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Nanoparticle diffusivity in narrow cylindrical pores
International Journal of Heat and Mass Transfer 114 (2017) 607–612
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Nanoparticle diffusivity in narrow cylindrical pores Efstathios E. Michaelides Department of Engineering, TCU, Fort Worth, TX 76132, USA
a r t i c l e
i n f o
Article history: Received 22 February 2017 Accepted 22 June 2017
a b s t r a c t The hydrodynamic force on particles increases significantly when the particle movement is in the vicinity of solid boundaries. The increased drag decreases the mobility of the particles close to the wall, and slows down the Brownian movement. Since the Brownian movement of particles is the cause of particle dispersion and diffusion, it follows that the diffusion of particles close to a wall is slower than the diffusion in an unbounded fluid. Using a Monte Carlo method, the isothermal Brownian dispersion of spherical nanoparticles close to the wall of narrow cylindrical pores is simulated in Newtonian, isothermal fluids and the cross-section averaged particle diffusion is calculated. The pore to particle radii ratio in these simulations is in the range 5 < R/a < 140 and the range of nanoparticle radii is 3 < a < 100 nm. It is observed that the effect of the pore walls is a significant reduction of the average particle diffusivity. The diffusivity reduction is strictly a geometric effect. It depends on the size of the particles and the pore-to-particle diameter ratio and does not depend on the fluid and particle thermodynamic and transport properties, such as relative density and fluid viscosity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The mass transfer of micro- and nano-particles through narrow membrane pores is met in several physical processes ranging from nanoparticle separation and fractionation, to drug delivery, to measurements of nanofluid diffusivity. It is well-known that, when particles pass through narrow tubes, they experience a higher fluid drag, which is caused by the constraint of the boundaries of the flow. In such cases, the velocity of the particles is reduced, oftentimes significantly [1]. The velocity retardation, due to the proximity of boundaries, also applies to the Brownian movement, which is the physical mechanism for the diffusion of particles. The retardation of the Brownian movement causes decreased diffusivity in pores and this has been referred to in the past as ‘‘hindered transport” and ‘‘hindered diffusivity” [2,3]. In studying the dynamics of translocation of gold nanoparticles, Goyal et al. [4] observed experimentally that the average particle diffusivity decreased by a factor of five. While studying the electrophoretic migration of nanoparticles through membrane pores, Han et al. [5] also observed that the diffusivity of nanoparticles within the nano-pores decreased by a factor of 20–23. They attributed the dramatic decrease of nanoparticle diffusivity to an increase of the fluid viscosity inside the pores. Similarly, Lan and White [6] observed diffusion hindrance of nanoparticles in conical pores, when they applied a pressure-reversal technique to capture the nanoparticles. Several other researchers also observed the E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
slower motion of solid particles as well as of fluid macromolecules inside narrow tubes and membrane pores [7–10]. The slower movement of these particles has been attributed to electric charge interactions (for charged particles), and the increased fluid viscosity. Since nanoparticle diffusivity is a very important transport property for the movement and heat transfer of nanofluids [11] it is important to determine the changes of diffusivity because of the proximity of flow boundaries. When the particles are very fine, the Brownian movement causes their axial and radial dispersion inside pores. The interactions of particles with the solid boundaries and the confinement of the fluid near the pore boundaries cause a higher hydrodynamic force on the particles, which shows as higher fluid drag and particle retardation. This phenomenon was first investigated by Faxen [12] who derived an analytical power-low solution for the hydrodynamic force of spherical particles within parallel planes. Happel and Brenner [13], among others, performed a more detailed analysis of the motion of spherical particles in the vicinity of solid boundaries and derived expressions for the hydrodynamic drag enhancement in the longitudinal and radial directions. Summaries of the analytical and experimental studies on the hydrodynamic interactions of spherical particles with solid and fluid boundaries are given in [1,14]. It is apparent in all these studies that the hydrodynamic drag on the spherical particles increases significantly when they approach a solid surface. A numerical study of the Brownian movement [15] used the results for the enhanced hydrodynamic drag to explain the retardation of spherical particles, driven by thermophoresis, close to a plane surface.
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The diffusivity of particles in fluids is the result of the Brownian movement, which is caused by the random and incessant movement of the fluid molecules. The so-called, Einstein-Stokes diffusivity [16] for spherical particles is given by the expression:
D0
d dt
*! !+ xx kB T ; ¼ 6palf 2
ð1Þ
where kB is the Boltzmann constant; T is the absolute temperature of the fluid; a is the radius of the particle; and lf is the dynamic viscosity of the fluid. The Einstein-Stokes diffusivity is a transport property of the fluid-particle system and has been derived for dilute systems of spherical particles in fluids [16]. The diffusivity of particles in dense systems is lesser and is a strong function of the volumetric concentration of the fluid-particle system. In a Newtonian framework the random Brownian movement of the particles may be considered to be the action of a random body force, which continuously acts on the particles and affects their motion in a random way. It has been proven that the equivalent force, which causes the random movement of spherical particles of radius a and a magnitude of dispersion equal to the StokesEinstein diffusivity, is [17–19]: !
F Br
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12pkB T alf ¼R ; Dt !
