J Wheeler
email: johnwh@liv.ac.uk

Anisotropic rheology during grain boundary diffusion creep and its relation to grain rotation, grain boundary sliding and superplasticity

Keywords: creep, deformation mechanisms, mechanics of materials, grain boundary diffusion, microstructure, superplastic deformation

The response of periodic microstructures to deformation can be analysed rigorously and this provides guidance in understanding more complex microstructures. When deforming by diffusion creep accompanied by sliding, irregular hexagons are shown to be anisotropic in their rheology. Analytic solutions are derived in which grain rotation is a key aspect of the deformation. If grain boundaries cannot support shear stress, the polycrystal viscosity is extremely anisotropic. There are two orthogonal directions of zero strength: sliding and rotation cooperate to allow strain parallel to these directions to be accomplished without any dissolution or plating. When a linear velocity/shear stress relationship is introduced for grain boundaries, the anisotropy is less extreme, but two weak directions still exist along which polycrystal strength is controlled only by the grain boundary "viscosity". Irregular hexagons are characterised by 4 parameters. A particular subset of hexagons defined by 2 parameters, which includes regular hexagons as well as some elongate shapes, shows singular behaviour. Grain shapes that are close to that of the subset may exhibit large grain rotation rates and have no well-defined rheology unless there is a finite grain boundary viscosity. This new analysis explains why microstructures based on irregular but near equi-axed grains show high rotation rates during diffusion creep and it provides a framework for understanding strength anisotropy during diffusion creep.

Introduction

In crystalline materials, grain boundary diffusion creep (Coble creep in metals and ceramics, or solution-aided mass transfer in rocks) is an important deformation mechanism [1][2][3]. It plays a role in superplastic behaviour of metals, and is important in how the Earth deforms on long timescales [START_REF] Drury | Earth's Mantle: Composition, Structure and Evolution[END_REF]. It is established that grain boundary sliding must accompany diffusion to maintain compatibility between grains [START_REF] Raj | [END_REF]6].

Grain rotations are an integral aspect of diffusion creep [7]. They are particularly important as they are commonly cited as the reason (at least in part) why grains remain quite equi-axed at high strains during superplastic deformation. In Wheeler page 2 geology grain rotations are important for that reason and more. In rocks there is commonly a lattice preferred orientation (LPO) which gives rise to elastic anisotropy and hence anisotropy in seismic wave velocities. Thus LPO can be detected deep in the Earth via monitoring seismic waves. Grain rotations during diffusion creep can in principle destroy seismic anisotropy by rotating different grains in different ways. This has been postulated as a mechanism for destroying LPO but critical experiments are lacking. Wheeler [8] gave a numerical model for rotations during diffusion creep and shows that they may be limited in extent under particular imposed deformations.

Consequently grain rotations should be the focus of further research.

The flow law for diffusion creep shows that strain rate is proportional to stress, and grain size has a major effect: for Coble creep strain rate is proportional to the inverse cube of grain size. Much less has been said about the possible anisotropy of rheology, although it is predicted to occur [9,10]. Anisotropy may be the result of elongate grain shapes, which mean diffusion path lengths are different in different directions. Anisotropy is important because it can give rise to strain localisation. For example, envisage an anisotropic polycrystal with local areas in "soft" orientations relative to an imposed shear stress. These soft orientations will focus strain which, depending on the link between microstructure and rheology, may evolve to be yet weaker and hence localise strain further.

Numerical models of diffusion creep encompass polycrystals with irregular grain shapes [2,11] and include the combined operation of stress-induced diffusion, grain rotation, and grain boundary sliding, sometimes with a shear stress term [12], and the effects of different imposed deformations (pure and simple shear etc.) [8].

In general, the behaviour of a large complex array of grains must be dealt with by numerical modelling. Grain-scale analytic studies can address only simple microstructures, but provide more insight into the reasons for particular behaviours.

Examples include parallelepipeds [9], regular hexagons [START_REF] Raj | [END_REF], cubes [13], orthorhombic shapes [10], cylinders [6], elongate hexagons with a mirror plane [12], and regular hexagons in shear [14].

In this contribution I present the analytic solution for the rheology and grainrotation behaviour of a periodic microstructure made of one grain shape (an irregular hexagon). There are 4 motivations for this study.

1. An analytic solution for a regular microstructure can provide a basis for understanding behaviour of irregular microstructures (which can be regarded in a Wheeler page 3 general fashion as "perturbed" regular microstructures). In particular numerical simulations of irregular microstructures show that relative grain rotations are much larger when grains are equi-axed than when they are elongate [8]: the analysis here will show why that is the case.

2. There are various separate treatments of hexagonal microstructures with different imposed deformations in the literature and it is not clear how these analyses link together to form a coherent whole: the full 2D solution for arbitrary hexagonal shapes and general deformations has never been presented. As I will show, the full theory provides substantial new insight.

3.

The analytic solution provides a framework for understanding rheological anisotropy as a function of grain shape, and the effects on rheology of shear stress along grain boundaries.

4.

The analytic solution provides a test for the correctness of numerical methods.

I begin the next section with a kinematic analysis: I show how an imposed strain rate field will relate to local grain rotations, dissolution/precipitation rates and sliding rates along boundaries in a periodic microstructure. This is followed by deriving the general equations for force balance, which are applied to diffusion creep with and without shear stress on grain boundaries. I then show how these equations are solved (for any grain shape and any imposed deformation rate) to derive the angular velocity of grains, the stress, and the viscosity tensor for the polycrystal. Specific grain shapes, or families of shapes defined by one parameter, are used to illustrate the general predictions. Finally I discuss the general significance of this work in understanding diffusion creep.

Periodic microstructure: geometry and kinematics

In this section I establish the basic description of the hexagonal tessellation and show how an imposed deformation rate tensor can, together with a specified angular velocity, be used to predict growth/dissolution rates along each grain boundary. Figure 1 shows a microstructure made by tessellating a single grain shape, that of an irregular hexagon. Each hexagon has 3 pairs of parallel faces, denoted by vectors B i . These are linked by vectors R i defining the translational symmetry of the tessellation. The choice of vectors is made so that the actual shape is the same if there is a cyclic interchange of the three boundary vectors (123 → 312 etc.), which is advantageous for analysis since it implies that many equations should be invariant under such cyclic interchanges.

It can be seen that:

R 1 = B 2 @B 3 R 2 = B 3 @B 1
(1)

R 3 = B 1 @B 2
There are 2 distinct types of triple junction in this microstructure, TJ a and TJ b which are not related to each other by translational symmetry.

