A kinetic model for anisotropic reactions in
amorphous solids
Wei Hong*
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA
ABSTRACT
When mechanical constraints are present, solid-state reactions often induce deviatoric inelastic
strains in addition to volume change. Existing models either attribute such deformation to the
plastic flow driven by the stress exceeding a non-measurable kinetic-dependent yield strength, or
need to introduce a deviatoric-stress-dependent chemical potential.
By employing the
transformation strain to characterize the state of reaction, this letter formulates a kinetic model
via averaging the reaction rate at all possible orientations. The model is illustrated through the
constrained lithiation-delithiation process of silicon as an example.
With just one fitting
parameter, the model can quantitatively capture the experimental results.
The model only
hypothesizes linear kinetics, and does not need to introduce kinetic-dependent plasticity or
modify basic thermodynamic quantities. This approach can also be applied to other material
systems, as well as extended to the nonlinear kinetics of far-from-equilibrium reactions.
1
Solid-state reactions are seldom regarded a new topic in material synthesis, but when it comes
to the strains or stresses induced by reactions, little has been explored beyond the similarities
with liquid- or gas-state reactions. With the acceptance of another reactant through diffusion or
other means of transportation, a solid reactant may expand in volume and/or change in shape.
The common approach of modeling such processes is to assign a stress- and kinetics-independent
transformation strain to the resultant.1-4 The actual state of deformation is then calculated based
on linear elasticity (or elasto-plasticity) as if the stress was applied after the reaction. Such a
decoupled approach may be applicable to a reaction of which a transformation strain is clearly
defined, e.g. one with a background crystalline reactant that retains its coherency throughout the
process, as sketched in Figure 1a. However, this approach may be problematic if either the
reactant or the resultant is amorphous, and thus the transformation strain or its orientation cannot
be uniquely determined by the state of reaction.
As illustrated by Figure 1b, without
significantly rearranging the spatial distribution of the reactant atoms, the resultants can bear
very different transformation strains, depending on the relative positions of the inserted atoms.
Variants of the resultants as those sketched in Figure 1b are chemically identical, but
mechanically different if the material is stressed or constrained.
To model the deformation involved in such reactions, it has been assumed that the
transformation strain of an amorphous reaction is purely volumetric, and a separate process of
plastic flow generates deviatoric inelastic strain, when the equivalent stress exceeds a threshold –
the yield strength. However, the yield strength needs to be taken as composition and kinetics
dependent.3,4 Such an assumption is perhaps originated from the description of liquid reactions,
in which the state of reaction can be fully described by a scalar variable. For solid reactants, the
assumption is less convincing.
The plentiful observations on reactions which transform
2
crystalline solids into amorphous resultants5-11, and complete within atomically sharp phase
boundaries12, may serve as counter evidences of this assumption: the amorphization would
require plastic flow that is non-affine at atomic level which defies the applicability of continuum
notions in the first place. It is natural to believe that such deformation takes place right at the
insertion of the second reactant, instead of after the reaction.
a
b
Figure 1.
Schematics of reaction-induced deformation.
(a) For a coherent crystalline-to-
crystalline reaction, the transformation strain is well-defined.
(b) For an amorphous-to-
amorphous reaction, the deviatoric strain is arbitrary from the chemical consideration only – with
the same relative positions and average atomic distances, the resultants may differ by a
deviatoric strain.
3
The ambiguity could be clarified if one allows additional variables for the state of reaction. To
differentiate between resultant variants like those sketched in Figure 1b, one needs a state
variable which contains the orientation information. While the choice is more or less arbitrary,
in this letter we select the transformation strain tensor ε t as the state variable. The rate of the
reaction, represented by the time derivative of the transformation strain, ε t , is a function of the
electrochemical driving force for the reaction  and the stress tensor σ .
To simplify
representation, we further decompose ε t into the volumetric and deviatoric parts:
ijt 
v
3
 ij  ~ijt .
(1)
Although the isotropy of the material requires the volumetric strain rate to be dependent only on
the electrochemical driving force and the hydrostatic stress, the deviatoric strain rate ~ε t could be
~ . In a simple case when the dependence is linear,
dependent on the deviatoric stress σ
~ijt  Tijkl~kl ,
(2)
where T is a fourth rank tensor relating the two deviatoric tensors, and the repeated indices
indicate a summation. The isotropy of the amorphous material requires T to contain only one
scalar parameter  , and the transformation strain rate to be parallel to the stress deviator:
~ijt  ~ij .
(3)
Although the kinetic relation (3) shares the same form as that of a Newtonian fluid, here  is
not a material constant, and could be dependent on the concentration of the mobile reactant or
the energetic driving force.
Instead of driven by shear stresses as a viscous fluid, the
transformation strain should be regarded as a part of the reaction process, and is nonzero only
when the reaction is taking place. It should be noted that relations in the form of Eq. (3) also
appear in most existing theories of related phenomena,1,3,4,13 but the difference lies in the
4
coefficient  and the underlying physical interpretation. For example, the similar expression in
the reactive flow theory is interpreted as the plastic flow driven by a stress exceeding the yield
strength.1,2 In some other models, the reactant chemical potential needs to be modified to
include a contribution from deviatoric stress,13,14 and thus the scalar reaction rate (or volumetric
strain rate) is also affected. In contrast to the existing theories, the current model makes no
hypothesis other than linear kinetics and linear elasticity, as detailed in the following discussion.
At the atomistic scale, the resultants of solid-state reactions are seldom isotropic. In the
extreme case when two atoms meet and react, the resultant group is non-spherical and always
takes an orientation. For amorphous solids, atomistic studies also show that it is energetically
more favorable for a group of few resultant atoms to have an orientation than to be spherical or
isotropic.15 In the absence of a directional field (e.g. electric field or stress), the orientations of
different atomic groups are randomly taken and thus the average properties often exhibit
macroscopic isotropy. Here we will focus on the case when non-negligible deviatoric stresses
are present, so that the orientation distribution could be affected. We will use the Li-Si system as
a timely example, but the methodology is generally applicable to almost all solid-state systems
with amorphous reactants or resultants.
For ease of description, we imagine a group of very few Si atoms and introduce an elemental
deformation gradient tensor to represent the deformation caused by reaction
1t 0