ð2Þ
!
where R is a vector whose components are Gaussian random numbers with zero mean and unit standard deviation; and Dt is the time interval when the random force is applied on a particle. When the random force is integrated over a long period of time t (t >> Dt) the results for D0 are independent of the choice of the interval Dt and this continuous force yields the Stokes-Einstein diffusion coefficient of Eq. (1) [20,21]. In the continuum mechanics approach that is followed in this paper, the Brownian force acts continuously on the particles, while its magnitude changes at every step of the MC !
simulation following the vector with random components, R . Given the choice of this vector, the time-average of the Brownian force is zero and the variance of the force is 12pakBlf/Dt. It must be noted that one may follow an alternative stochastic approach and formulate the particle velocity and position equations as stochastic differential equations. This type of approach is often used for the study of macromolecule properties and polymeric fluids [21]. The continuum mechanics, discrete particle approach, which makes use of a Lagrangian equation for the particle motion, is favored in this paper, because it is simpler to formulate and easier to implement. This study examines numerically the hydrodynamic effect of cylindrical pore walls on the Brownian movement of nanoparticles inside the pores, and the overall effect of the pore walls on the area-average diffusivity of the nanoparticles. The MC simulation method is used to determine the dispersion of a large ensemble of nanoparticles in narrow cylindrical tubes of various diameters, using the Lagrangian computations. Suitable time- and ensemble-averages enable us to determine the time-averaged local movement of the nanoparticles, their dispersion, the local particle diffusivity and the pore-averaged diffusivity. The hindered diffusion of particles is expressed as the dimensionless ratio of the actual diffusion to the diffusion of the same particle-fluid system in an infinite domain. 2. Increased drag due to a boundary We consider a spherical nanoparticle of radius a immersed in a Newtonian fluid of density qf and viscosity lf. The velocity of the fluid is denoted by u and that of the particle by v. For most Newtonian fluids the flow is Stokesian, (Rep << 1, with
Rep = 2aqf(u v)/lf). The system is isothermal, which implies that the fluid viscosity is uniform, and the thermophoretic force is zero. Because the system is dilute, the diffusiophoresis also vanishes. The fluid inside the pore is at rest and, therefore, the space- and time-averages of u vanish. Under these conditions, the Lagrangian equation of motion of the particle becomes:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12pkB T alf mp 2 ¼ R 6pf Kn alf ðDtÞ dt 2!
d x
!
!
!
v K þ mp ðqp qf Þ g ;
ð3Þ
!
where v is the instantaneous velocity of the particles; fKn is a correction factor that accounts for the slip at the solid liquid interface that depends on the Knudsen number and is similar to Cunningham’s factor; the last term represents the gravitational force on the particle; and K is the hydrodynamic drag enhancement coefficient because of the proximity to the solid boundary [12,13,1,15]. Given that the fluids considered here are liquids, the mean free path of the fluid is very low in comparison to the radii of the nanoparticles considered in this study. Hence, Kn << 1 and fKn 1. For nanoparticles, the gravitational force is proportional to a3, several orders of magnitude less than the other terms of Eq. (3), and is usually neglected in the Lagrangian simulations [22,23,16]. Fig. 1 depicts the geometrical setting of the problem. The particle of radius a is located at a distance h from the wall of the cylinder, whose radius is R. The length of the cylinder, L, is much greater than the radius R, so the particle is inside the cylinder for the entire duration of the Lagrangian simulation. The coordinate system (r, z) is also shown in the Figure. Because the fluid hydrodynamics associated with the movement of the particle are different when the particle translates perpendicular or parallel to the wall, the drag enhancement coefficient, K, has components that are different in the two directions, r and z. Happel and Brenner [13] provide an analytical expression for the component Kz, which is pertinent to the translation of the sphere parallel to the wall:
Kz ¼ 1 þ
9a : 16h
ð4Þ
For the movement of the sphere perpendicular to the wall, Happel and Brenner [13] give the values of the pertinent drag enhancement coefficient in a tabular form. Based on these values a correlation was developed to be used in the Lagrangian computations:
2α h
z
R r
Fig. 1. A particle of radius a, translating in a cylinder of radius R at a distance h from the wall.
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K r ¼ exp
2:1827a2 h
2
0:5026a þ : h
ð5Þ
When compared to the tabular data, the average error of the correlation is less than 1% and the maximum error is 4.2%. By the inclusion of this empirical equation in the model, both the anisotropic nature of the drag force and its dependence on the distance from the wall are inserted as input into the Brownian movement of nanoparticles. The physical mechanism for the enhanced drag is the proximity of the solid boundary and the inability of the fluid to penetrate this boundary as it is displaced by the particle movement. The fluid displacement creates a flow field around the particle, which is symmetric in an infinite flow domain. The presence of a boundary, which is impenetrable to the fluid, does not allow the flow field around the particle to expand unimpeded beyond the boundary and this is manifested as the enhanced drag force on the particle [13,14]. In the mechanistic simulations, the particle ‘‘senses” the presence of the cylindrical walls because of the enhanced drag exerted that impedes its motion. Only the hydrodynamic particle-wall interaction force is in the function K(h). If any other particle-wall interactions occur, e.g. electric, magnetic, van der Waals, etc., they must be accounted separately in the simulations. Early numerical studies on the hindered diffusion of large molecules and small particles discovered that, when the diffusivity changes spatially, particle configurations of high diffusivity are depleted as a result of vigorous random walks away from such configurations, compared with more feeble random walks returning from the surrounding configurations with a smaller diffusivity [24]. Unless a correction is made to the basic algorithm that models the Brownian movement, regions in the pore with higher particle mobility – close to the pore center – will be depleted of nanoparticles, while regions close to the boundary will accumulate the nano particles. This violates the Boltzmann equilibrium condition. Grassia et al. [25] resolved the particle accumulation problem by introducing a drift velocity in the Brownian movement model. The drift velocity is directed to the center of the pore, preserves the Boltzmann distribution of particles, and has a magnitude:
VD ¼
kB T 6palf
1 ; div K
ð6Þ
This drift velocity in the numerical algorithms restores the Boltzmann equilibrium of particles in the flow field and does not cause artificial accumulation of particles [25]. The introduction of the drift velocity on the stochastic model for the restoration of the Boltzmann distribution for small particles and nanoparticles has been verified with more recent numerical studies, such as those by Lau and Lubenski [26] and Morse [27]. It must be noted that Eq. (3) applies to a single particle translating in a fluid and does not account for the presence of other particles in the computational domain. Thus, any inter-particle forces are absent in the MC simulations. The implication of this is that the results of the simulations only apply to dilute particle fluid systems, for which the volumetric concentration of solids is less than approximately 6.5% [1,13,14,17]. The Einstein-Stokes diffusivity of particles in fluids is an equilibrium property of the particle-fluid system [16] which also applies only to dilute systems at very low volumetric concentrations. The MC simulations performed in this study predict very well the Stokes-Einstein diffusivity, under the condition K(h) = 1, and are appropriate to calculate the modifications to the Einstein-Stokes diffusivity caused by the constrained motion of the particles in the vicinity of flow boundaries. In the absence of body forces such as the Brownian force, the velocity of a very small particle would approach asymptotically the fluid velocity and, hence, the particle will come to rest. The Brownian force of Eq. (2) induces small, random perturbations on
the relative velocity of the nanoparticles. The time-average movement and the dispersion of the spherical particles always determine the local diffusivity of the nanoparticles. This method to determine the local diffusivity may be applied in all flow domains including infinite domains and the interior of pores. 3. The computational model Several MC Lagrangian simulations were performed on ensembles of nanoparticles to determine the effects of the increased hydrodynamic force on the dispersion and the diffusivity of nanoparticles in stagnant, isothermal fluids. For the computations, the equation of motion of the particles was re-written as follows: !
