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68
yflK 539.612
NANOPARTICLE ADHESION AND FLOW PROPERTIES OF COHESIVE NANOPOWDERS Jürgen Tomas
Mechanical Process Engineering,
The Otto-von-Guericke-UniversityMagdeburg, Germany
Represented by a member of the Editorial Board Professor E. Tsotsas
Key Words and Phrases: adhesion forces; cohesion; constitutive models; particle mechanics; powder mechanics; powder compressibility; powder flow properties; titania nanopowder; van der Waals forces.
Abstract: The fundamentals of cohesive powder consolidation and flow behaviour using a reasonable combination of particle and continuum mechanics are explained. By means of the model “stiff particles with soft contacts” the influence of elastic-plastic repulsion in particle contacts is demonstrated. With this as the physical basis, the stationary yield locus, instantaneous yield loci and consolidation loci, flow function, consolidation and compression functions by the flow properties of a very cohesive titania nanopowder (surface diameter ds = 200 nm) are presented. These models are used to evaluate shear cell test results as constitutive functions for computer aided design of process apparatuses for reliable powder flow.
Symbols
a - separation, nm Indices
A - area, particle contact area, m2
CH - Hamaker constant, J b - bulk
d - particle size, ^m c - compressive
E - modulus of elasticity, kN/mm2 K - total contact
F - force, N e -effective
p - pressure, kPa el - elastic
pf - plastic yield strength of particle con- H - adhesion
tact, MPa r - radius, nm 8 - porosity k - elastic-plastic contact consolidation i - internal
l - liquid M -centre of Mohr circle
coefficient N - normal
ka - elastic-plastic contact area coefficient pl - plastic
kp - plastic contact repulsion coefficient R - radius of Mohr circle
9 - angle of friction, deg s - solid
p - density, kg/m3 sls - solid-liquid-solid
ct - normal stress, kPa st - stationary
Ü! - major principal stress, kPa VdW - van der Waals
ct0 - isostatic tensile strength, kPa Z - tensile
t - shear stress, kPa 0 - initial, zero point.
Introduction
The well-known flow problems of cohesive particulate solids in storage and transportation containers, conveyors or process apparatuses - mainly mentioned by Jenike [1] -leads to bridging, channelling, oscillating mass flow rates and particle characteristics
associated with feeding and dosing problems. Taking into account this list of selected technical problems and hazards, it is worth to deal with the fundamentals of particulate solids consolidation and flow, i.e., to develop a reasonable combination of particle and continuum mechanics. This method appears to be appropriate to derive constitutive functions on physical basis in the context of micro-macro transition of particle-powder behaviour.
Particle Contact Constitutive Model
The well-known failure hypotheses of Tresca, Coulomb-Mohr, Drucker and Prager, the yield locus concept of Jenike [1] and Schwedes [2, 3] were supplemented from Molerus [4, 5] by the cohesive steady-state flow criterion. The consolidation and non-rapid, frictional flow of fine and cohesive particulate solids was explained by the adhesion forces at particle contacts [4]. His advanced theory is the suitable basis of the extended model approach [27 to 30] which is shown now.
In principle, there are four essential mechanical deformation effects in particle-surface contacts and their force-response behaviour can be explained as follows:
(1) elastic contact deformation (Hertz [6], Huber [7], Mindlin [8], Dahneke [9], Deijaguin (DMT theory) [10], Johnson (JKR theory) [11], Thornton [12] and Sadd [13]) which is reversible, independent of deformation rate and consolidation time effects and valid for all particulate solids;
(2) plastic contact deformation with adhesion (Derjaguin [14], Krupp [15], Schubert [16], Molerus [4, 5], Maugis [17], Walton [18] and Thornton [19]) which is irreversible, deformation rate and consolidation time independent, e.g. mineral powders;
(3) viscoelastic contact deformation (Yang [20], Rumpf [21] and Sadd [13]) which is reversible and dependent on deformation rate and consolidation time, e.g. soft particles as bio-cells;
(4) viscoplastic contact deformation (Rumpf [21]) which is irreversible and dependent on deformation rate and consolidation time, e.g. nanoparticles fusion.