Suppose that a deformation rate D, homogeneous on the large scale, is imposed on this microstructure. Deformation at the scale of one grain will be accomplished by dissolution, precipitation and sliding at grain boundaries, with no internal strain in the grain, but possibly with rotation of the grain. In general the movement of each grain is characterised by a velocity w at the origin and an angular velocity ω (positive anticlockwise) [2], so that the velocity of any point x in that grain is given by:

v = w + ωSx (2)
where S is the skew tensor

S = 0 @1 1 0 d e (3)
Although the behaviour at grain scale is heterogeneous, consider different grains G1 and G2. It is to be expected that any two points x G1 and x G2  related by a translational symmetry vector R will have velocities related by

v G2 @v G1 = DR (4)
If there is any rotation, there must be a single value of angular velocity for all grains.

So when two grains are related by R we must have

w G2 + ωS x + R `a@w G1 @ωSx = DR so w G2 @w G1 = D @ωS `aR
The important kinematic quantity is the relative velocity of two grains at their contact.

Using Eq. ( 2) for grains G1 and G2 but with the same value of x we find that, regardless of the boundary position, Wheeler page 5

∆v = w G2 + ωS x @w G1 @ωSx = D @ωS `aR (5)
This can be split into normal and transverse relative velocities for each boundary i, using the unit vector N i normal to the boundary and unit vector T i parallel to it, with the signs being governed by the "anticlockwise" construction of the vectors B i (Fig. 1).

u i = N i AD @ωS `aR i (6)
(positive indicates moving apart of the two grains) and

t i =T i AD @ωS `aR i (7)
(positive indicates clockwise or dextral shear component along boundary). Note that if a deformation rate is imposed, the angular velocity is the only unknown kinematic variable.

Force balance

For this regular tessellation there are some relationships between the local forces on grain boundaries and the average (macroscopic) stress which prove useful later. It is assumed that the material deforms slowly, so there are no accelerations or angular accelerations. This means that there is no net force and no net moment on any grain [2]. Let F i be the net force along boundary i pointing into the grain. Then, by translational symmetry and force balance, the boundary opposite to i has force -F i acting on it. Thus, force balance is assured. Suppose we have some arbitrarily complicated traction distribution along each grain boundary i, specified by f i (s) acting outwards from the grain, where s is distance along the boundary. The moment due to this is given by Z f s `aB x ds (8) where the 2D cross product (a scalar) is indicated Because of translational symmetry, the force distribution on one member of the pair must be the opposite of that on the other. So the contribution of a pair of boundaries to the moment is

a B b = a x b y @a y b x = b ASa (9)
Z 2 bdys f s `aB x ds = Rf s `aB x ds + R@f s `aB x + R `ads = R@f s `aB R ds = F B R ( 10
)
where F is defined as the net force on the boundary, pointing into the grain. The net moment on the grain is obtained by summing over the 3 pairs of boundaries:

0 =X i = 1 3 F i B R i ( 11
)
What is the average value of the stress, which could be used to describe the large-scale behaviour of the material as if it were homogeneous? We take compressive stresses as positive. It is known that the average value of stress within a region can be written in terms of forces f α at the boundary of the region, regardless of the complexity of stress within that region [15, 16 p. 389] in 3D

Rσ αβ dV = D f α x β dS
where dV is a volume element and dS a surface area element, and in 2D

Rσ αβ dA = D f α x β ds
where dA is an area element and ds is a boundary length element, or, where ⊗ indicates the outer (tensor) product of two vectors,

RσdA = Df N x ds (12)
f is the force resolved along an outwards-facing normal. Let the 2D region under consideration be one grain. Then the boundary consists of three pairs of parallel faces.

Because of translational symmetry, the force distribution on one member of the pair must be the opposite of that on the other. So

f N x ds = Rf s `aN x ds + R@f s `aN x + R `ads = R@f s `aN R ds = @F N R
Hence the average stress in the grain is obtained by summing over the 3 pairs of boundaries (see also [START_REF] Rodin | [END_REF]):

σ = 1 A fffff X i = 1 3 F i N R i ( 13
)
The stress tensor must be symmetric, but this is assured as a consequence of Eq. ( 11), noting that a⊗b is a symmetric tensor iff a × b = 0.

Equations for grain boundary diffusion

The previous two sections are quite general insofar as they make no assumptions about the details of stress distributions. The aim in this section is to show how precipitation rates (velocity differences at grain boundaries) in grain boundary diffusion creep relate to forces on the boundaries. Diffusion is driven by gradients in chemical potential (in metals, these are linked to gradients in vacancy concentration) which, in diffusion creep, are in turn related to gradients in normal stress at interfaces: for details see Ford et al. [2]. The diffusive current along a boundary is given by

C = @LVw ∂σ n ∂s ffffffffffff ( 14
)
where L is the Onsager diffusion coefficient, w is the effective grain boundary width, and V the molar volume and C is current in mol/m/s. σ n is the component of stress normal to the boundary (compressive positive). The Onsager coefficient is used so that the same mathematical framework can apply to diffusion of vacancies (as in metals) or of ionic species dissolved in an aqueous grain boundary film (as in rocks), compare for example Cocks et al. [START_REF] Cocks | Advances in Applied Mechanics[END_REF] eqn. 2.4. If grains move apart at velocity u, mass balance together with Eq. ( 14) dictates that For convenience we define Z = LV 2 w. In a complicated microstructure, grains can rotate relative to each other in which case u is a linear function of s and σ n is cubic in s. However, in the microstructure discussed here, there are no differential rotations and σ n is quadratic. Consequently we can integrate (15) to obtain

u = @V ∂C ∂s ffffffffff = LV 2 w ∂ 2 σ n ∂s 2 ffffffffffffffff (15)
σ n s `a= σ s + σ e 2 ffffffffffffffffffffff @ uB 2 8Z fffffffffff + σ e @σ s B fffffffffffffffffffffff s + u 2Z ffffffff s 2 (16)
where σ s and σ e are the normal stress values at the start and end of the boundary, B is the boundary length and s is distance along the boundary with the centre point as origin (so it runs from -B/2 to B/2). The current is then

C = Z V fffff σ s @σ e B fffffffffffffffffffffff @ u V fffff s ( 17
)
There are only two types of node which are not related by symmetry, labelled TJ a and TJ b on Fig. 1. There are thus two (unknown) values of normal stress at nodes, which we define as σ a and σ b . Writing out Eq. ( 17) for each boundary we have, for normal stresses and currents:

C 1 s `a= Z V fffff σ a @σ b B 1 ffffffffffffffffffffffff @ u 1 V ffffff s C 2 s `a= Z V fffff σ a @σ b B 2 ffffffffffffffffffffffff @ u 2 V ffffff s C 3 s `a= Z V fffff σ a @σ b B 3 ffffffffffffffffffffffff @ u 3 V ffffff s
The net current out of a triple junction should be zero, so for TJ a , noting that it is at the start point of each of the three boundary vectors,

