparable with common liquid electrolytes of NIBs.
[1] Towards Greener and more Sustainable Batteries for Electrical Energy Storage, D. Larcher,
J. M. Tarascon, Nature Chem. 7 (2015) 19.
[2] Superionic Glass-ceramic Electrolytes for Room-temperature Rechargeable Sodium Batteries,
A. Hayashi, K. Noi, A. Sakuda, M. Tatsumisago, Nature Commun. 3 (2012) 856.
[3] The Preparation, Structure, and Conductivity of Scandium-substituted NASICONs, M. A. Subra-
manian, P. R. Rudolf, A. Clearfield, J. Solid State Chem. 60 (1985) 172.
[4] Sodium and Oxygen Disorder in a Scandium-substituted Nasicon: A Time of Flight Neutron Powder
Diffraction Study of Na2.5Zr1.8Sc0.2Si1.3P1.7O12, P. J. Squattrito, P. R. Rudolf, P. C. Hinson,
A. Clearfield, K. Volin, J. Jorgensen, Solid State Ion. 31 (1988) 31.
[5] Scandium-substituted Na3Zr2(SiO4)2(PO4) Prepared by a Solution-assisted Solid-state Reaction
Method as Sodium-ion Conductors, Q. Ma, M. Guin, S. Naqash, C.-L. Tsai, F. Tietz, O. Guillon,
Chem. Mater. 28 (2016) 4821.
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1


Scandium-Substituted Na3Zr2(SiO4)2(PO4) as Superior Sodium-Ion Conductors

Striolo, Alberto

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Transport mechanisms in heterogeneous pore networks: 
from molecular to meso-scale observations 
 
Alberto Striolo* 
University College London, Department of Chemical Engineering 
London, United Kingdom 
* a.striolo@ucl.ac.uk 
 
Fluids under confinement exhibit interesting peculiarities and deviations from bulk behaviour. 
Understanding transport mechanisms and predicting transport properties is important for further 
developing applications such as catalysis and gas separation, and it is becoming essential for 
developing unconventional hydrocarbon reservoirs. Much can be learned from molecular simulations. 
We conducted equilibrium, and sometimes non-equilibrium molecular dynamics simulations for 
fluids, pure and mixed, confined within extremely narrow pores. We considered aqueous systems 
containing ethanol, H2S, and methane, as well as organic systems containing alkanes of various 
molecular weights and CO2. We will attempt to quantify and generalize the results, which suggest that 
depending on fluid composition, pore properties, and pore-fluid preferential interactions, a number of 
different mechanisms dictate the transport properties of the confined fluids. The results sometimes 
suggest radically different behaviour compared to bulk fluids, which is often the case when we analyse 
the solubility of gases (methane and H2S) in confined water. While detailed experimental verification 
of the observations can be problematic, quasi-elastic neutron scattering (QENS) has proven useful. We 
will present recent attempts at reconciling observations form atomistic molecular dynamics 
simulations and QENS, which suggest that synergistic combinations of experiments and simulations 
yield a powerful tool to discover transport mechanisms in confinement. However, to further practical 
applications one needs to up-scale these insights. We will present a kinetic Monte Carlo approach, 
which yields insights useful in particular for quantifying the rate-limiting mechanisms that currently 
hinder the production of shale gas. 
 
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Transport mechanisms in heterogeneous pore networks: from molecular to meso-scale observations

Beke, D. L.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
 
Atomistic interpretation of the interface transfer coefficients for 
interdiffusion in AB binary phase separating system 
 
D.L. Beke1* 
 
1Department of Solid State Physics, University of Debrecen, Hungary  
*dbeke@science.unideb.hu 
 
During intermixing in A/B diffusion couple the actual compositions (c’α and c’β) on the left as well as 
on the right hand side of the α/β interface gradually will decrease (from unity) as well as increase 
(from 0) until the equilibrium cα and cβ will be reached (α and β refer to the A- and B-rich phases, 
separated by the interface). The diffusion flux of A atoms across this interface, JI, is determined from 
the expression of atomic fluxes given in the modified Martin model [1,2]. Assuming that JI, is 
proportional to the deviations, Δα=c’α - cα and Δβ=cβ - c’β, from the equilibrium       
JI=(1/Ω)[KIαβΔα + KIβαΔβ), 
where Ω=ΩA=ΩB is the atomic volume. It is shown that the KIαβ and KIβα interface transfer 
coefficients are positive and equal to each other for symmetric miscibility gap and can be given as 
KIβα= KIαβ=K= zvaΓIαβcαξ. 
Here ΓIαβ  is the jump frequency from α to β phase across the interface, a is the lattice parameter zv is 
the vertical coordination number, ξ=[1+exp(ZV{cα-cβ}/kT)], V is the well-known solid solution 
parameter, kT has its usual meaning. The above expression justifies the conjecture, frequently used in 
the literature (see e.g. [3]), that only one interface transfer coefficient is enough for the description of 
mass transfer across an interface. For short diffusion times the finite value of ΓIαβ  will restrict the 
flux, leading to linear kinetics. It will also be shown that for the estimation of the critical interface shift 
(giving the transition from the interface to diffusion control, i.e. from linear to parabolic kinetics) the  
xc≅aexp[-ZM(1-2cβ)/kT] 
relation can be used, where a is the lattice spacing and the exponential factor is the ratio of factors 
describing the composition dependence of the jump frequencies in the β phase and across the 
interface, respectively. Thus xc is independent of the value of V and only the composition dependence 
of the jump frequencies is important. For composition independent jump frequencies xc ≅ a, (i.e. it can 
not be detected), while for the case when the diffusivity changes by seven orders of magnitude from 
pure A to B [2,4], xc is about 150nm.   
      