F t   0 t2
0 0

0

0,
t2 
(4)
where 1t and t2 are the principal stretches of transformation, functions of the nominal Li
concentration C . The components of F t are written in the local coordinate system which aligns
with the principal directions, and it is also assumed that the transformation of the Li-Si group has
5
transverse isotropy, which represents the case of 1:1 Li-Si atomic ratio as suggested by atomistic
simulations15. The transformation F t can take general anisotropy and the resulting macroscopic
rate equation still takes the form as in Eq. (3), but the coefficient  will depend on more
parameters. On the other hand, the available experimental results are mostly unidirectional and
insufficient to distinguish these anisotropies. The transverse anisotropy is thus taken as a timely
approximate. In the context of finite deformation, the local inelastic strain rate due to lithiation
is given by
 1t
 t
 1
1

ε l  F t  F t   0

0

0
t2
t2
0

 d1t
0
 t

 1dC


0   C  0


t2 
 0

t2 
0
dt2
t2 dC
0

0 


0 .

dt2 
t2dC 
(5)
In general, the derivatives d1t dC and dt2 dC are dependent on the composition as well as the
direction of the reaction (e.g. lithiation or delithiation), and may be identified through atomistic
simulations.15,16 Here for simplicity, we take an approximation by assuming both d1t 1t dC and
dt2 t2dC to be material constants and write them in terms of the volumetric contribution  and
deviatoric contribution  :
dt2
 
d1t

  .