C 1 @ B 1 2 fffffff f g + C 2 B 2 2 fffffff f g + C 3 @ B 3 2 fffffff f g = 0
Similarly for TJ b ,

@C 1 B 1 2 fffffff f g @C 2 B 2 2 fffffff f g @C 3 B 3 2 fffffff f g = 0
These two equations can be expanded and rearranged to show that

σ a = σ b ( 18
)
and The latter confirms that precipitation rates weighted by boundary length sum to zero as expected since mass is conserved. From previous sections we see that only the net force on a boundary is relevant for the evaluation of moment and average stress. For a generic boundary, the orthogonal force is, from Eq. ( 16)

u 1 B 1 + u 2 B 2 + u 3 B 3 = 0 (19)
F ? = Z @ B 2 fffffff B 2 fffffff σ n s `ads = σ s + σ e 2 ffffffffffffffffffffff B @ u 1 12Z ffffffffffff B = σ a B @ u 1 12Z ffffffffffff B
3 so for each of the 3 boundaries

F i ? = σ a B i @ u i 12Z ffffffffffff B i 3 ( 20
)
Notice that σ a contributes in proportion to the length of the boundary and has the character of a pressure. The isotropic component of imposed stress cannot be determined from the imposed deformation rate. Later in this contribution the calculated stress will be adjusted to have zero trace. For this reason, the value of σ a is irrelevant -it will cancel out in the adjustment -so from now on it is set to zero.

Equations for grain boundary sliding

It is common to assign zero shear strength to grain boundaries as part of diffusion creep models, but there is no loss of generalisation by assigning a viscosity as done for example by [9,[START_REF] Kim | [END_REF]. It is assumed here that there exists a material constant ζ which relates shear stress to sliding velocity

τ i = ζt i
There is experimental evidence for a linear relationship at least for some temperatures and timescales [20]. Multiplying by the boundary length gives:

F i || = ζt i B i ( 21
)
If t i > 0, we have dextral movement along the boundary; but then the contribution of F i || to the force on the grain is in the direction of minus T i (Fig. 1). We now have equations which give the forces on the boundaries in terms of the tangential and normal components of relative velocities at each boundary, which are in turn related to the deformation rate tensor and the angular velocity by Eq. ( 5). Wheeler page 10

Solution of equations

The aim here is to derive expressions for the angular velocity of grain rotation and for anisotropic rheology, the two key concepts discussed in this contribution.

Angular velocity

The only unknown quantity in this analysis is the angular velocity, and the only constraint not automatically satisfied is that of zero moment (11). We therefore have one equation in one unknown. Combining Eqs. ( 6), ( 7), ( 20) and ( 21) the force vector on boundary i is

F i = @ B i 3 12Z ffffffffffff N i AD @ωS `aR i f g N i @ ζB i T i AD @ωS `aR i b c
T i (22) Each boundary pair contributes F i B R i to the moment. Noting that

N B R = @T AR
and

T B R = N AR
we rewrite Eq. ( 11) as

0 =X i = 1 3 B i 3 12Z ffffffffffff N i AD @ωS `aR i f g T i AR i @ ζB i T i AD @ωS `aR i b c N i AR i H J I K
This is linear in ω and is rearranged to give

ω = X i = 1 3 B i 3 12Z fffffffffffff N i ADR i b c T i AR i @ζB i T i ADR i b c N i AR i F G X i = 1 3 B i 3 12Z fffffffffffff N i ASR i b c T i AR i @ζB i T i ASR i b c N i AR i F G fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff (23)
Note that this expression is cyclic in the indices 1..3 as required from the definitions of B i (Fig. 1).

Viscosity tensor

The angular velocity ( 23) is substituted back into Eq. ( 22) to give the forces on each boundary, which in turn substitute in Eq. ( 13) to give the stress as a function of deformation rate. The stress is adjusted to be trace-free since the hydrostatic Since D is volume-conserving, the relationship between stress and strain can be reduced to a linear dependence of a trace-free stress tensor on a symmetric tracefree strain rate tensor. This is of the form

@σ αβ = η αβγδ e A γδ ( 24
)
making it clear that the dependence is via a fourth-rank viscosity tensor [9], which can also be expressed as its inverse, analogous to an elastic compliance tensor [10].

Compressive stress is taken as positive here. The tensor η is argued to have the same symmetries as the elastic stiffness [21],

η αβγδ = η βαγδ ,η αβγδ = η αβδγ ,η αβγδ = η γδαβ (25)
In 2D a symmetric tensor with zero trace only has two independent components, for example

σ = σ xx σ xy σ xy @σ xx f g
and can be represented by a 2D vector (Appendix 1 properties (e.g. Fig. 4b).

My aim is to illustrate the effect of different grain shapes on rheology and grain rotation. I use 4 sets of hexagonal shapes governed by a single parameter λ, such that λ = 0 represents a regular hexagon in each set (Figure 2 and Table 2).

Quantities are non-dimensionalised for simplicity, as follows: let a be a characteristic grain size parameter, and ε A a strain rate. Then:

B C = B a fffff D C = D ε A ffffff σ C = σ Z a 3 fffffff H C = HZ a 3 fffffffffff ω C = ω ε A fffff ζ C = 36ζ Z a 2 fffffff (27)

Behaviour when boundaries do not support shear stress

This is the usual situation to be considered [2,[START_REF] Raj | [END_REF]22]; the terms in ζ are set to zero.

Kinematics

Figure 3 illustrates the response of the four types of shape in Figure 2 to pure shear (with extension parallel to y-axis) and simple shear (top-to-right, parallel to x axis). In the latter deformation, the principal axis of extension is 45° clockwise of the x axis. Fig. 3(b) shows that for pure shear, more elongate grains (larger λ) may show slower rotations but not necessarily (e.g. Hex4). Fig. 3(c,d) shows different rotation behaviours for simple shear. It is notable that some grains rotate the same way as the imposed shear, others counter to it. For shapes near that of a hexagon, very large angular velocities are predicted, with magnitudes far in excess of any invariant expressing magnitude of the strain rate. For example Hex1 in simple shear has grains rotating clockwise at 2.23 for an imposed deformation rate of 1. Figure 3(a,c) also shows the instantaneous precipitation rates on each of the three faces. In pure shear, grains in Hex1 show precipitation on the top and symmetric dissolution on the two side faces as expected. In simple shear (Fig. 3((c) and (d)), however, all precipitation rates are zero, because sliding and grain rotation together are sufficient to take up the deformation, for that instant. This would also be the case if the deformation were a pure shear imposed at 45° to the axes, because the rotational component of deformation does not affect the precipitation rates. This "sliding only" response is the case for any value of λ in Hex1 and in Hex4. It has major implications for the strength of the polycrystal, discussed in section 6.2.