These force-displacement models are shown as characteristic constitutive functions in Fig. 1. Based on these theories, a general approach for the time and deformation rate dependent and combined viscoelastic, elastic-plastic, viscoplastic, adhesion and dissipative behaviours of a spherical particle contact was derived [28,29] and is briefly explained here - the comprehensive review of Tomas [41] comprises all the derivations in detail:
During approaching of the spheres, the adhesion force FH0 without any additional contact deformation, see Fig. 2, a), can be modelled as a single rough sphere-sphere-contact, additionally, with a characteristic hemispherical micro-roughness height or radius hr < d instead of particle size d [25, 26]:
^ ^H,sls hr
FH0 =—2----------
12aF=0
d / hr 1 +- r
2 ( + hr / aF=o )
^H,sls hr (i)
12aF=o
If an external compressive normal force FN is acting at a soft contact of two isotropic, stiff, linear elastic, mono-disperse spherical particles the previous contact point is deformed to a small contact area, Fig. 2, and the adhesion force between these two particles is increasing, Krupp [15], Rumpf et al. [21] and Molerus [4, 5]. After this loading and elastic deformation, Fig. 2, b), the contact starts at pmax = pf with plastic yielding. This elastic-plastic contact deformation response in Fig. 2, c) is given by the particle
a) nonlinear elastic, adhesion
b) linear plastic, adhesion dissipative
Walton
hr
c) nonlinear viscoelastic
(-) adhesion force
d) nonlinear plastic
perfect softening
e) nonlinear elastic, dissipative f) linear viscoplastic
g) hnear plastic, nonhnear h) linear plastic, nonhnear
elastic, dissipative, adhesion elastic, adhesion, dissipative,
viscoelastic, viscoplastic
Fig. 1 Force - displacement diagram of constitutive models of contact deformation of smooth spherical particles in normal direction without/with adhesion (compression +, tension -). The basic models for elastic behaviour were derived by Hertz [6], for viscoelasticity by Yang [20], for constant adhesion by Johnson et al. [11] and for plastic behaviour by Thornton and Ning [19] and Walton and Braun [18] and for plasticity with variation in adhesion by Mol-erus [4] and Schubert et al. [16]. This has been expanded stepwise to include nonlinear plastic contact hardening and softening. Energy dissipation was considered by Sadd et al. [13] and time dependent viscoplasticity by Rumpf et al. [21]. Considering all these theories, one obtains a general contact model for time and rate dependent viscoelastic, elastic-plastic, viscoplastic, adhesion and dissipative behaviours, Tomas [28, 29]
Fig. 2 Particle contact approach, elastic, elastic-plastic deformation and detachment. After loading with an external force FN the spherical contact is elastically compacted to a plate-plate-contact and shows the Hertz [6] elliptic pressure distribution b). With increasing normal load this contact starts at the yield point pmax = pf with plastic yielding c). The micro-yield surface is reached and this maximum pressure has not been exceeded. A hindered plastic field is formed at the contact with a circular constant pressure pmax and an annular elastic pressure distribution dependent on radius rK,el, full lines in c)
contact force equilibrium between attraction (-) and elastic as well as soft plastic repulsion (+) or force response (rK coordinate of annular elastic contact area):
rK
XF = 0 = -FH0 - pVdWnrK -FN + pf ™K,pl + 2n j pel (rK)rKdrK . (2)
rK,pl
Superposition provided, this leads to a very useful linear force displacement model (for ka « const) with the particle centre approach of both particles hK [28], shown in Fig. 3
as elastic-plastic boundary (averaged radius of particle 1 and 2 r12 = (1/r1 +1/r2)
FN + FH0 =nr1,2pf (kA -Kp )hK. (3)
Thus, the contact stiffness decreases with smaller size d = 4r12 (or microroughness radius) of cohesive powders and nanoparticles, predominant plastic yielding behaviour provided [28]:
kN,pl = -FN = nr1,2 pf (KA -Kp ). (4)
dhK
The plastic repulsion coefficient kp describes a dimensionless ratio of attractive Van der Waals pressure pVdW to repulsive particle micro-hardness pf for a plate-plate model:
K = pVdW = CH,sls (5)
K p = =7 3 (5)
pf 6naF=0 pf
The characteristic adhesion distance in Eqs. (1) and (5) lies in a molecular scale a = aF=0 « 0,3...0,4 nm. It depends mainly on the properties of liquid-equivalent packed adsorbed layers and can be estimated for a molecular interaction potential minimum -dU / da = F = 0 = Fat + Frep or force equilibrium. It depends mainly on the prop-
erties of liquid-equivalent packed adsorbed layers and can be estimated for a molecular interaction potential minimum -dU / da = F = 0 = Fat + Frep or force equilibrium [22].