and
t
t
2dC 3 2
1dC 3
(6)
The free-energy landscape of the reaction is illustrated schematically by Figure 2, whereas the
actual reaction space should be three dimensional. In the absence of deviatoric stress or any
other directional driving force, the energy profile is isotropic, as sketched in Figure 2a. When
~ , the free energy of the resultant is orientation
the material is under a non-zero deviatoric stress σ
dependent, as sketched in Figure 2b. Here we assume the driving forces to be much lower than
6
thermal fluctuation, and employ linear kinetics for the rate of reaction. Locally for a small
atomic group, the transformation strain within unit time is given by Eq. (5), and its orientation by
the azimuth and zenith angles  ,  of the local x1 -axis in the global coordinate system. The
linear kinetics assumption dictates the rate of reaction and thus the probability density of
transformation in orientation  ,  to be proportional to its electro-chemo-mechanical driving
force, Pr ,   C  det F t σl  ,  : ε l , in which  is the electrochemical driving force in
the absence of stress, and σ l is the stress tensor σ transformed into the local coordinates through
the rotation matrix Q ,  , σl  Q  σ  QT . It should be pointed out that the driving force is for
reaction only,3,4 which differs from the electrochemical potential for the migration of reactants.
The factor det F t accounts for the finite volume change during the reaction, as the energy terms
are written in the reference configuration, i.e. with respect to the number of background Si
atoms. After normalization, the probability density could be written explicitly as
a
b
B2
B
A
A
B1
~0
σ
~0
σ
Figure 2. Sketches of the energy landscape of a solid-state reaction. a) In a hydrostatic stress
state, the energy profile is isotropic and the material undergoes a pure volumetric transformation.
7
~ , the free energy is lowered for anisotropic reactants
b) Under non-negligible deviatoric stress σ
at particular orientations, due to the contribution from elastic strain energy.
ijl
l





1



v
ij 
C  1   v  
1
C ,
Pr ,  

4C  1   v   ijl ijl sindd 4   1     
v
kk
3
l l
ij ij
(7)
where  v  det F t  1 is the volumetric strain due to reaction. With the aid of the probability
distribution (7), we can calculate the expectation of the macroscopic inelastic strain rate due to
reaction
l
ijt   Qkikm
Qmj Pr , sindd .
(8)
Substituting Eq. (7) into (8) and carrying out the integrals over the surface of a unit sphere, we
arrive at


~
2




1




3
v
ij
ijt  C   ij 
,
10   1      
3
v
nn
3


(9)
As expected, the statistical analysis of the reactions in different orientations recovers the form
from symmetry consideration, Eqs. (1) and (3), in which the unknown coefficient  is now given
explicitly by

3 2
1   v C ,
10
(10)
where     1   v  kk 3 is the total (electro-chemo-mechanical) driving force for the
reaction.
The total driving force  is related to, but should not be confused with the
electrochemical potential for the diffusion of reactants. It could be easier to understand it by
imagining a virtual source that directly injects the reactants to the place of reaction without any
8
transportation process.3,4 Different from a Newtonian flow, the deviatoric strain is only activated
in a reaction process, when C  0 . On the other hand, the volumetric strain rate v  C  or the
commonly invoked isotropic reaction rate C is independent of the deviatoric stress state. As a
kinetic equation, Eq. (9) should be regarded as an extension to the commonly used linear kinetic
relation of isotropic reactions. Unlike existing models which result in similar mathematical
forms from different asumptions,13 Eq. (9) is only a consequence of linear kinetics and no
modification on any thermodynamic quantities (e.g. the chemical potential) is needed. In the
extreme case of an equilibrium or quasi-static state, the model would have no effect over that
governs by thermodynamics. To complete the mathematical formulation, a stress-strain relation
of linear elasticity (or elasto-plasticity) is needed in addition to Eq. (9).
In the case when  is not explicitly rate dependent, Eq. (9) can be reduced to a rateindependent form
d ijt
dC