Before that, though, the limit λ→0 demands discussion.

Singular solutions and associated grain shapes

For all types of shape in Fig. 3 the limit λ→0 is a regular hexagon. One would expect, then, that for a particular imposed deformation a unique angular velocity will be predicted as λ→0 for all 4 types of hexagon. In fact, in pure shear there may be a limiting value for ω as λ→0 (Hex1, Hex4), or not (Hex2, Hex3). The same is true for simple shear (Fig. 3(d)) so it appears there is no unique solution for the angular velocity of a regular hexagon during diffusion creep. This may seem surprising but is easily explained. The numerator and denominator in Eq. ( 23) both contain terms T i AR i . Each term is the dot product of a grain boundary vector with the translation vector which relates it to its symmetric equivalent. But in a regular hexagon, each boundary is perpendicular to its translation vector (Fig. 1) -so all these dot products disappear, and Eq. ( 23) takes the indeterminate form 0/0. This result has not been noted by other authors who used specific deformations and assumed zero angular velocity. Physically, it is because in each pair of boundaries, the two are directly opposite each other. Thus, not only forces but also moments cancel automatically.

This singular behaviour is not restricted to regular hexagons. Consider a hexagon with two sides defined by arbitrary vectors B 1 and B 2 . Now define the third side by

B 3 = B 1 AB 2 B 1 B B 2 fffffffffffffffffffffff S B 1 @B 2 b c (28)
It is trivial to show that T i AR i = 0 for all sides. Consequently, recalling ζ = 0, the denominator in Eq. ( 23) is zero. An arbitrary irregular hexagon is defined here by 6 Wheeler page 14 independent parameters; ignoring size and orientation this means the basic shape is defined by 4 parameters. Eq. ( 28) shows that a subset, defined by 2 parameters, of all hexagonal shapes will yield singular behaviour in the model discussed here. An example is shown in Fig. 2.

The existence of this singular subset is significant. Small perturbations of shape from any shape in that subset will give rise to a great range of angular velocities, because the denominator in Eq. ( 23) will be small. One thus expects a great variety of rotation behaviours for quite subtly different grain shapes. Microstructures which bear some similarity to "ordinary" hexagonal shapes will not show such extreme angular velocities.

Regular hexagons are the simplest idealisation of equi-axed grains, but they belong to the singular subset. This is the underlying explanation for why irregular networks of (on average) equi-axed grains give rise to such a wide range of angular velocities (Fig. 1 in [8]).

Dynamics

The viscosity tensor can be obtained by calculating the stress for 2 different strain rate tensors using Eqs. ( 13), ( 22) and ( 23), normalising the trace of the stress to zero in each case, and then using Eq. ( 26). The result is a function of the 6 variables that define B 1..3 , and is too long to write out in full. Symbolic routines in Maple via Matlab, however, allow analysis of the situation, in particular an examination of the viscosity anisotropy. Before describing some specific examples, I discuss the result that for a completely general set of boundaries, when they cannot support shear stress, det H = 0 (29)

so that H has one eigenvalue equal to zero. This is a remarkable result, because it implies that any microstructure formed by periodic irregular hexagons has one direction in double-angle space which has zero viscosity. In actual space 2 directions at right angles will have zero viscosity. The result implies that there exists a particular strain rate tensor for which the stress is zero, and consequently the force on each boundary is zero, so that from Eq. ( 20), all precipitation rates are zero. The deformation in this case is accomplished purely by sliding and grain rotation: two examples have already been shown of imposed deformations giving rise to this response (simple shear of Hex1 and Hex4, Fig. 3). Since in this section we are considering inviscid grain boundaries, no stress is required to accomplish deformation Wheeler page 15 by sliding. Equation (29) implies that there are 2 "zero-strength" directions for any polycrystal made of irregular hexagons. We can determine the "zero strength" directions from the eigenvectors of H. It is more illuminating, though, to use the kinematic equations ( 6) to determine the "zero strength" direction, by solving the three equations u 1..3 = 0 for D. If two of these equations are satisfied, the third will be automatically, because of mass conservation (Eq. ( 19)). Writing out the equations for the first two boundaries, we have:

N 1 AD @ωS `aR 1 = 0 N 2 AD @ωS `aR 2 = 0
D has 3 unknown components, so we can set ω to zero to obtain 2 equations in 3

unknowns. Solving gives a value for D (except for an arbitrary constant of multiplication) as a function of the grain shape as defined by B 1..3 . Appendix 2 shows that the solution is:

D w = GS ( 30
)
where G is a symmetric tensor given by:

G = B 2 B B 3 b c B 1 N B 1 + B 3 B B 1 b c B 2 N B 2 + B 1 B B 2 b c B 3 N B 3 (31)
Note that G is cyclic in the indices 1 to 3 as expected from the initial definitions of the vectors (Fig. 1). The weak directions can be determined from the symmetric part of D w , namely

E w = D w + D w T 2 fffffffffffffffffffffffffff (32)
The eigenvectors of E w give the two weak directions.

Example Hex1

This is a symmetric hexagon, elongate parallel to the x-axis. We find: The eigenvector corresponding to the weak directions is (0, 1) in double-angle space, corresponding to two axes at 45° to the x-axis in real space, regardless of the value of λ. This can be understood by noting that the weak axes must be at 90° to each other, but must also satisfy the symmetry of the grain. This has vertical and horizontal mirror planes, so only 0°, 45° and 90° are allowed. 0° and 90° are strong directions.

H C = 1 72 ffffffff 1 + 3
The limit as λ→0 is

H C = 1 72 ffffffff 1 0 0 0 d e (34)
which will be compared with the limit for a different parameterisation now.

Example Hex2

This is a distorted hexagonal shape with no mirror symmetry. Here we have:

H C = 1 216 fffffffffff f g λ 2 @
3 p w w w w w w w w w w w w w w w w w λ @ 3 p w w w w w w w w w w w w w w w w w λ 3

h j i k ( 35
)
The non-zero eigenvalue is

H s C = 1 216 fffffffffff λ 2 + 1 72 ffffffff ( 36
)
and the eigenvector of H corresponding to the weak direction is, in double-angle space, (37)

Note that for large λ these approach 45° to the coordinate axes, but this example shows that in general the weak directions relate in a complicated fashion to the grain shape, unless that grain has mirror symmetry (Hex1).