Provided that these molecular contacts are stiff enough compared with the soft particle contact behaviour, this separation aF=0 is assumed to be constant. The particle surface behaviours are influenced by mobile adsorption layers due to molecular rearrangement. The Hamaker constant CH,sls [23] includes these solid-liquid-solid interactions of continuous media. Thus CH,sls can be calculated due to Lifshitz theory and depends on dielectric constants and refractive indices [22, 24].
The elastic-plastic contact area coefficient ka represents the ratio of plastic particle contact deformation area Apl to total contact deformation area AK = Apl + Ael including
a certain elastic displacement [28]
2 +1 Apl i 13 hK,f (6)
KA = - + 3 — = 1 -, (6)
33 AK 3V hK
with the centre approach hK,f for incipient yielding at point Y in Fig. 3, pel(rK = 0) = pmax = = pf
hK,f = d [E^ , (7)
and the averaged modulus of elasticity E* of both particles 1and 2 (v Poisson’s ratio).
E* = 2
Л 2Л 2\ 1 1-V1 + 1-V^
, E1 E2 ,
V 1 У
(8)
Constant mechanical bulk properties provided, the finer the particles the smaller is again the yield point hK,f which is shifted towards zero centre approach. Thus, an initial
Fig. 3 Force - displacement diagram of recalculated characteristic contact deformation of very cohesive titania particles as spheres, surface diameter dS = 200 nm, surface moisture
XW = 0,4 %. The origin of this diagram hK = 0 is equivalent to the characteristic adhesion separation for direct contact aF=0. After loading 0 - Y the contact is elastically compacted with an approximated circular contact area, Fig. 2, b) and starts at the yield point Y at pmax = pf with plastic yielding, Fig. 2, c). Next, the combined elastic-plastic yield boundary of the plate-plate contact is achieved, Eq. (3). This displacement is expressed by annular elastic Ael (thickness rK,el) and circular plastic Apl (radius rK,pl) contact area, Fig. 2, c). After unloading between the points U - A the contact recovers elastically according to Eq. (9) to a displacement hKA. The reloading curve runs from point A to U to the displacement hKU, Eq. (10). If one applies a certain pull-off force FNZ = - FHA as given in Eq. (11) but here negative, the adhesion boundary line at failure point A is reached and the contact plates fail and detach with the increasing distance a = aF=0 + hK a - hK , Fig. 2 panel d). This actual particle separation is considered for the
calculation by a hyperbolic adhesion force curve Fn z = -Fh a k a~3 of the plate-plate model.
This hysteresis behaviour could be shifted along the elastic-plastic boundary and depends on the pre-loading or, in other words, on pre-consolidation level FNU. Thus, the variation in adhesion forces Fh a between particles depend directly on this frozen irreversible deformation, the so-called contact pre-consolidation history FH(FN), see next Fig. 4
pure elastic contact deformation Apl = 0, ka = 2/3, has no relevance for cohesive nanoparticles and should be excluded. But after unloading beginning at point U along curve U - E, Fig. 3, the contact recovers elastically in the compression mode and remains with a perfect plastic displacement hK,E. For this pure plastic contact deformation Ael = 0 and AK = Apl, ka = 1 is obtained.
Below point E left the tension mode begins. Between U - E - A the contact recovers probably elastically along a supplemented Hertzian parabolic curvature up to displacement hK,A:
the adhesion (failure) boundary at point A is reached and the contact plates are failing and detaching with the increasing distance a = aF=0 + hKA - hK . This actual particle separation can be considered for the calculation by means of a long-range hyperbolic adhesion force curve FN,Z = -FHA x a-3 with the plate-plate model as given in Eq. (5).