! dv 1 ! ½½F Br þ mp ðqp qf Þ g 6palf ¼ mp dt
¼
1 ! ½F T 6palf mp
!
v K
!
v K;
ð7Þ
!
Since the total force, F T , is considered constant during the integration interval, dt, this equation has an exact solution, which is as follows:
! 9lf K v ðt þ dtÞ ¼ v ðtÞ exp 2 dt 2a qp " !# ! 9l f K FT 1 exp 2 dt : þ 6plf aK 2a q p !
!
ð8Þ
In the last equation, the parentheses indicate arguments of functions and the square brackets are used for the multiplication operation. The drift velocity, VD, of Eq. (6) must be added to this solution to satisfy the Boltzmann equilibrium condition for the particles. The resulting particle velocity also has an exact solution for the position of the particle. With the addition of the effect of the drift velocity, the new position of the nanoparticle at time t + dt is:
! ~ 9lf K 2a2 qp FT v ðtÞ þ exp 2 dt ~ dt 9l f K 6plf aK 2a qp ! 9lf K aqp~ FT exp 2 dt þ ~ V D dt: þ 2a qp 27pl2f K 2
~ xðt þ dtÞ ¼ ~ xðtÞ
ð9Þ
Because Eqs. (8) and (9) are exact, the integration time interval, dt, does not have to be very small for the calculations of the velocity and the new position of the particles to be accurate. In the calculations conducted for this study, dt was between 0.1 and 0.5 of the characteristic time of the nanoparticles, sp = 2a2qp/9lf. The value of the parameter Dt in the equation for the Brownian force (Eq. (2)) was taken to be equal to dt in all the computations (dt = Dt). By conducting several independent computations, it was ensured that the numerical results of this and similar studies are independent of the choice of the time interval dt [15,20]. For the MC calculations the motion of an ensemble of at least 4000 particles was considered. The uncertainty associated with this size of this ensemble is 1/(4000)1/2 = 2%. This choice is sufficient to yield statistically significant results for the MC simulations and for the calculation of the effective transport properties of particles, such as the hindered diffusion in pores. It must be noted that the characteristic dimension for the determination of particle diffusion is the radius of the particle a. This yields the characteristic diffusion time for the particles, a2/D1, or:
sD ¼
6pa3 lf : kB T
ð10Þ
E.E. Michaelides / International Journal of Heat and Mass Transfer 114 (2017) 607–612
In all the simulations, the total time of the simulation of a particle was 500,000sp or higher, which, implies that the total time of the simulations was by far higher than sD. This choice of the simulation time ensures that the random Brownian force acts on every particle for a time long enough to generate meaningful results for the diffusion process in the narrow tubes. This deduction is supported by the fact that all the simulations, when applied to infinite fluid domains, reproduced with less than 2% error the StokesEinstein diffusivity, D1. The first series of computations was conducted with particles in an unbounded fluid domain by taking K(h/a) = 1.0. It was ensured that, at the end of each run, the computed diffusivity of the parti-
1
0.8
0.6 20 nm 50 nm
0.4
0.2
! !
Fig. 2 depicts typical results for a MC simulation with 1000 particles of a = 50 nm radius. The initial position of all the particles is the center of the pore, r(0) = 0, and the radius of the pore, R, is 10 particle radii (R = 500 nm = 0.5 lm). The ordinate of the figure is the cardinal number of each particle, 1–1000. The total time of simulation for every particle was equal to 500,000sp. The final positions of the particles are normalized with respect to the particle radius, r/a. It is observed in this figure that the particles are dispersed almost uniformly around their initial position. While most of the nanoparticles end up close to their initial position, at r = 0, the Brownian movement carries several of them away from their initial position toward the boundaries. Fig. 3 depicts the computed local diffusivity of two types of nanoparticles with radii 20 and 30 nm – normalized using the Einstein-Stokes diffusivity, D1 – along a cylindrical pore of radius 0.5 lm. It is observed that the normalized diffusivity very close to the wall of the pore almost vanishes because of the very high hydrodynamic force exerted on the nanoparticles in that region. The diffusivity at the centerline of the cylindrical pore is higher,
-0.3
-0.1
0.1
0.5
Fig. 3. The local relative diffusivity of 20 and 30 nm nanoparticles in a pore of radius 0.5 lm, as a function of the radial position of the particles.
1.20E-11
1 0.9
1.00E-11
0.8 8.00E-12
0.7
6.00E-12
0.6 0.5
Normalized Diffusivity
0.4
Diffusivity
4.00E-12 2.00E-12
0.3 0.2
0.00E+00 0
20
40
60
80
100
Pore to Parcle Radii Rao, R/α Fig. 4. Average normalized diffusivity of 30 nm particles in pores of several radii. The particle diffusivity in an unbounded medium is: D1 = 1.33 * 1011 m2/s.
approximately 80% of the diffusivity value in an unbounded medium. It is apparent that the presence of the pore boundaries influences significantly the local diffusivity of the nanoparticles; that the diffusivity in all the positions within the pore is less than the value obtained from the Einstein-Stokes equation, D1, for an unbounded fluid; and that the local diffusivity reduction is very high in the vicinity of the pore boundaries.