The viscosity limit as λ→0 is

H C = 1 72 ffffffff 0 0 0 1 d e ( 38
)
with weak directions parallel to x and y axes. This value differs in orientation from that derived for Hex1 (Eq. ( 34)), illustrating that no unique value of viscosity exists for the rheology of an array of regular hexagons (or for other "singular" shapes).

The parameterisation using λ does not necessarily represent the subsequent evolution of this shape; it is intended just to illustrate the responses of grains which have shapes as a function of one parameter. Wheeler page 17

Behaviour when boundaries support shear stress

If ζ is non-zero, the denominator in Eq. ( 23) is non-zero for any shape, and now the viscosity for regular hexagons is well-defined and isotropic:

H C = 1 + ζ C 72 fffffffffffffffff 1 0 0 1 d e (39)
This is the simplest illustration that grain boundary viscosity increases the overall viscosity of the polycrystals,

To examine the general way in which ζ affects rheology, consider the derivation of the stress tensor for a particular deformation rate. From Eq. ( 23) the numerator and denominator of ω are linear in

ζ ω = J 0 + J 1 ζ A 0 + A 1 ζ fffffffffffffffffffffffffffff ( 40
)
where the Js are a function of geometry and deformation rate, and the As a function of geometry. From Eqs. ( 22) and ( 13), we see that

σ = A `a+ A `aζ + ζ A `a+ A `aω b c
Where (A) is shorthand for an (unspecified) function of geometry and deformation rate; this is sufficient to show that σ σ σ σ is of the form (quadratic in ζ)/(linear in ζ), and hence the viscosity is:

H = H 0 + H 1 ζ + H 2 ζ 2 1 + A 1 ζ fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff (41)
Here H 0 is the viscosity in the absence of grain boundary viscosity. For large ζ, H is linear in ζ. The viscosity of a polycrystal made of irregular hexagons is in general still strongly anisotropic.

Example Hex1

The elongate hexagon has mirror symmetry so, as in the case for zero ζ, the two weak directions are at 45° to the coordinate axes in actual space; in double angle space the eigenvectors of H are parallel to the coordinate axes and the off-diagonal terms are zero.

Example Hex2

The viscosity tensor as a function of ζ * and λ is too long to write out, so for illustration substitute λ = (1/4)√3, the value being chosen because then the length of side 3 (vector B 3 ) is 5/4 that of the other two sides which leads to relatively simple exact expressions for the viscosity tensor.

H 11 C = 1 3456 fffffffffffffff 945 + 63013ζ C + 63294 ζ C b c 2 315 + 926ζ C ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff H 12 C = @ 1 288 fffffffffff 315 @1489ζ C + 1434 ζ C b c 2 315 + 926ζ C ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff H 22 C = 1 72 ffffffff 315 + 1871ζ C + 858 ζ C b c 2 315 + 926ζ C ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff (43)

Time evolution

My aim here is to give a complete analytic treatment of the instantaneous response of a periodic microstructure in diffusion creep. Building on this, the evolution through time can be modelled, as two examples show (Fig. 5). These were produced using a numerical modelling program developed previously for diffusion creep [2], extended to incorporate periodic microstructures and boundary conditions [8]. Analytic solutions cannot be obtained for this because of the way that new grain boundary and triple junction positions are calculated [2]. There is no inconsistency between the numerical and analytic modelling. The analytic treatment here was motivated by the need to understand the results of the numerical simulations, which are frequently counter-intuitive.

Grain rotation

Grain rotation is an integral part of the deformation. For nearly hexagonal shapes, large angular velocities are predicted, especially when no shear stress is assumed along grain boundaries. This is because the regular hexagon shape actually yields a singular solution. I suggest that complicated microstructures can be considered -to a degree -as perturbed versions of periodic hexagonal microstructures. The unbounded angular velocities predicted here for nearly regular hexagons then provide an explanation for the wide range of angular velocities predicted in numerical models of irregular grain networks when those grains are on average equi-axed [8]. Equally importantly, if grains do for any reason become elongate, the angular velocities are much reduced in many cases (e.g. Fig. 3).

Anisotropic viscosity

It has been shown that a periodic microstructure made from a single irregular hexagonal grain shape has marked viscosity anisotropy during diffusion creep. The anisotropy is manifest in two orthogonal strong directions, and two orthogonal weak directions at 45° to the strong directions. In the absence of grain boundary viscosity, the weak directions have zero strength. Coaxial stretching parallel to a weak direction thus requires, for an instant, no stress. Simple shear with a shear plane at 45° to a weak direction (i.e. shear plane parallel to a strong direction) also requires no stress. When grains are elongate the periodic microstructure discussed here is anisotropic, with or without grain boundary shear stresses. It is to be expected then that irregular aggregates of shaped grains will be anisotropic, and this will have implications for the way in which strain partitions and perhaps localises during diffusion creep and superplasticity.

F

Relationship to previous work

Lifshitz [9] pointed out that diffusion creep is likely to give rise to anisotropic viscosity, with the emphasis on volume diffusion creep but a preliminary treatment of grain boundary diffusion creep in his Appendix 3. He made no explicit mention of grain rotation and so it remains unclear how this would affect the predictions.

Greenwood [23] provided a precise expression for the anisotropic response of an orthorhombic grain during Nabarro-Herring creep, but under special stress orientations. Greenwood [10] provided expressions for the viscosity tensor under more general stress orientations. In that work it was assumed that creep strength varies smoothly as the stress axes are rotated, so that there are no maxima or minima at intermediate positions. This contribution, however, shows that there may be marked strength minima at 45 degrees to strength maxima, so it would be interesting to see how the different approaches could be reconciled. Again, the effects of grain rotation must be considered.

Separate strands of research have examined certain restricted types of hexagonal arrays under restricted deformation conditions. For hexagons with mirror symmetry in pure shear [12], the analysis here is in agreement. Specifically, the stretched hexagonal shapes are of the form Hex1, with the parameter λ related to the finite strain. In pure shear, the H 11 component (from Eq. ( 42)) defines the stress and it can be shown that this is equivalent to equations 11, 13 and 14 of Kim et al. [12].

However the latter analysis did not encompass different strain directions, and hence does not address anisotropy. sliding surfaces (their Fig. 2.3). They then analyse movement accommodated by diffusion along such serrated surfaces. There appears to be a distinction made between the slip surfaces and apparently rigid polycrystals on each side. If, however, all grain boundaries are assigned the same properties, they must all be involved in sliding and accommodating strain and hence the picture presented by those authors becomes modified. Kim et al. [14] Fig. 2. Four different types of perturbed regular hexagonal shape, c.f. Table 1.