Additionally, if one considers a single elastic-plastic particle contact as a conservative mechanical system without heat dissipation, the energy absorption equals the lens-shaped area between both unloading and reloading curves A - U in Fig. 3:
With Eqs. (9) and (11) for FH,A and (10), (3) for _FN>U, one obtains finally the specific or mass related energy absorption Wmdiss = kWdiss / mp , which includes the averaged particle mass mP = 4/37rr]32ps. In addition, the resultant Eq. (3) includes a characteristic contact number in the bulk powder (coordination number k » n/s [4]):
This specific energy of 3 to 85 J/g for the titania powder example mentioned was dissipated during one unloading - reloading - cycle in the bulk powder with an average pressure of only CTM,st = 2... 18 kPa (or major principal stress ct1 = 4.. .33 kPa).
The slopes of elastic-plastic yield and adhesion boundaries in Fig. 3 are characteristics of irreversible particle contact stiffness or compliance. Consequently, if one eliminates the deformation hK the linear adhesion-normal force function FH = f(FN) in Fig. 4 is obtained [28].
(9)
But along the symmetric curve A - U the contact may be reloaded:
(10)
If one applies a certain pull-off force FNZ = - FHA, here negative,
FH,A = FH0 + nr1,2 pVdW hK,A
(11)
hK ,U hK,U
^diss - j FN,reload(hK) dhK - j FN,unload (hK )dhK •
(12)
hK, A
*K,A
[KAhK,U -Kp (hK,U - ^K,A )]
(13)
----fn - (1 + k)FH0 + kFn •
(14)
K A -K p
The dimensionless elastic-plastic contact consolidation coefficient (strain characteristic) k is given by the slope of adhesion force FH influenced by predominant plastic contact failure.
к - -
ka - кг
(15)
This elastic-plastic contact consolidation coefficient k is a measure of irreversible particle contact stiffness or softness as well. A shallow slope implies low adhesion level FH » FH0 because of stiff particle contacts, but a large slope means soft contacts, or i.e., a cohesive powder flow behaviour. This model considers, additionally, the flattening of soft particle contacts caused by the adhesion force kFH0. Thus, the total adhesion force consists of a stiff contribution FH0 and a contact strain influenced component
k(FH0 + fn ) > Fig. 4.
This Eq. (14) can be interpreted as a general linear particle contact constitutive model, i.e. linear in forces, but non-linear concerning material characteristics. The intersection of function (14) with abscissa (FH = 0) in the negative extension range of consolidation force Fn is surprisingly independent of the Hamaker constant CH,sls, Fig. 4:
FN,Z - -—aF-0 hr pf
2 + A
3 3A
1+-
d / hr
2 (l + V aF-0 )
* - - aF-0 hr Pf • (16)
Considering the model prerequisites for cohesive powders, this minimum normal (tensile) force limit FN,Z combines the opposite influences of a particle stiffness, microyield strength pf » 3af or resistance against plastic deformation and particle distance distribution. The last-mentioned is characterised by roughness height hr as well as molecular centre distance aF=0. It corresponds to an abscissa intersection g1Z of the constitutive consolidation function, Eq. (28) and Fig. 7.
Fig. 4 Adhesion force - normal force diagram of recalculated particle contact forces of titania according to Eq. (4) using data of Fig. , surface diameter dS = 200 nm, surface moisture XW = 0,4 %. This is equivalent to idealised mono-molecular adsorption layers being in equilibrium with ambient air temperature of 20 °C and 50 % humidity. The points characterise the pressure levels of the yield loci YL 1 to YL 4 according to Fig. 7. A characteristic line for k = 0,77 of a very cohesive powder is included and shows directly the correlation between strength and force enhancement with pre-consolidation, Eq. (31)
к
p
Cohesive Powder Flow Criteria
Using the elastic-plastic particle contact constitutive model Eq. (14) the failure conditions of particle contacts are formulated [30]. It should be noted that the stressing pre-history of a cohesive powder flow is stationary (steady-state) and delivers significantly a cohesive stationary yield locus in the x-a-diagram of Fig. 5, [28]:
Tst = tan 9 st(ast +a0). (17)
This shear zone is characterised by a dynamic equilibrium of simultaneous contact shearing, unloading and failing, creating new contacts, loading, reloading, unloading and shearing again. The stationary yield locus is the envelope of all Mohr-circles for steady-state flow (critical state line) with a certain negative intersection of the abscissa
H0
So d2
(18)
This isostatic tensile strength a0 of an unconsolidated powder without any particle contact deformation is obtained from the adhesion force FH0, Eq. (1), with the initial porosity of very loose packing 60 = 1 - Pb,0 / Ps, Eq. (34).