8 Final radial posion of parcle, r/α
0.3
Radial Posion, r/R
Diffusivity, m 2/s
4. Computational results and discussion
0 -0.5
Normalized Diffusivity
cles, 1/2 hx x i/t, agrees with the Einstein-Stokes diffusivity of Eq. (1) within the uncertainty limits of the computations. For the determination of the average particle diffusivity inside the cylindrical pores computations were performed with particle initial radial positions on several distances from the wall. Depending on the diameter of the pore, these computations were conducted at 5–41 radial positions. The higher number of radial computations was used for the larger pores. For the properties of the fluid and the particles the properties of water and aluminum were used in all the simulations. The temperature in all computations was taken as T = 300 K, where the water viscosity is 548 ⁄ 106 Pa s.
Normalized Diffusivity, D/D ∞
610
6 4 2 0 -2 -4 -6 -8
Fig. 2. The dispersion of 1000 nanoparticles of 50 nm radii from an initial radial position r(t = 0) = 0.
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1.8E-11 1.6E-11
Normalized Diffusivity Diffusivity
0.80
1.4E-11 1.2E-11
0.60
1.0E-11 8.0E-12
0.40
6.0E-12
Diffusivity, m 2/s
Normalized Diffusivity
1.00
4.0E-12
0.20
2.0E-12 0.0E+00
0.00 0
10
20
30
40
50
Parcle Radius, α Fig. 5. The effect of particle size on the diffusivity at R/a = 20.
A cross-section average diffusivity is defined for the entire cross-sectional area of the pore as follows:
Dav ¼
Z
1
pR
2
0
R
2prDdr:
ð11Þ
It is apparent from the results of Fig. 3 that for all pores: Dav/ D1 < 1. Fig. 4 depicts the ratio Dav/D1 for 30 nm spherical particles in cylindrical pores of different sizes. The normalized diffusivity decreases at the lower values of R/a and it is expected to vanish as R/a = 1. The normalized cross-section average diffusivity approaches asymptotically the value 1 when the pore to particle radii ratio becomes very large, that is: lim(Dav/D1) = 1 as R/a ? 1. Because the Brownian movement is stronger for smaller particles and the Stokes-Einstein diffusivity is inversely proportional to the particle radius, it is expected that the particle radius, a, by itself affects significantly the diffusivity of particles in cylindrical pores. The effect of the size of spherical particles on the average diffusivity reduction was examined by conducting a number of MC simulations where the ratio R/a was kept constant at different values, while the radii of the pore and of the particles varied. A set of the results of these simulations is depicted in Fig. 5. The data in the figure pertain to calculations conducted with a constant ratio R/a = 20 and variable a and R. It was observed in the figure, as well as from the other computations with different pore-to-particle radii ratios, that the normalized particle diffusivity increases with a when the ratio R/a is constant. Several computations were also performed for different types of fluids (e.g. different density and viscosity) and different types (density) of the nanoparticles. Within the numerical error of the simulations, it was observed that the average diffusivity reduction did not change appreciably in these computations. This leads to the conclusion that the hindered diffusivity of nanoparticles is a purely geometric effect that depends on the size of the particles and the ratio of pore-to-particle radii, and is not appreciably influenced by the thermodynamic and transport properties of the fluidparticle system. There was also no appreciable difference when the time of the MC simulations was varied, as long as the total simulation time was above 10,000sp. The results in Figs. 2–5 were obtained after a total simulation time of at least 500,000sp for each particle. From the results of all the simulations, a correlation for the average diffusivity reduction was derived for the two significant variables R/a and a:
Dav ¼ D1
This correlation applies in the ranges 5 < R/a < 140; and 3 nm < a < 100 nm.
a0:603 h a i 1 2:3573 0:225 ln þ 0:217 : R 1 nm
ð12Þ
5. Conclusions The increased hydrodynamic force on particles in the vicinity of solid boundaries is the reason for the slowing of the movement of particles close to pore walls, including Brownian movement. As a result, the diffusion of particles close to a pore wall is significantly slower than the diffusion in an unbounded fluid. The poreaveraged spherical particles’ diffusivity is also less than the diffusivity of the same particles in an unbounded fluid and depends strongly on the size of the particles and on the pore-to-particle diameter ratio. The diffusivity reduction is a geometric effect and does not depend on the properties of the fluid and the particles. Based on the mechanistic MC simulations, which determine accurately the spherical particle diffusivity in any domain, an algebraic correlation was derived for the diffusivity reduction of nanoparticles with radii in the range 3 nm < a < 100 nm, when they diffuse in cylindrical pores with pore to particle radii ratio in the range 5 < R/a < 140. Conflict of interest The author declares that there is no conflict of interest. Acknowledgements This research was supported by the W.A. ‘‘Tex” Moncrief Chair of Engineering at Texas Christian University. References [1] E.E. Michaelides, Hydrodynamic force and heat/mass transfer from particles, bubbles and drops-the Freeman Scholar Lecture, J. Fluids Eng. 125 (2003) 209– 238. [2] W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J. (1987) 1409–1425. [3] R.B. Bird, Five decades of transport phenomena, AIChE J. 50 (2004) 273–287. [4] G. Goyal, K.J. Freedman, M.J. Kim, Gold nanoparticle translocation dynamics and electrical detection of single particle diffusion using solid-state nanopores, Anal. Chem. 85 (2013) 8180–8187. [5] R. Han, G. Wang, S. Qi, C. Ma, S.E. Yeung, Electrophoretic migration and axial diffusion of individual nanoparticles in cylindrical nanopores, J. Phys. Chem. C 116 (2012) 18460–18468. [6] W.J. Lan, H.S. White, Diffusional motion of a particle translocating through a nanopore, ACS-Nano 6 (2014) 1757–1765. [7] Y.-Q. Li, Y.-B. Zheng, R.N. Zare, Electrical optical, and docking properties of conical nanopores, ACS-Nano 6 (2014) 993–997.