Q B B 1 = B 1 B B 2 b c B 3 B B 1 b c B 2 B B 3 b c @ B 3 B B 1 b c B 2 B B 1 b c B 3 B B 2 b c = 0 Hence Q is parallel to B 1 , so D w R 1 is parallel to B 1 .
Vertical height of regular hexagon is 1, so its sides are length 1/√3. The last figure

shows a "singular" hexagonal shape. Pairs of faces are directly opposite each other, so that moments as well as forces will always balance, giving rise to singular behaviour.

  effect on the creep rate. The actual analysis is implemented via symbolic mathematics in Maple via Matlab (code available from the author). It is easy to show that the antisymmetric part of D plays no role in the stress, as follows: express the antisymmetric part of D as αS. Since ω is linear in D, the antisymmetric part contributes α to ω and therefore contributes αS -αS = 0 to D -ωS , which is the only form in which D appears in the equations for force and stress.

p w w w w w w w w w w w w w w w w w λ

3 p w w w w w w w w w w w w w w w w w ,λ b c which in actual space corresponds to two orthogonal directions F 3 + λ 2 q 3 p w w w w w w w w w w w w w w w w

 3323 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w fffffffffffffffffffffffffff w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w +

3 p w w w w w w w w w w w w w w w w w + 2λ b c 3 p w w w w w w w w w w w w w w w w w + λ + 3 p w w w w w w w w w w w w w w w w w + 3λ 3 p w w w w w w w w w w w w w w w w w 3 p w w w w w w w w w w w w w w w w w λ 2 + 6 3 p w w w w w w w w w w w w w w w w w + 18λ + 3 3 p w w w w w w w w w w w w w w w w w λ 2

 33333332 the general form of viscosity dependence on ζ * while, for H 11 , the denominator is a factor of the numerator, leaving a linear dependence on ζ. For ζ = 0, H 11 has the limit as in Eq. (33) and H 22 becomes zero. Fig 4(a) illustrates the dependence of the eigenvalues on ζ * for λ = 2. There is a particular value at which the viscosity is isotropic, but as ζ increases, anisotropy remains.

  Substituting ζ * = 0 gives a viscosity consistent with Eq. (35) with the appropriate value of λ. Fig. 4(b) shows how both eigenvalues increase with ζ * . Because this grain shape is not orthorhombic, the eigenvectors are not constrained by symmetry and rotate with ζ * (Fig. 4(c)).

  [START_REF] Raj | [END_REF] show a regular hexagonal array and draw zigzag surfaces of connected grain boundary segments which they identify as Mode 1 and Mode 2

Fig. 1 .

 1 Fig. 1. The meaning of the main geometric vectors defined here.

Fig. 3 .

 3 Fig. 3. Each row relates to one of the perturbed hexagonal shapes. Column (a): the microstructure for a value of λ = 0.5, showing precipitation rates (numbers written beside each of 3 distinct sides) and angular velocity ω (rotation sense emphasised by curved arrow) for D = [-1/2 0; 0 1/2], stretching parallel to y axis. (b): angular velocity as a function of λ for the same D. Cross marks value used in (a). (c): the microstructure for a value of λ = 0.5, showing kinematic quantities as in (a) but for D = [0 1/2; 0 0], top-to-right simple shear parallel to x axis. (d): angular velocity as a function of λ for the same D. Cross marks value used in (c).

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) Eigenvalues of the viscosity tensor as a function of ζ * for a particular shape, Hex1 with λ = 2. The grain shape is shown bottom right. (b) Nested polar plots showing the viscosity as a function of the orientation of the maximum strain rate axis, for 4 values of ζ * . (c) As (a), for Hex2 with λ = (1/4)√3. (d) Angle of "weak" direction to x-axis as a function of ζ * for Hex2 as in (b). (e) Viscosity plots for Hex2; note that because Hex2 has no mirror plane, the orientation of the viscosity tensor can vary.

  Wheeler page 12 the stress and strain rate tensors are parallel to each other. Two different eigenvalues indicate anisotropy. A polar plot of viscosity versus strain axis illustrates these

	F o r
	P
	e
	e r
	R
	e
	v i e
	). It is convenient to define a 2 x 2 w
	viscosity matrix H so that -σ xx σ xy f g = H 11 H 12 H 21 H 22 h j i k e A xx e A xy f g	O n l	(26)
	The third Eq. (25) ensures that H is symmetric, and its 3 components contain all the y
	information about anisotropy. H does not transform like an ordinary tensor under
	rotations. However, note that if a symmetric matrix with zero trace is rotated by angle
	θ, its associated vector (Eq. (A1.1)) rotates by 2θ; hence the matrix H transforms
	under rotations like a rank 2 tensor except with double the angle. H has 2

eigenvectors, which indicate orientations in which the stress and strain rate tensors are parallel to each other. These eigenvectors are at 90° in this "double-angle" space. Therefore in real space there exist 4 orientations, spaced at 45° to each other, in which

  Wheeler page 20This behaviour arises because grain boundary sliding is always one aspect of diffusion creep, and there is always an orientation of strain in which sliding alone is a sufficient response, without dissolution and precipitation of material at grain boundaries. If finite shear strength is assigned to grain boundaries, anisotropy remains but now the polycrystal has non-zero viscosity in all directions. Both viscosity eigenvalues increase as the grain boundary viscosity ζ increases if other quantities are fixed.

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

  allow all grains to rotate and derive a flow stress for shear (their equation[START_REF] Kim | [END_REF] which can be shown to agree with Eq. (39) given here when there is grain boundary viscosity. This work does not reveal the singular nature of the regular hexagonal array in the absence of grain boundary viscosity, however; nor does it reveal the large and disparate angular velocities and strengths to be expected under different loading conditions and for grain shapes slightly different from regular.

	Noting that
	a B b = @b B a
	we have
				F o r F o r F o r
					P P P
					e e e
						e r e r e r
						R R R
						e e e
						v i e v i e v i e
						w w w
						O n l O n l O n l
					θ `aT = j h	y `a@P xx cos 2θ `a, P xy cos 2θ `a@P xy sin 2θ P xx cos 2θ P xy cos 2θ `a+ P xx sin 2θ `a, P xy sin 2θ `a k `a+ P xx sin 2θ `a i y y
	Representing P by a 2-element vector
	p c	=	f	P xx P xy	g	(A1.1)
	it is seen that
						Wheeler page 24

In summary, this contribution shows how the grain rotations and strength properties of a periodic array of irregular hexagons may be predicted in a unified fashion. Some previous work can be seen as particular examples of this more general theory. The new theory highlights the importance of strength anisotropy and of the effect of overall grain shapes on grain rotation during diffusion creep: it provides a platform for understanding the behaviour of more complicated microstructures.