From the above formulation for cohesive steady-state flow Eq. (17), the stress-dependent effective angle of internal friction 9 as introduced by Jenike [1], i.e., slope of cohesionless effective yield locus, follows in accordance with experimental experience [28]:
i ~ ^ ^
sin фе = sin 9st
ct1 +CT
0
CTi - sinфst -Сто
(19)
If the major principal stress cti reaches the stationary uniaxial compressive strength acst, the effective angle of internal friction amounts to 9 = 90° and for aj ^ follows 9 ^ 9st. For the combination of angle of internal friction 9 for incipient contact failure (slope of yield locus) with the stationary angle of internal friction <pst following relation is used [4, 30]:
tan 9st = (1 + k) tan 9; (20)
Fig. 5 Shear stress - normal stress diagram of yield characteristics of a cohesive powder. In general, the steady-state flow of a cohesive powder is cohesive. Hence, the total normal stress consists of an external contribution a, e.g. by weight of powder layers, plus (by absolute value) an internal contribution by pre-consolidation dependent adhesion (tensile stress aZ)
normal stress iJ in kPa
Fig. 6 Shear stress - normal stress diagram of yield loci (YL) and stationary yield locus (SYL) of titania powder, straight line regression fit > 0,97, surface diameter dS = 200 nm, solid density ps = 3870 kg/m3, shear rate vS = 2 mm/min, surface moisture XW = 0,4 % accurately analysed by Karl Fischer titration; angles of internal friction 9i = 25...37°, stationary angle of internal friction 9i = 54°, isostatic tensile strength ct0 = 0,33 kPa
The softer the particle contacts, the larger are the differences between these friction angles and consequently, the more cohesive is the powder response.
The instantaneous yield locus describes the limit of incipient plastic powder deformation or yield. A linear yield locus, Fig. 5, is obtained from resolution of a general square function [30], is simply to use (aM,st, CTRst centre and radius of Mohr circle for steady-state flow as parameter of powder pre-consolidation):
: = tan ф; (ct + <jz ) = tan Ф;
( CTR,st ^
CT +-------------CTM,st
sin 9;
(21)
/
It is worth to note here that only the isostatic tensile strength ctz for incipient yield depends directly on the consolidation pre-history and is given by:
CT
R,st
CT Z = _-------------CTM,st =
sin ф;-
sin Фst sin Ф;
-1
sin ф^
CTM,st + _-------------CT0 • (22)
sin Ф;
The smaller a radius stress for pre-consolidation ctw < CTR,st, the larger is the centre stress ctvm > CTM,st right of largest Mohr circle for steady-state flow in Fig. 5, and the smaller may be the powder tensile strength ctz. The so-called consolidation locus lies at the right hand side and represents the envelope of all Mohr circles for consolidation stresses with plastic powder failure, Fig. 5, i.e. the radius ctvr and centre ctvm stresses. Provided that the particle contact failure is equivalent to that between incipient powder flow and consolidation, one can write for a linear consolidation locus with negative slope -sirnpj which is symmetrically with the linear yield locus, Eq. (26):
CTVR = sin(-CTVM + ^iso). (23)
Due to this symmetry between yield and consolidation locus, one can directly estimate the isostatic powder compression cti = ct2 = ctvm = ctiso from Fig. 5 for the radius stress ctvr = 0:
Generally, when we use these radius ctr and centre stresses ctm, the essential flow parameters are compiled as one set of linear constitutive equations, i.e. for instantaneous consolidation, the consolidation locus (CL),
These yield functions are completely described only with three material parameters plus the characteristic pre-consolidation stress CTMst or average pressure influence, see Tomas [30]:
(1) 9! - incipient particle friction of failing contacts, i.e. Coulomb friction;
(2) 9st - steady-state particle friction of failing contacts, increasing adhesion by means of flattening of contact expressed with the contact consolidation coefficient k, or by friction angles (sin<pst - sin9) as shown in the next Eqs. (28) and (29). The softer
the particle contacts, the larger are the difference between these friction angles the more cohesive is the powder;
(3) ct0 - extrapolated isostatic tensile strength of unconsolidated particle contacts without any contact deformation, equals a characteristic cohesion force in an unconsolidated powder;
(4) CTMst - previous consolidation influence of an additional normal force at particle contact, characteristic centre stress of Mohr circle of pre-consolidation state directly related to powder bulk density. This average pressure influences the increasing isostatic tensile strength of yield loci via the cohesive steady-state flow as the stress history of the powder.