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Nanoparticle diffusivity in narrow cylindrical pores Efstathios E. Michaelides Department of Engineering, TCU, Fort Worth, TX 76132, USA
a r t i c l e
i n f o
Article history: Received 22 February 2017 Accepted 22 June 2017
a b s t r a c t The hydrodynamic force on particles increases significantly when the particle movement is in the vicinity of solid boundaries. The increased drag decreases the mobility of the particles close to the wall, and slows down the Brownian movement. Since the Brownian movement of particles is the cause of particle dispersion and diffusion, it follows that the diffusion of particles close to a wall is slower than the diffusion in an unbounded fluid. Using a Monte Carlo method, the isothermal Brownian dispersion of spherical nanoparticles close to the wall of narrow cylindrical pores is simulated in Newtonian, isothermal fluids and the cross-section averaged particle diffusion is calculated. The pore to particle radii ratio in these simulations is in the range 5 < R/a < 140 and the range of nanoparticle radii is 3 < a < 100 nm. It is observed that the effect of the pore walls is a significant reduction of the average particle diffusivity. The diffusivity reduction is strictly a geometric effect. It depends on the size of the particles and the pore-to-particle diameter ratio and does not depend on the fluid and particle thermodynamic and transport properties, such as relative density and fluid viscosity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The mass transfer of micro- and nano-particles through narrow membrane pores is met in several physical processes ranging from nanoparticle separation and fractionation, to drug delivery, to measurements of nanofluid diffusivity. It is well-known that, when particles pass through narrow tubes, they experience a higher fluid drag, which is caused by the constraint of the boundaries of the flow. In such cases, the velocity of the particles is reduced, oftentimes significantly [1]. The velocity retardation, due to the proximity of boundaries, also applies to the Brownian movement, which is the physical mechanism for the diffusion of particles. The retardation of the Brownian movement causes decreased diffusivity in pores and this has been referred to in the past as ‘‘hindered transport” and ‘‘hindered diffusivity” [2,3]. In studying the dynamics of translocation of gold nanoparticles, Goyal et al. [4] observed experimentally that the average particle diffusivity decreased by a factor of five. While studying the electrophoretic migration of nanoparticles through membrane pores, Han et al. [5] also observed that the diffusivity of nanoparticles within the nano-pores decreased by a factor of 20–23. They attributed the dramatic decrease of nanoparticle diffusivity to an increase of the fluid viscosity inside the pores. Similarly, Lan and White [6] observed diffusion hindrance of nanoparticles in conical pores, when they applied a pressure-reversal technique to capture the nanoparticles. Several other researchers also observed the E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
slower motion of solid particles as well as of fluid macromolecules inside narrow tubes and membrane pores [7–10]. The slower movement of these particles has been attributed to electric charge interactions (for charged particles), and the increased fluid viscosity. Since nanoparticle diffusivity is a very important transport property for the movement and heat transfer of nanofluids [11] it is important to determine the changes of diffusivity because of the proximity of flow boundaries. When the particles are very fine, the Brownian movement causes their axial and radial dispersion inside pores. The interactions of particles with the solid boundaries and the confinement of the fluid near the pore boundaries cause a higher hydrodynamic force on the particles, which shows as higher fluid drag and particle retardation. This phenomenon was first investigated by Faxen [12] who derived an analytical power-low solution for the hydrodynamic force of spherical particles within parallel planes. Happel and Brenner [13], among others, performed a more detailed analysis of the motion of spherical particles in the vicinity of solid boundaries and derived expressions for the hydrodynamic drag enhancement in the longitudinal and radial directions. Summaries of the analytical and experimental studies on the hydrodynamic interactions of spherical particles with solid and fluid boundaries are given in [1,14]. It is apparent in all these studies that the hydrodynamic drag on the spherical particles increases significantly when they approach a solid surface. A numerical study of the Brownian movement [15] used the results for the enhanced hydrodynamic drag to explain the retardation of spherical particles, driven by thermophoresis, close to a plane surface.
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The diffusivity of particles in fluids is the result of the Brownian movement, which is caused by the random and incessant movement of the fluid molecules. The so-called, Einstein-Stokes diffusivity [16] for spherical particles is given by the expression:
D0
d dt
*! !+ xx kB T ; ¼ 6palf 2
ð1Þ
where kB is the Boltzmann constant; T is the absolute temperature of the fluid; a is the radius of the particle; and lf is the dynamic viscosity of the fluid. The Einstein-Stokes diffusivity is a transport property of the fluid-particle system and has been derived for dilute systems of spherical particles in fluids [16]. The diffusivity of particles in dense systems is lesser and is a strong function of the volumetric concentration of the fluid-particle system. In a Newtonian framework the random Brownian movement of the particles may be considered to be the action of a random body force, which continuously acts on the particles and affects their motion in a random way. It has been proven that the equivalent force, which causes the random movement of spherical particles of radius a and a magnitude of dispersion equal to the StokesEinstein diffusivity, is [17–19]: !
F Br
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12pkB T alf ¼R ; Dt !
ð2Þ
!
where R is a vector whose components are Gaussian random numbers with zero mean and unit standard deviation; and Dt is the time interval when the random force is applied on a particle. When the random force is integrated over a long period of time t (t >> Dt) the results for D0 are independent of the choice of the interval Dt and this continuous force yields the Stokes-Einstein diffusion coefficient of Eq. (1) [20,21]. In the continuum mechanics approach that is followed in this paper, the Brownian force acts continuously on the particles, while its magnitude changes at every step of the MC !