Table 1 :

 1 Because N 1 is orthogonal to B 1 we have proved Eq. (A2.1) for i = 1. All expressions involved are cyclic with respect to indices 123, so it follows that Eq. (A2.1) holds for all i. Notation

	Tables
	Symbol	Meaning
	B	Vector defining grain boundary
	C	Current (flux x grain boundary width)
	D	Deformation rate tensor
	Dw	Deformation rate tensor giving rise to zero stress and zero grain rotation
	f	Force per unit length as function of position
	F H L N R	Force across boundary 2nd rank viscosity tensor (in "double angle" space) F Onsager diffusion coefficient o Unit vector perpendicular to (anticlockwise of ) B = ST r Vector translating boundary to symmetrically equivalent position
	s S t T u V	Distance along a boundary segment Skew tensor P Sliding velocity along boundary e Unit vector parallel to B Divergence velocity of two grains (precipitation rate at their contact) e r Molar volume
	w w G Z η η η η λ σ σ σ σ ω ζ	Effective grain boundary width Velocity of grain G measured at origin R =LV 2 w (units of m 3 /Pa/s) e 4th rank viscosity tensor Parameter defining a set of hexagon shapes, λ = 0 is regular v Stress i Angular velocity e w Ratio of shear stress to sliding velocity (units of Pas/m)
		O n l
		y

Table 2 :

 2 The non-dimensionalised side vectors defining four families of hexagons, each parameterised by λ.

	Regular hexagon	1 2 3

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Appendix 1: transformation properties of H

It is shown here that the viscosity tensor in actual space can be related to H which transforms like a tensor under rotations in a "double angle" space. Let P be any symmetric tensor with zero trace, so it has just 2 independent parts, P xx and P xy .

P

Thus, all tensors P can be represented as 2-vectors of the form p, which transform as if rotated by 2θ when the tensor P is rotated by θ. A linear relationship between two such tensors in actual space is represented by a 4 th rank tensor. In double angle space the linear relationship can be expressed by a second rank tensor which as a consequence of (A1.2) transforms like an ordinary tensor under rotations but with double the angle:

Appendix 2: equations for zero strength directions

Rather than give the derivation, it is sufficient to show that D w as defined using G in Eq. (30) satisfies

using any value of i. Consider the vector


Filip Kadlec

Christelle Kadlec

Tanya Paskova

Keith Evans

Effect of Fe doping on the terahertz conductivity of GaN single crystals

Keywords: Gallium nitride, terahertz spectroscopy, iron doping, self-heating PACS numbers: 78.30.Fs

Bulk single crystals of GaN with different degree of Fe doping were studied using time-domain terahertz spectroscopy at high temperatures. Features due to free carriers were observed in the complex permittivity spectra with a pronounced dependence on both doping and temperature. Fitting the spectra using the Drude model made it possible to deduce the defect ionization energy of 16 meV in the undoped sample while the spectra of doped samples are consistent with an ionization energy of 60 meV. Also, the free carrier concentrations at temperatures from 300 K to 900 K were estimated.

Introduction

Gallium nitride is a technologically important material with a wide range of both present and prospective applications. Since the success in epitaxial growth of GaN thin films (see Ref. [START_REF] Pearton | Gallium nitride processing for electronics, sensors and spintronics[END_REF] for a review), it has been largely employed in light-emitting diodes, LED displays, and lasers. In the last years, there has been a rapid development in technologies aiming at the production of bulk GaN; its future uses include e.g. high-power high-frequency operating field effect transistors, radars and power amplifiers [START_REF] Keller | Gallium nitride based high power heterojunction field effect transistors: Process development and present status at ucsb[END_REF]. The epitaxial growth often uses sapphire substrates, and the technology leads to an unintentional n-doping of GaN; O and Si are the main residual impurities at the origin of dc conductivity [START_REF] Moore | Identification of Si and O donors in hydride-vapor-phase epitaxial GaN[END_REF]. For compensation of these donors, doping by Fe can be used; Fe acceptors substitute for Ga and substantially increase the resistivity [START_REF] Kumagai | Fe-doped semi-insulating GaN substrates prepared by hydride vapor-phase epitaxy using GaAs starting substrates[END_REF]. In order to achieve optimal properties of the doped crystals, a deeper knowledge of the underlying physical properties of bulk GaN, namely their conductivity mechanisms, is required. From the point of view of high-power applications which may lead to self-heating [START_REF] Khan | GaN-AlGaN heterostructure field-effect transistors over bulk GaN substrates[END_REF], the properties at higher temperatures are of particular importance. However, standard transport measurements are challenging owing to difficulties in producing low resistivity contacts to the highlyresistive GaN material.

Terahertz (THz) spectroscopy is an alternative technique and a convenient tool for studying the dynamical conductivity of GaN. It was shown that the free carriers follow the Drude model [START_REF] Zhang | Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN[END_REF] with a plasma frequency in the THz range. Small deviations from the Drude model were reported and studied [START_REF] Tsai | Terahertz response of GaN thin films[END_REF], and the THz spectra were shown to depend on the carrier concentration [START_REF] Guo | Terahertz carrier dynamics and dielectric properties of GaN epilayers with different carrier concentrations[END_REF]. In the present paper, we study experimentally the THz response of bulk GaN crystals with varying Fe doping and at variable temperatures.

Sample preparation

The samples were grown in a vertical hydride vapor phase epitaxy hot-wall reactor at a sub-atmospheric pressure. The single crystals were deposited at a rate of 150 µm/h on cplane (0001) sapphire substrates with an AlN template layer, in the downstream part of the reactor where GaCl and NH 3 , transported by the N 2 gas react. The substrates were maintained at a temperature of T ≈ 1000 ℃. As a Fe source for doping, a variable flow of bis(cyclopentadienyl)iron was introduced into the reactor via N 2 carrier gas. Once the substrates removed, the GaN crystals were cut and polished to the shape of slabs with optical quality surfaces and thicknesses ranging from 0.5 to 1.2 mm. Four different samples were measured: one undoped, No. 1 (AH939), and three doped, Nos. 2-4 (AE492, AE535 and AE782; these fabrication labels were used also in refs. [START_REF] Fang | Deep centers in semi-insulating Fe-doped native GaN substrates grown by hydride vapour phase epitaxy[END_REF][START_REF] Freitas | Semi-insulating gan substrates for high-frequency device fabrication[END_REF]). Their Fe atom concentrations were determined by means of secondary ion mass spectrometry as 3 × 10 17 , 7 × 10 17 and 3 × 10 18 cm -3 , respectively.