These physically based flow parameters are necessary to derive the uniaxial compressive strength ctc which is simply found from the linear yield locus, Eq. (26) and Fig. 5, for ctc = 2ctr (ct2 = 0 and ctr = ctm) as a linear function of the major principal stress ct1;
Equivalent to this linear function of the major principal stress ct1 and using again Eq. (26), the absolute value of the uniaxial tensile strength ctz1 is also found for ctz1 = = Ctr (cti = 0 and Ctr = - ctm):
Both flow parameters ctc and ctz1 depend on the pre-consolidation level of the shear zone which is expressed by the applied consolidation stress for steady-state flow cti. A considerable time consolidation under this major principal stress cti after one day storage at rest is also shown in Fig. 7. Equivalent linear functions are also used to describe these time consolidation effects [30].
(25)
for incipient yield, the yield locus (YL),
(26)
and for steady-state flow, the stationary yield locus (SYL):
CTR,st = sin9st (CTM,st +CT0 )
(27)
Fig. 7, [28]:
c (1 + sin ф^ )(1 - sin Ф; ) 1 (1 + sin ф^ )(1 - sin Ф; ) 0
2^Шф^ - sinф; ) 2sinф^ (1 + sinф;)
----:-----гг-----:-rCT1 +—--------- -------—---- -----CT,
(28)
(29)
consolidation stress a1 in kPa
Fig. 7 Powder strength - consolidation stress diagram of constitutive consolidation function of titania, straight line regression fit = 0,99, dS = 200 nm, XW = 0,4 %
Powder Flowability and Compressibility
In order to assess the flow behaviour of a powder, Eq. (28) shows that the flow function due to Jenike [1] ffc =ct1/ ctc is not constant and depends on the preconsolidation level ct1 . Approximately, one can write for a small intercept with the ordinate ctc 0, Fig. 7, the stationary angle of internal friction is equivalent to the effective angle 9st « 9e and Jenike’s [1] formula is obtained:
f (1 + sin 9e )(1 - sin 9i) (30)
c 2(sin9e - sin9i)
Thus, the semi-empirical classification by means of the flow function introduced by Jenike [1] is adopted here with considerations for certain particle behaviour, Table 1.
Obviously, the flow behaviour is mainly influenced by the difference between the friction angles, Eq.(30), as a measure for the adhesion force slope k in the general linear particle contact constitutive model, Eq. (15). Thus one can directly correlate k with flow function ffc [30]:
K= 1 + (2 ffc - 1)sinф;
tanфг (2 ffc -1 + sinф; )
1-
1 + (2ffc - 1)sinФ; '2
2 ffc - 1 + sin Ф;
-1. (31)
1
Flowability assessment and elastic-plastic contact consolidation coefficient K(^i = 30°)
flow function ffc к-values 9st in deg evaluation Examples
0 о о 0,01006.0,107 30,3.33 free flowing dry fine sand
4...10 0,107.0,3 7 3 3 3 easy flowing moist fine sand
2...4 0,3.0,77 6 4 7 3 cohesive dry powder
1.2 0,77...œ 0 С* 6 4 very cohesive moist powder
< 1 œ - non flowing moist powder
A characteristic value k = 0,77 for ^ = 30° of a very cohesive powder is included in the adhesion force diagram, Fig. 4, and shows directly the correlation between strength and force increasing with pre-consolidation, Table 1. Due to the consolidation function, a small slope designates a free flowing particulate solid with very low adhesion level because of stiff particle contacts but a large slope implies a very cohesive powder flow behaviour because of soft particle contacts, Fig. 7.