simulation following the vector with random components, R . Given the choice of this vector, the time-average of the Brownian force is zero and the variance of the force is 12pakBlf/Dt. It must be noted that one may follow an alternative stochastic approach and formulate the particle velocity and position equations as stochastic differential equations. This type of approach is often used for the study of macromolecule properties and polymeric fluids [21]. The continuum mechanics, discrete particle approach, which makes use of a Lagrangian equation for the particle motion, is favored in this paper, because it is simpler to formulate and easier to implement. This study examines numerically the hydrodynamic effect of cylindrical pore walls on the Brownian movement of nanoparticles inside the pores, and the overall effect of the pore walls on the area-average diffusivity of the nanoparticles. The MC simulation method is used to determine the dispersion of a large ensemble of nanoparticles in narrow cylindrical tubes of various diameters, using the Lagrangian computations. Suitable time- and ensemble-averages enable us to determine the time-averaged local movement of the nanoparticles, their dispersion, the local particle diffusivity and the pore-averaged diffusivity. The hindered diffusion of particles is expressed as the dimensionless ratio of the actual diffusion to the diffusion of the same particle-fluid system in an infinite domain. 2. Increased drag due to a boundary We consider a spherical nanoparticle of radius a immersed in a Newtonian fluid of density qf and viscosity lf. The velocity of the fluid is denoted by u and that of the particle by v. For most Newtonian fluids the flow is Stokesian, (Rep << 1, with
Rep = 2aqf(u v)/lf). The system is isothermal, which implies that the fluid viscosity is uniform, and the thermophoretic force is zero. Because the system is dilute, the diffusiophoresis also vanishes. The fluid inside the pore is at rest and, therefore, the space- and time-averages of u vanish. Under these conditions, the Lagrangian equation of motion of the particle becomes:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12pkB T alf mp 2 ¼ R 6pf Kn alf ðDtÞ dt 2!
d x
!
!
!
v K þ mp ðqp qf Þ g ;
ð3Þ
!
where v is the instantaneous velocity of the particles; fKn is a correction factor that accounts for the slip at the solid liquid interface that depends on the Knudsen number and is similar to Cunningham’s factor; the last term represents the gravitational force on the particle; and K is the hydrodynamic drag enhancement coefficient because of the proximity to the solid boundary [12,13,1,15]. Given that the fluids considered here are liquids, the mean free path of the fluid is very low in comparison to the radii of the nanoparticles considered in this study. Hence, Kn << 1 and fKn 1. For nanoparticles, the gravitational force is proportional to a3, several orders of magnitude less than the other terms of Eq. (3), and is usually neglected in the Lagrangian simulations [22,23,16]. Fig. 1 depicts the geometrical setting of the problem. The particle of radius a is located at a distance h from the wall of the cylinder, whose radius is R. The length of the cylinder, L, is much greater than the radius R, so the particle is inside the cylinder for the entire duration of the Lagrangian simulation. The coordinate system (r, z) is also shown in the Figure. Because the fluid hydrodynamics associated with the movement of the particle are different when the particle translates perpendicular or parallel to the wall, the drag enhancement coefficient, K, has components that are different in the two directions, r and z. Happel and Brenner [13] provide an analytical expression for the component Kz, which is pertinent to the translation of the sphere parallel to the wall:
Kz ¼ 1 þ
9a : 16h
ð4Þ
For the movement of the sphere perpendicular to the wall, Happel and Brenner [13] give the values of the pertinent drag enhancement coefficient in a tabular form. Based on these values a correlation was developed to be used in the Lagrangian computations:
2α h
z
R r
Fig. 1. A particle of radius a, translating in a cylinder of radius R at a distance h from the wall.
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K r ¼ exp
2:1827a2 h
2
0:5026a þ : h
ð5Þ
When compared to the tabular data, the average error of the correlation is less than 1% and the maximum error is 4.2%. By the inclusion of this empirical equation in the model, both the anisotropic nature of the drag force and its dependence on the distance from the wall are inserted as input into the Brownian movement of nanoparticles. The physical mechanism for the enhanced drag is the proximity of the solid boundary and the inability of the fluid to penetrate this boundary as it is displaced by the particle movement. The fluid displacement creates a flow field around the particle, which is symmetric in an infinite flow domain. The presence of a boundary, which is impenetrable to the fluid, does not allow the flow field around the particle to expand unimpeded beyond the boundary and this is manifested as the enhanced drag force on the particle [13,14]. In the mechanistic simulations, the particle ‘‘senses” the presence of the cylindrical walls because of the enhanced drag exerted that impedes its motion. Only the hydrodynamic particle-wall interaction force is in the function K(h). If any other particle-wall interactions occur, e.g. electric, magnetic, van der Waals, etc., they must be accounted separately in the simulations. Early numerical studies on the hindered diffusion of large molecules and small particles discovered that, when the diffusivity changes spatially, particle configurations of high diffusivity are depleted as a result of vigorous random walks away from such configurations, compared with more feeble random walks returning from the surrounding configurations with a smaller diffusivity [24]. Unless a correction is made to the basic algorithm that models the Brownian movement, regions in the pore with higher particle mobility – close to the pore center – will be depleted of nanoparticles, while regions close to the boundary will accumulate the nano particles. This violates the Boltzmann equilibrium condition. Grassia et al. [25] resolved the particle accumulation problem by introducing a drift velocity in the Brownian movement model. The drift velocity is directed to the center of the pore, preserves the Boltzmann distribution of particles, and has a magnitude:
VD ¼
kB T 6palf
1 ; div K
ð6Þ
This drift velocity in the numerical algorithms restores the Boltzmann equilibrium of particles in the flow field and does not cause artificial accumulation of particles [25]. The introduction of the drift velocity on the stochastic model for the restoration of the Boltzmann distribution for small particles and nanoparticles has been verified with more recent numerical studies, such as those by Lau and Lubenski [26] and Morse [27]. It must be noted that Eq. (3) applies to a single particle translating in a fluid and does not account for the presence of other particles in the computational domain. Thus, any inter-particle forces are absent in the MC simulations. The implication of this is that the results of the simulations only apply to dilute particle fluid systems, for which the volumetric concentration of solids is less than approximately 6.5% [1,13,14,17]. The Einstein-Stokes diffusivity of particles in fluids is an equilibrium property of the particle-fluid system [16] which also applies only to dilute systems at very low volumetric concentrations. The MC simulations performed in this study predict very well the Stokes-Einstein diffusivity, under the condition K(h) = 1, and are appropriate to calculate the modifications to the Einstein-Stokes diffusivity caused by the constrained motion of the particles in the vicinity of flow boundaries. In the absence of body forces such as the Brownian force, the velocity of a very small particle would approach asymptotically the fluid velocity and, hence, the particle will come to rest. The Brownian force of Eq. (2) induces small, random perturbations on
the relative velocity of the nanoparticles. The time-average movement and the dispersion of the spherical particles always determine the local diffusivity of the nanoparticles. This method to determine the local diffusivity may be applied in all flow domains including infinite domains and the interior of pores. 3. The computational model Several MC Lagrangian simulations were performed on ensembles of nanoparticles to determine the effects of the increased hydrodynamic force on the dispersion and the diffusivity of nanoparticles in stagnant, isothermal fluids. For the computations, the equation of motion of the particles was re-written as follows: !