[1] G. Martin: Phys Rev. B 41, 2279-2283 (1990), p.2279 
[2] D.L. Beke, Z. Erdélyi: Phys. Rev. B 73, 035426-1-7 (2006),   
[3] U. Gössele, K.N. Tu, J, AppL Phys. 53, 3252-3260 (1982),  
[4] Z. Erdélyi, M. Sladecek, L -M. Stadler, I. Zizak, G. A. Langer, M. Kis-Varga, D. L. Beke, and B. Sepiol, 
Science 306, 1913-1915 (2004)  
 
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Atomistic interpretation of the interface transfer coefficients for interdiffusion in AB binary phase separating system

Erdélyi, Z.

Gajdics, B.

Tomán, J. J.

Radnóczi, G.

Bokányi, E.

Misják, F.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
Size dependent spinodal decomposition in Cu-Ag nanoparticles 
 
Z. Erdélyi2, B. Gajdics, J.J. Tomán, G. Radnóczi1, E. Bokányi1, F. Misják1 
1Research Centre for Energy Research, Hungarian Academy of Sciences, H-1525 Budapest, 
PO Box 49, Hungary 
2Department of Solid State Physics, University of Debrecen, P.O. Box 2, H-4010 Debrecen, 
Hungary 
 
Nanoparticles are of considerable interest, owing to their size-dependent properties, different 
from those of bulk materials. For revealing internal atomic processes in them, individual 
nanoparticles of Cu-Ag alloys were grown by direct current (DC) magnetron sputtering. 
Phase-separation during growth in Cu-Ag particles was found to be size- and composition-
dependent. Particles below 5 nm in diameter grow as a solid solution of the components for 
all compositions (15–80 at% Ag). In the low Ag content range (15 and 30 at% Ag) phase-
separation occurs only for particles above 5 nm in diameter. The separation into Cu-rich and 
Ag-rich domains, when observed, takes place by spinodal decomposition for all particle sizes. 
In particles undergoing incomplete coalescence, phase-separation occurs even if the diameter 
of the colliding particles is below 5 nm. In the higher Ag content range (60-80 at%), however, 
no phase-separation is observed until coalescence sets in. Lattice parameter measurements in 
alloy particles of 30 at% Ag revealed that the miscibility gap in individual particles varies 
between 70 and 90 at%. Calculation of the composition dependence of the critical length for 
spinodal decomposition based on the Cahn-Hilliard theory provided quantitative explanation 
for the observed phenomena. [1] Besides, computer simulations using the Stochastic Kinetic 
Mean Filed model (skmf.eu, open source) [2] have also been performed which confirmed the 
results of the analytical calculations. 
 
[1] G. Radnóczi, E. Bokányi, Z. Erdélyi, F. Misják, ACTA MATERIALIA 123: pp. 82-89. 
(2017) 
[2] Erdélyi Z, Pasichnyy M, Bezpalchuk V, Tomán JJ, Gajdics B, Gusak AM, COMPUTER 
PHYSICS COMMUNICATIONS 204: pp. 31-37. (2016) 
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1


Size dependent spinodal decomposition in Cu-Ag nanoparticles

Grebenkow, D. S.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
Universal formula for the mean first passage time in planar domains 
 
D.S. Grebenkov1,2* 
 
1 Laboratoire de Physique de la Matière Condensée,CNRS -- Ecole Polytechnique,  
University Paris-Saclay, 91128 Palaiseau, France 
 
2 Interdisciplinary Scientific Center Poncelet (ISCP), CNRS, 
Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia 
*Email : denis.grebenkov@polytechnique.edu  
 
We present a general exact formula for the mean first passage time (MFPT), , from a fixed point 
inside a planar simply connected domain to a connected escape region  on the boundary  
[1]. The underlying mixed Dirichlet-Neumann boundary value problem is conformally mapped onto 
the unit disk, solved exactly, and mapped back (Fig. 1). The resulting formula (1) for the MFPT is 
valid for an arbitrary space-dependent diffusion coefficient , while the leading logarithmic term in 
Eq. (3) is explicit, simple, and remarkably universal. In contrast to earlier works [2], we show that the 
natural small parameter of the problem is the harmonic measure  of the escape region, not its 
perimeter. The conventional scaling of the MFPT with the area of the domain is altered when diffusing 
particles are released near the escape region. These findings change the current view of escape 
problems and related chemical or biochemical kinetics in complex, multiscale, porous or fractal 
domains, while the fundamental relation to the harmonic measure opens new ways of computing and 
interpreting MFPTs. 
 