3
 ij 
3 2
1   v ~ij .
10
(11)
Furthermore, if the kinetics of the system is limited by a constant reaction rate (e.g. during
galvanostatic lithiation of silicon), the total driving force  may be regarded as constant. As
neither  nor  could be directly measured, it is more convenient to leave the ratio  2  as a
single fitting parameter to be determined by comparing to experiments.
As a simple example to illustrate the model and compare with existing experiments, we
calculate the lithiation process of a Si film constrained laterally by the underlying rigid substrate.
For simplicity, we will start with the thin-film limit, in which both the deformation and the Li
concentration are homogeneous throughout the film. The principle values of inelastic strain are
the through thickness components  1t and the lateral component  2t . Here for simplicity, we
9
neglect plasticity or any inelastic deformation other than that induced by reaction. Due to the
lateral constraint of the rigid substrate, the lateral elastic strain  2e   2t .
Therefore, for
relatively small  2t , the lateral stress component
E 2t
,
2  
1 
(12)
where E is Young’s modulus and  Poisson’s ratio. Combining Eqs. (11) and (12), we arrive at
the differential equation for  2 :
d 2
E

dC
1 
  1   v  2 2 
 
.
10
 
3
(13)
In the case of small volumetric strain,  v  1 , this simple dynamic system has an attractor at
 2   103   2 . Depending on the sign of  , or whether the system is under a lithiation or
delithiation process,  2 takes a negative or positive value, respectively, corresponding to the
plateau stresses observed in experiments.
At any substantial Li concentration level, the volume expansion due to lithiation is actually
non-negligible. However, if the stress change due to the geometric effect of volume expansion is
much smaller than that due to constraint, i.e. the dimensionless combination E 2   10 , the
attractor for stress  2   103   2 1   v  still exists, just the value would tapper off with
volume expansion at higher Li concentrations.
Equation (13) contains 6 parameters, most of which are directly measurable. Here, we take the
representative values for the Young’s modulus E  80 GPa and Poisson’s ratio   0.22 of
amorphous silicon.17 The parameter  is related to the volumetric transformation strain  v C  .
From the commonly used linear relation,18-20 we obtain approximately   0.5 and
10
 v  exp x   1 .
(For direct comparison, we have adopted the dimensionless measure of
concentration, the x in Li xSi .) The only parameter that is not directly measurable is the ratio
 2  . By taking the value  2   0.8 GPa-1 and numerically integrating Eq. (13), we reached a
good agreement with the experimental data19 for both lithiation and delithiation processes, as
plotted in Figure 3. If we take the value of the deviatoric transformation strain,   2.4 , from
the result of atomistic simulations between the composition 0  x  1 ,15 the fitting parameter
 2   0.8 GPa-1 corresponds to a driving force of approximately 0.9eV per Si atom from Si to
LiSi, which is within the reasonable range for lithiation reactions. The reaction is thought to be
accomplished through numerous small steps between Si and LiSi, each of which driven by a
fraction of the total energy drop (0.9eV), and the linear kinetics assumption is not violated. It
should also be noted that the driving force  changes its sign from lithiation to delithiation, and
consequently   2 is also negative during delithiation.
Theory
Experiment
2
 2 (GPa)
1
0
-1
-2
0
0.5
1
x in Lix Si
1.5
2
Figure 3. Calculated stress-composition loop during a galvanostatic lithiation-delithiation cycle
of a lateral constrained Si sample with homogeneous deformation. The experimental data (dots)
11
are extracted from Ref. 17. With only one fitting parameter, the model recovers the stress
evolution during both lithiation and delithiation processes.
Here for the purpose of illustration, we have taken many parameters to be composition