! dv 1 ! ½½F Br þ mp ðqp qf Þ g 6palf ¼ mp dt
¼
1 ! ½F T 6palf mp
!
v K
!
v K;
ð7Þ
!
Since the total force, F T , is considered constant during the integration interval, dt, this equation has an exact solution, which is as follows:
! 9lf K v ðt þ dtÞ ¼ v ðtÞ exp 2 dt 2a qp " !# ! 9l f K FT 1 exp 2 dt : þ 6plf aK 2a q p !
!
ð8Þ
In the last equation, the parentheses indicate arguments of functions and the square brackets are used for the multiplication operation. The drift velocity, VD, of Eq. (6) must be added to this solution to satisfy the Boltzmann equilibrium condition for the particles. The resulting particle velocity also has an exact solution for the position of the particle. With the addition of the effect of the drift velocity, the new position of the nanoparticle at time t + dt is:
! ~ 9lf K 2a2 qp FT v ðtÞ þ exp 2 dt ~ dt 9l f K 6plf aK 2a qp ! 9lf K aqp~ FT exp 2 dt þ ~ V D dt: þ 2a qp 27pl2f K 2
~ xðt þ dtÞ ¼ ~ xðtÞ
ð9Þ
Because Eqs. (8) and (9) are exact, the integration time interval, dt, does not have to be very small for the calculations of the velocity and the new position of the particles to be accurate. In the calculations conducted for this study, dt was between 0.1 and 0.5 of the characteristic time of the nanoparticles, sp = 2a2qp/9lf. The value of the parameter Dt in the equation for the Brownian force (Eq. (2)) was taken to be equal to dt in all the computations (dt = Dt). By conducting several independent computations, it was ensured that the numerical results of this and similar studies are independent of the choice of the time interval dt [15,20]. For the MC calculations the motion of an ensemble of at least 4000 particles was considered. The uncertainty associated with this size of this ensemble is 1/(4000)1/2 = 2%. This choice is sufficient to yield statistically significant results for the MC simulations and for the calculation of the effective transport properties of particles, such as the hindered diffusion in pores. It must be noted that the characteristic dimension for the determination of particle diffusion is the radius of the particle a. This yields the characteristic diffusion time for the particles, a2/D1, or:
sD ¼
6pa3 lf : kB T
ð10Þ
E.E. Michaelides / International Journal of Heat and Mass Transfer 114 (2017) 607–612
In all the simulations, the total time of the simulation of a particle was 500,000sp or higher, which, implies that the total time of the simulations was by far higher than sD. This choice of the simulation time ensures that the random Brownian force acts on every particle for a time long enough to generate meaningful results for the diffusion process in the narrow tubes. This deduction is supported by the fact that all the simulations, when applied to infinite fluid domains, reproduced with less than 2% error the StokesEinstein diffusivity, D1. The first series of computations was conducted with particles in an unbounded fluid domain by taking K(h/a) = 1.0. It was ensured that, at the end of each run, the computed diffusivity of the parti-
1
0.8
0.6 20 nm 50 nm
0.4
0.2
! !
Fig. 2 depicts typical results for a MC simulation with 1000 particles of a = 50 nm radius. The initial position of all the particles is the center of the pore, r(0) = 0, and the radius of the pore, R, is 10 particle radii (R = 500 nm = 0.5 lm). The ordinate of the figure is the cardinal number of each particle, 1–1000. The total time of simulation for every particle was equal to 500,000sp. The final positions of the particles are normalized with respect to the particle radius, r/a. It is observed in this figure that the particles are dispersed almost uniformly around their initial position. While most of the nanoparticles end up close to their initial position, at r = 0, the Brownian movement carries several of them away from their initial position toward the boundaries. Fig. 3 depicts the computed local diffusivity of two types of nanoparticles with radii 20 and 30 nm – normalized using the Einstein-Stokes diffusivity, D1 – along a cylindrical pore of radius 0.5 lm. It is observed that the normalized diffusivity very close to the wall of the pore almost vanishes because of the very high hydrodynamic force exerted on the nanoparticles in that region. The diffusivity at the centerline of the cylindrical pore is higher,
-0.3
-0.1
0.1
0.5
Fig. 3. The local relative diffusivity of 20 and 30 nm nanoparticles in a pore of radius 0.5 lm, as a function of the radial position of the particles.
1.20E-11
1 0.9
1.00E-11
0.8 8.00E-12
0.7
6.00E-12
0.6 0.5
Normalized Diffusivity
0.4
Diffusivity
4.00E-12 2.00E-12
0.3 0.2
0.00E+00 0
20
40
60
80
100
Pore to Parcle Radii Rao, R/α Fig. 4. Average normalized diffusivity of 30 nm particles in pores of several radii. The particle diffusivity in an unbounded medium is: D1 = 1.33 * 1011 m2/s.
approximately 80% of the diffusivity value in an unbounded medium. It is apparent that the presence of the pore boundaries influences significantly the local diffusivity of the nanoparticles; that the diffusivity in all the positions within the pore is less than the value obtained from the Einstein-Stokes equation, D1, for an unbounded fluid; and that the local diffusivity reduction is very high in the vicinity of the pore boundaries.