 Original problem in any planar domain          Conformal mapping  Modified problem in the disk 
 
                                                                             
                                                                                                            
                                                               
 
 
                                                                                                 
 
Figure 5: An arbitrary simply connected domain  (left) can be mapped onto the unit disk (right) by a 
conformal mapping  such that the origin of the disk is mapped onto the starting point . The escape 
region (in red) is mapped onto the arc , where  is the harmonic measure of  
seen from . 
 
 
 
 
 
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The universal formula for the MFPT reads 
                                                      
(1) 
where 
                                                                                            
(2) 
The asymptotic expansion of Eq. (1) over the natural small parameter, the harmonic measure , 
yields 
                                                                             
(3) 
where the explicit formulas for the coefficients ,  are given in [1], and 
 is the harmonic mean of the diffusion coefficient . The leading logarithmic 
term substitutes the conventional scaling  with the normalized perimeter  [2]. 
 
References 
[1] D. S. Grebenkov: Universal formula for the mean first passage time in planar domains, Phys. 
Rev. Lett. 117, 260201 (2016).  
[2] D. Holcman and Z. Schuss, The Narrow Escape Problem, SIAM Rev. 56, 213-257 (2014). 
 
 
2


Universal formula for the mean first passage time in planar domains

Guskov, A.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
Spinodal decomposition of solutions during crystallization 
 
A. Guskov  
 
ISSP RAS, Chernogolovra, Russia 
 
Motivation research: Explanation and prediction of the distribution of components in the 
crystallization solutions causes great difficulties. The modern theory of phase transitions cannot 
explain the results of many experiments of interphase mass transfer. One reason for this is the 
assumption that during crystallization the solution is in the metastable state. The decomposition of the 
solution occurs by binodal scenario in this case. Crystallization nuclei form and grow in solution. The 
purpose of this study to show that in many cases the solution during crystallization is in an unstable 
state [1,2]. The unstable condition leads to decomposition the solution by spinodal scenario. The 
unstable solution decomposes continuously in the whole volume in this case. The distribution 
component at the spinodal decomposition is determined by arbitrarily small perturbations of 
temperature or concentration.  
Methodology & Theoretical Orientation: Experimental demonstration of spinodal 
decomposition of the solution is conducted video shooting process of decomposition of an aqueous 
solution of bromthymol blue while its crystallization. Locally - configuration thermodynamic model is 
used to explain the state changes of the solution during the phase transition [2,3]. This model allowed 
adding additional coordinate - mixing energy in the equilibrium phase diagram. Boundary of the 
spinodal area of the solution is built into the new coordinates. Spinodal defines curve the dynamic 
equilibrium that must be considered in the calculations of the processes of interphase mass transfer.  
Conclusion & Significance: Spinodal decomposition of the solution explains the process of 
formation of a periodic distribution of the eutectic composites. The layer of the unstable solution is 
localized in front of the unstable interface. The unstable solution decomposes into phases, which have 
a composition close to the eutectic composition of the solid phases. The period of alternation of these 
phases is set by the period of instability of the interface [4]. Experiments show that the formation of 
dendrites in the mushy zone (Fig. 1) and extremum of the component concentration close to interface 
also occurs in the spinodal decomposition scenario. The possibility of spinodal decomposition of the 
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solution during its crystallization significantly alters the representation of interphase mass transfer. 
Many cases of the redistribution component during the phase transition cannot be explained without 
taking into account of spinodal decomposition. 
 
Figure 6: Formation of a dendrites branches at the borders of the eutectic structure under crystallization. 
 
References 
[1]  Guskov A.P., Nekrasova L.P., Ershov A.E., Kogtenkova O.A. The decomposition of the 
solution in the interface under directional solidification. Materilovedenie - Materials Science,  no. 10, pp. 
10-15, (2013) (in Russian). 
[2] A. Guskov, L. Nekrasova. Decomposition of Solutions in Front of the Interface Induced by 
Directional Crystallization. Journal of Crystallization Process and Technology., 3, 170 – 174, (2013). 
[3] Guskov A.P. The decomposition of the solution during the formation of eutectic composites. 
COMPOSITES and NANOSTRUCTURES, Volume 8, N3, pp. 2-15, (2016). (in Russian) 
[4] A.Guskov, Qualitative Analysis of Interface Behavior under First Order Phase 
Transition. Journal of Crystallization Process and Technology, 2, pp. 25-29 (2012). 
 
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Spinodal decomposition of solutions during crystallization

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