Obviously, the finer the particles the “softer” are the contacts and the more cohesive is the powder [27, 28]. Kohler [31] has experimentally confirmed this thesis for alumina powders (a-Al2O3) down to the sub-micron range (ctc0 * const = 2 kPa, d50 median particle size in ^m):
ffc * 2,2d5°062. (32)
A survey of uniaxial compression equations was given by Kawakita [32]. Thus in terms of a moderate cohesive powder compression, to draw an analogy to the adiabatic
gas law pVKad = const, a differential equation for isentropic compressibility of a powder dS = 0, i.e. remaining stochastic homogeneous (random) packing without a regular order in the continuum, is derived, beginning with:
rfpj, = ndp = n «toM* . (33)
Pb p CTM,st +CT0
The total pressure including particle interaction p = aMst + ct0 should be equivalent to a pressure term with molecular interaction (P + «Vdw/Vm)(Vm -b)= RT in van der
Waals equation of state to be valid near gas condensation point. A “condensed” loose powder packing is obtained pb = pb0, if only particles are interacting without an external consolidation stress aMst = 0, e.g. particle weight compensation by a fluid drag, and Eq. (33) is solved:
Pb
pb,0
CTn +a
M,st
Y
(34)
Therefore, this physically based compressibility index n = 1/Kad lies between n = 0,
i.e. incompressible stiff bulk material and n = 1, i.e. ideal gas compressibility. Considering the predominant plastic particle contact deformation in the stochastic homogeneous packing of a cohesive powder, following values of compressibility index are recommended in Table 2.
Compressibility index of powders, semi-empirical estimation for ct1 = 1...100 kPa
index n evaluation examples flowability
0.0,01 0,01.0,05 0,05.0,1 0,1.1 incompressible low compressibility compressible very compressible gravel fine sand dry powder moist powder free flowing cohesive very cohesive
Generally, the influence of micro-properties as particle contact stiffness on the macro-behaviour as powder flow properties, i.e. cohesion, flowability and compressibility, can be directly shown in Fig. 8.
a) particle contact deformation
b) particle adhesion
c) powder yield loci
cohesive /SYL
d) consolidation functions
compliant cohesive
consolidation stress Oj
e) powder constitutive models SYL
ai 0 average pressure о
f) compression function
.compliant compressible
S =-■e
¿4 j
1L
V*
stiff, incompressible
0 consolidation stress Cj
Fig. 8 Characteristic constitutive functions of stiff and compliant particle contact behaviours, free flowing and cohesive powder behaviours, and finally, stiif incompressible and soft compressible powders [40]. Increasing contact compliance determine decreasing slope of the elastic-plastic yield boundary (limit) and increasing inclination of the adhesion boundary or limit. As the result, the slope of the normal force-adhesion force function increases. Next, the difference between the stationary angle and angle of internal friction of the powder becomes larger. Consequently, the slope of the powder consolidation function increases and the powder is more compressible
Conclusions
A complete set of physically based equations for steady-state flow, incipient powder consolidation and yielding, compressibility and flowability has been shown. Using this, the yield surfaces due to theory of plasticity may be described with very simple linear expressions:
ФYL, SYL, CL - 0 -
аR - sinф{ (аM - аM,st ) - аR,s
yield locus (YL)
aR
,st - sin9st (Mst +ct0 ) stationary yield locus (SYL)
consolidation locus (CL)
Tr - sin9j (M +CTM,st )-CTR,st
(35)
The consolidation and yield loci and the stationary yield locus are completely described only with three material parameters, i.e., angle of internal friction 9, stationary angle of internal friction <pst, isostatic tensile strength of an unconsolidated powder ct0 plus the characteristic pre-consolidation (average pressure) influence aM,st. The compressibility index n as an additional constitutive bulk powder parameter was introduced and the classification 0 < n < 1 recommended. A direct correlation between flow function ffc and elastic-plastic contact consolidation coefficient k was derived.