8 Final radial posion of parcle, r/α
0.3
Radial Posion, r/R
Diffusivity, m 2/s
4. Computational results and discussion
0 -0.5
Normalized Diffusivity
cles, 1/2 hx x i/t, agrees with the Einstein-Stokes diffusivity of Eq. (1) within the uncertainty limits of the computations. For the determination of the average particle diffusivity inside the cylindrical pores computations were performed with particle initial radial positions on several distances from the wall. Depending on the diameter of the pore, these computations were conducted at 5–41 radial positions. The higher number of radial computations was used for the larger pores. For the properties of the fluid and the particles the properties of water and aluminum were used in all the simulations. The temperature in all computations was taken as T = 300 K, where the water viscosity is 548 ⁄ 106 Pa s.
Normalized Diffusivity, D/D ∞
610
6 4 2 0 -2 -4 -6 -8
Fig. 2. The dispersion of 1000 nanoparticles of 50 nm radii from an initial radial position r(t = 0) = 0.
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1.8E-11 1.6E-11
Normalized Diffusivity Diffusivity
0.80
1.4E-11 1.2E-11
0.60
1.0E-11 8.0E-12
0.40
6.0E-12
Diffusivity, m 2/s
Normalized Diffusivity
1.00
4.0E-12
0.20
2.0E-12 0.0E+00
0.00 0
10
20
30
40
50
Parcle Radius, α Fig. 5. The effect of particle size on the diffusivity at R/a = 20.
A cross-section average diffusivity is defined for the entire cross-sectional area of the pore as follows:
Dav ¼
Z
1
pR
2
0
R
2prDdr:
ð11Þ
It is apparent from the results of Fig. 3 that for all pores: Dav/ D1 < 1. Fig. 4 depicts the ratio Dav/D1 for 30 nm spherical particles in cylindrical pores of different sizes. The normalized diffusivity decreases at the lower values of R/a and it is expected to vanish as R/a = 1. The normalized cross-section average diffusivity approaches asymptotically the value 1 when the pore to particle radii ratio becomes very large, that is: lim(Dav/D1) = 1 as R/a ? 1. Because the Brownian movement is stronger for smaller particles and the Stokes-Einstein diffusivity is inversely proportional to the particle radius, it is expected that the particle radius, a, by itself affects significantly the diffusivity of particles in cylindrical pores. The effect of the size of spherical particles on the average diffusivity reduction was examined by conducting a number of MC simulations where the ratio R/a was kept constant at different values, while the radii of the pore and of the particles varied. A set of the results of these simulations is depicted in Fig. 5. The data in the figure pertain to calculations conducted with a constant ratio R/a = 20 and variable a and R. It was observed in the figure, as well as from the other computations with different pore-to-particle radii ratios, that the normalized particle diffusivity increases with a when the ratio R/a is constant. Several computations were also performed for different types of fluids (e.g. different density and viscosity) and different types (density) of the nanoparticles. Within the numerical error of the simulations, it was observed that the average diffusivity reduction did not change appreciably in these computations. This leads to the conclusion that the hindered diffusivity of nanoparticles is a purely geometric effect that depends on the size of the particles and the ratio of pore-to-particle radii, and is not appreciably influenced by the thermodynamic and transport properties of the fluidparticle system. There was also no appreciable difference when the time of the MC simulations was varied, as long as the total simulation time was above 10,000sp. The results in Figs. 2–5 were obtained after a total simulation time of at least 500,000sp for each particle. From the results of all the simulations, a correlation for the average diffusivity reduction was derived for the two significant variables R/a and a:
Dav ¼ D1
This correlation applies in the ranges 5 < R/a < 140; and 3 nm < a < 100 nm.
a0:603 h a i 1 2:3573 0:225 ln þ 0:217 : R 1 nm
ð12Þ
5. Conclusions The increased hydrodynamic force on particles in the vicinity of solid boundaries is the reason for the slowing of the movement of particles close to pore walls, including Brownian movement. As a result, the diffusion of particles close to a pore wall is significantly slower than the diffusion in an unbounded fluid. The poreaveraged spherical particles’ diffusivity is also less than the diffusivity of the same particles in an unbounded fluid and depends strongly on the size of the particles and on the pore-to-particle diameter ratio. The diffusivity reduction is a geometric effect and does not depend on the properties of the fluid and the particles. Based on the mechanistic MC simulations, which determine accurately the spherical particle diffusivity in any domain, an algebraic correlation was derived for the diffusivity reduction of nanoparticles with radii in the range 3 nm < a < 100 nm, when they diffuse in cylindrical pores with pore to particle radii ratio in the range 5 < R/a < 140. Conflict of interest The author declares that there is no conflict of interest. Acknowledgements This research was supported by the W.A. ‘‘Tex” Moncrief Chair of Engineering at Texas Christian University. References [1] E.E. Michaelides, Hydrodynamic force and heat/mass transfer from particles, bubbles and drops-the Freeman Scholar Lecture, J. Fluids Eng. 125 (2003) 209– 238. [2] W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J. (1987) 1409–1425. [3] R.B. Bird, Five decades of transport phenomena, AIChE J. 50 (2004) 273–287. [4] G. Goyal, K.J. Freedman, M.J. Kim, Gold nanoparticle translocation dynamics and electrical detection of single particle diffusion using solid-state nanopores, Anal. Chem. 85 (2013) 8180–8187. [5] R. Han, G. Wang, S. Qi, C. Ma, S.E. Yeung, Electrophoretic migration and axial diffusion of individual nanoparticles in cylindrical nanopores, J. Phys. Chem. C 116 (2012) 18460–18468. [6] W.J. Lan, H.S. White, Diffusional motion of a particle translocating through a nanopore, ACS-Nano 6 (2014) 1757–1765. [7] Y.-Q. Li, Y.-B. Zheng, R.N. Zare, Electrical optical, and docking properties of conical nanopores, ACS-Nano 6 (2014) 993–997.
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