independent. It is believed that if these parameters, such as E C  ,  C  , and  v C  are properly
measured or calculated,15,16 even better agreement with experiments could be achieved.
Compared to the model of reactive plastic flow,19 these composition-dependent material
parameters only depends on the thermodynamic state rather than the kinetics or rate of reaction,
and could thus be much easily measured ex-situ.
After all, the change in the overall shape of
the curve will be small.
Although the current model of anisotropic reaction needs much less fitting parameters, it
recovers the experimental observation of 1D constrained lithiation of amorphous Si equally well
as the model of reactive flow.3,4,19 Moreover, it is expected that the two models will give similar
predictions in other stress states, as the transformation strain in the current model and the plastic
strain in flow theories are taken to be proportional to the stress deviator. However, as the two
models have different physical origins, it is still possible to distinguish them experimentally. Let
us still consider the lithiation process as an example. The reactive flow model considers the
plateau of stress to be a consequence of plastic yielding. In other words, every point along the
plateau of the curves in Figure 3 corresponds to a state in which the solid has already yielded. In
the current model, on the other hand, the material is deformed elastically throughout the entire
process (if the stress is below the yield strength of the solids). Despite the assumption that the
yield strength is reaction dependent,19 a yielded solid is easily distinguishable from one in elastic
regime. For example, one can apply an instantaneous mechanical load in situ to the sample
during lithiation / delithiation (e.g. by bending the cantilever backward). The additional load
12
will result in a permanent plastic deformation according to the reactive flow theory, while the
current model predicts a full recovery upon unloading if the additional load is not enough to
yield the sample mechanically.
Another major difference lies in the initial steep part of a lithiation or delithiation curve which
is thought to be elastic in the reactive flow model, while in the current model the inelastic
deviatoric deformation takes place whenever a stress deviator is present. Complicated by the
formation of the solid electrolyte interface layer, the initial lithiation stage during the first cycle
of a virgin sample may not be the best candidate for comparison.
If we look at the
lithiation/delithiation curves of subsequent cycles,20 the initial part tends to be more curved as
predicted by the current model (Figure 4a). Moreover, we suggest measurements to be carried
out over a small lithiation-delithiation cycle, in which the composition change is so small that the
resultant stress is much lower than the plateau level of a complete cycle. As shown in Figure 4b,
the current model predicts a stress-composition hysteresis loop that gradually shifts in stress
values, due to the reaction-induced deviatoric deformation. In contrast, in the absence of plastic
flow, the reactive flow model simply predicts overlapping lithiation and delithiation curves (or
straight lines), and no shift (or relaxation) in subsequent cycles. Once this prediction is verified,
such experiments can be used to determine some of the composition-dependent parameters or
kinetic parameters, so that no fitting parameter will be needed in the model, and more accurate
prediction could be achieved.
13
a
b
2
1.5
1
 2 (GPa)
1
 2 (GPa)
2
0
-1
0.5
0
-0.5
-1
-2
0
0.5
1
x in Lix Si
1.5
2
-1.5
0.45
0.5
0.55
x in Lix Si
0.6
0.65
Figure 4. Stress evolution in repeated lithiation-delithiation cycles. a) Relatively large cycles of
Li concentration x  0 ~ 2 , with slight shifted state of charge between each cycle. b) Small
cycles around x  0.55 , in which the stress clearly exhibits inelastic hysteresis and relaxation.