This approach has been used to evaluate the powder flow properties concerning various particle size distributions (nanoparticles to granules), moisture contents (dry, moist and wet) and material properties (minerals, chemicals, pigments, waste, plastics, food etc.), which have been tested and evaluated for more than the last 20 years [27]. Thus, these models are directly applied to evaluate the test data of a new oscillating shear cell [33, 34,35] and a press-shear-cell in the high-level pressure range from 50 to 2000 kPa for liquid saturated, compressible filter cakes [36, 37,38] and for dry powders [39]. Additionally, the force - displacement behaviour during stressing and the breakage probability are useful constitutive functions to describe the mechanics of agglomerates to assess the physical product quality [43]. These contact models are also needed to simulate the shear dynamics of cohesive powders using the discrete element method (DEM) and to calibrate these simulations by shear cell measurements [44].
The influence of particle surface properties, e.g. as contact stiffness, on the powder flow properties can be directly interpreted, Fig. 8, and practically used to design particulate products in process industries [42, 45].
Acknowledgements
The author would like to acknowledge his co-workers Dr. S. Aman, Dr. T. Groger, Dr. W. Hintz, Dr. Th. Kollmann and Dr. B. Reichmann for providing relevant information and theoretical tips. The advices from H.-J. Butt [46] and S. Luding [47] with respect to the fundamentals of particle and powder mechanics were especially appreciated during the collaboration of the project “shear dynamics of cohesive, fine-disperse particle systems“ of the joint research program “Behaviour of Granular Media“ of German Research Association (DFG).
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Адгезия наночастиц и текучесть когезионных нанопорошков
Ю. Томас
Кафедра «Механические процессы и аппараты»
Университет Отто-фон-Герике, Магдебург, Германия
Ключевые слова и фразы: адгезионные силы; когезия; конститутивная модель; механика порошков; механика сыпучих сред; сжимаемость порошков; силы Ван-дер-Ваальса; текучесть порошков; функция свойств; ТЮ2-нанопорошок.
Аннотация: Описаны основы свойств упрочнения и текучести
когезионных порошков на базе объединения механики частиц и механики непрерывной среды. С помощью модели «жесткая частица с мягкими контактами» показано влияние упруго-пластичного отталкивания при контакте частиц. На этих физических основах представлены стационарная линия текучести, индивидуальные линии текучести, линии упрочнения, функция текучести сильно когезионного ТЮ2-нанопорошка (гидравлический диаметр ^ = 200 нм). Модели используются для оценки результатов сдвиговых испытаний. Эти функции свойств используются также для компьютерных расчетов технологических аппаратов на надежное истечение порошка.
Nanopartikelhaftung und Fliesseigenschaften kohäsiver Nanopulver
Zusammenfassung: Die Grundlagen des Verfestigungsverhaltens und des Fließverhaltens kohäsiver Pulver werden erläutert, in dem die Partikelmechanik und die Kontinuumsmechanik sinnvoll miteinander verknüpft werden. Mittels des Modells „steife Partikel mit weichen Kontakten“ wird der Einfluß der elastisch-plastischen Abstoßung im Partikelkontakt gezeigt. Auf dieser physikalischen Grundlage werden der stationäre Fließort, die Momentanfliessorte, die Verfestigungsorte, die Fließfunktion, die Verfestigungsfunktion und die Kompressionsfunktion anhand der Fließeigenschaften eines sehr kohäsiven TiO2-Nanopulvers (Sauterdurchmesser ds = 200 nm) demonstriert. Die Modelle dienen der Bewertung der Ergebnisse von Scherversuchen. Diese Eigenschaftsfunktionen werden wiederum gebraucht, um verfahrenstechnische Apparate fließgerecht und computergestützt auszulegen.
Adhésion des nanoparticules et fluidité des nanopoudres cohésives
Résumé: Sont décrites les bases des propriétés de la consolidation et de la fluidité des poudres cohésives à la base de la réunion de la mécanique des particules et celle du milieu continu. A l’aide du modèle «particule dure avec des contacts souples» on a montré l'influence de la répulsion flexible et plastique lors du contact des particules. Sur ces bases physiques sont proposées la ligne stationnaire de la fluidité, les lignes individuelles de la fluidité, les lignes du renforcement, la fonction de la fluidité de la nanopoudre TiO2 très cohésive (le diamètre hydraulique ds = 200 nm). Les modèles sont employés pour l’évaluation des résultats des essais de décalage. Ces fonctions sont utilisées de nouveau pour le calcul des appareils technologiques sur l’écoulement sûr de la poudre à l’aide de l’ordinateur.