In summary, this letter presents a kinetic model for the anisotropic deformation induced by
reactions in amorphous solids. The model uses the inelastic transformation strain to characterize
the state of reaction, including the orientation information of the resultant, and makes no
hypothesis beyond linear kinetics. The model carries much fewer parameters than existing
theories describing similar phenomena, and most parameters can be directly measured or
calculated through atomistic simulations. The model is a natural extension to the linear kinetic
law of reactions, and needs no modification on thermodynamics or linear elasticity. The model
clearly differentiates the inelastic transformation strain due to reaction and the plastic strain due
to stress. The model is illustrated through the example of constrained lithiation and delithiation
of Si thin film, and compared to published experiments. With only one fitting parameter, the
prediction of the model reaches a satisfactory agreement with experimental measurements.
Some further experiments have been suggested to validate the current model and distinguish it
from existing reactive flow theories. We are eagerly waiting for experimental verifications.
14
Although Li-Si is used as a sample system here to illustrate the theory, it is believe that the
current model can also be applied to other material systems of amorphous solid-state reactions,
especially those with large volumetric changes or under large deviatoric stresses. The current
approach may be further extended to model the nonlinear kinetics of far-from equilibrium
reactions, although the resulting kinetic relations is expected to be more complex than that given
herein.
AUTHOR INFORMATION
Corresponding Author
*E-mail: whong@iastate.edu.
ACKNOWLEDGMENT
WH acknowledges the National Science Foundation for the support through grant CMMI1000748.
REFERENCES
(1) Bower, A. F., Guduru, P. R., Sethuraman, V. A., J. Mech. Phys. Solids 2011, 59 (4), 804828.
(2) Liu, X. H. et al, Nano Lett. 2011, 11, 3312-3318.
(3) Brassart, L.; Suo, Z. Int. J. Appl. Mech. 2012, 4 (03), 1250023.
(4) Brassart, L; Suo, Z. J. Mech. Phys. Solids 2013, 61 (1), 61-77.
(5) Li, F.; Zhang, L.; Metzger, R. M. Chem. Mater. 1998, 10 (9), 2470-2480.
15
(6) Li, H.; Huang, X. J.; Chen, L. Q.; Zhou, G. W.; Zhang, Z.; Yu, D. P.; Mo, Y. J.; Pei, N.
Solid State Ionics 2000, 135 (1–4) 181-191.
(7) Gong, D.; Grimes, C.; Varghese, O. K.; Hu, W.; Singh, R. S.; Chen, Z.; Dickey, E. C. J.
Mater. Res. 2001, 16 (12), 3331-3334.
(8) Obrovac, M. N.; Christensen, L. Electrochem. Solid-State Lett. 2004, 7 (5) A93-A96.
(9) Chan, C. K.; Peng, H. L.; Liu, G.; McIlwrath, K.; Zhang, X. F.; Huggins, R. A.; Cui, Y.
Nat. Nanotechnol. 2008, 3 (1) 31-35.
(10) Key, B.; Bhattacharyya, R.; Morcrette, M.; Seznec, V.; Tarascon, J. M.; Grey, C. P. J.
Am. Chem. Soc. 2009, 131 (26) 9239-9249.
(11) Wan, W. H.; Zhang, Q. F.; Cui, Y.; Wang, E. G. J. Phys.: Condens. Matter 2010, 22 (41)
415501.
(12) Chon, M. J.; Sethuraman, V. A.; McCormick, A.; Srinivasan, V.; Guduru, P. R. Phys.
Rev. Lett. 2011, 107 (4) 045503.
(13) Levitas, V. I.; Attariani, H. Sci. Rep. 2013, 3, 1615.
(14) Cui, Z.; Gao, F.; Qu, J. J. Mech. Phys. Solids 2012, 60 (7), 1280-1295.
(15) Shenoy, V. B.; Johari, P.; Qi, Y. J. Power Sources 2010, 195 (19), 6825-6830.
(16) Zhao, K. J.; Wang, W. L.; Gregoire, J.; Pharr, M.; Suo, Z. G.; Vlassak, J. J.; Kaxiras, E.
Nano Lett. 2011, 11 (7) 2962-2967.
16
(17) Freund, L.B.; Suresh, S., Thin film materials, Cambridge University Press, Cambridge,
UK, 2003, p. 96.
(18) Sethuraman, V. A.; Chon, M.J.; Shimshak, M.; Van Winkle, N.; Guduru, P. R.
Electrochem. Comm 2010, 12, 1614-1617.
(19) Zhao, K; Tritsaris, G. A.; Pharr, M.; Wang, W. L.; Okeke, O.; Suo, Z.; Vlassak, J. J.;
Kaxiras, E. Nano Lett. 2012, 12 (8), 4397-4403.
(20) Nadimpalli, S. P. V.; Sethuraman, V. A.; Bucci, G.; Srinivasan, V.; Bower, A. F.;
Guduru, P. R. J. Electrochem. Soc. 2013, 160, A1885-A1893.
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