Experimental details

Our custom-made experimental setup for time-domain THz spectroscopy was based on a Ti:sapphire laser oscillator providing 810 nm, 80 fs pulses at a 76 MHz repetition rate. The pulses excited an interdigitated photoconducting emitter [START_REF] Dreyhaupt | High-intensity terahertz radiation from a microstructured large-area photoconductor[END_REF] which generated linearly polarized THz pulses. A weak part of the laser beam was directed on a 1 mm thick ⟨110⟩ cut ZnTe crystal for electro-optic sampling of the transmitted THz waves. The whole THz beam was enclosed in a vacuum chamber (about 20 mbar) in order to avoid water vapor absorption. The samples were placed in a furnace with optical ports, using a clear aperture of 3 to 7 mm, and heated up to T = 900 K with steps of 100 K, except for the sample No. 1 where the temperature of 400 K was replaced by 360 and 430 K. The instrument covered a frequency range of 0.25-2.5 THz.

Results and evaluation

From the complex transmittance, we calculated numerically the spectra of complex refractive index N (ν) which we converted to permittivity ε(ν) ≡ ε ′ + iε ′′ = N 2 (ν) and conductivity spectra, σ(ν) = -2πνiε 0 ε(ν); ε 0 denotes the vacuum permittivity. Symbols in Fig. 1 show the real parts, ε ′ (ν) and σ ′ (ν). Signatures of free carrier transport can be clearly seen-even a relatively weak conductivity shows up as a non-vanishing σ ′ in the low-frequency part. This feature is the strongest for the undoped sample where it clearly gives rise to a Drude peak and it and decreases upon Fe doping. For the highest doped sample (No. 4), no signs of free charge carriers are visible at any temperature.

As a common trend, the ε ′ (ν) spectra display a monotonic increase upon heating. For all Fe doped samples, they are almost flat, except for small experimental artifacts and/or noise. In the upper frequency parts of the spectra of samples (No. 3, 47) with higher doping (Fig. 1e,g), an increase with frequency is seen, similar to previous observations [START_REF] Guo | Terahertz carrier dynamics and dielectric properties of GaN epilayers with different carrier concentrations[END_REF]. We attribute this to a contribution of a broad phonon absorption beyond our accessible frequency range.

This term also influences the σ ′ (ν) (i.e., ε ′′ (ν)) spectra in their high-frequency part. From the fitting described below it appears clearly that without such a term, the conductivity increase above ν > 1 THz would not be observed.

The most interesting information, however, is related to the low-frequency parts of σ ′ (ν); it is that about free charge carriers. The temperature dependence of these low-frequency parts is strongly influenced by doping. While it decreases with T in the sample No. 1 (Fig. 1b), an increase upon heating is observed in the lower Fe doped samples No. 2 and No. 3 (Fig. 1d,f); in the latter case, heating up is necessary to bring up the Drude-like contribution. Finally, for the sample No. 4, the free carrier conductivity is not visible at any temperature (Fig. 1h).

We fitted the ε(ν) spectra of the samples displaying signs of electrical conduction. In view of the observed features, we found the same model as used in Ref. [START_REF] Guo | Terahertz carrier dynamics and dielectric properties of GaN epilayers with different carrier concentrations[END_REF] to be the best suited (ω = 2πν):

ε(ω) = ω 2 p ω 2 -iωγ + f o ω 2 0 -ω 2 -iωΓ + ε ∞ (1)
with the plasma frequency given by ω 2 p = N e 2 0 /m 0 m * ε 0 . Here, e 0 and m 0 denote the free electron charge and mass, respectively. Eq. 1 contains two frequency-dependent terms: (i) a Drude-like plasma of free charge carriers with a concentration N , effective mass m * and a mean collision rate γ and (ii) an overdamped harmonic oscillator with oscillator strength f 0 , eigenfrequency ω 0 and damping Γ, describing the broad features at higher frequencies. The constant ε ∞ accounts for contributions of all modes far beyond the THz region.

First, the complex room-temperature spectrum of the undoped sample (No. 1) alone was fitted; the result is shown by solid lines in Fig. 1a,b. The extrapolated value of static conductivity is σ DC ≈ 1(Ω•cm) -1 . Owing to the pronounced slope in the real part, it is possible to reliably determine the angular plasma frequency ω p = 8.3 × 10 12 rad s -1 and the scattering time τ ≡ 1/γ = 160 fs. Assuming that the free charge carriers are electrons with an effective mass [START_REF] Vurgaftman | Band parameters for nitrogen-containing semiconductors[END_REF] of m * = 0.2, one can calculate their concentration, N (300 K) = 4×10 15 cm 3 and mobility µ = 1440 cm 2 /V• s. These results are in agreement with values determined by THz spectroscopy and reported earlier for undoped GaN [START_REF] Zhang | Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN[END_REF].

Further, we performed a common fit of the spectra at all temperatures. We assumed a linear temperature dependence of ε ∞ and the oscillator tranverse frequency was fixed to ω 0 /2π = 5 THz; its strength was assumed to be constant in T and its damping Γ linearly proportional to T . Thus, it was possible to describe the high-frequency part of the spectra. In the weak-doping approximation, the concentration of free electrons can be written as N (T ) = N 0 exp(-E D /k B T ) where E D is the defect ionization energy and k B the Boltzmann constant. The collision rate γ was allowed to vary with T without limitations. The solid lines in Fig. 1a,b show the fit matching well the experimental data; the best match is found for a value of E D = 16 ± 3 meV. Solid lines in Fig. 2 show the obtained functions τ (T ) and N (T ).

The spectra of the sample No. 2 with low Fe doping display much smaller spectral features than those of the undoped sample (No.1). Thus, it was necessary to fit them together; we used the same model (Eq. 1) and temperature dependence of parameters. We found the room-temperature concentration of electrons to be N (300 K) = 1.4 × 10 14 cm 3 , while the defect ionization energy is substantially higher, E D = 60 ± 10 meV. This leads to a steeper increase of N upon heating (Fig. 2b). The model does not agree very well with the experimental data at the highest temperatures (T > 700 K), which is probably a sign of a supplementary absorption mechanism in this temperature range. Therefore, the values of the scattering time τ become unreliable and they are not shown in Fig. 2a. Nevertheless, since the model describes well the conductivity spectra in the lowest part of the THz range, it should still provide a useful estimate of the free carrier concentration up to the highest temperatures.

In the sample No. 3, the even higher level of doping suppresses features linked to the free carrier absorption at room-temperature (Fig. 1e,f). However, upon heating above 500 K, σ ′ increases at lower frequencies. Due to its low values, the fit performed for this sample does not provide unambiguously the concentration of the free electrons. Therefore, we assumed a concentration inversely proportional to the Fe doping level, i.e. N (300 K) = 6 × 10 13 cm 3 . In this way, with the same value of E D = 60 meV, a good agreement with the measured spectra was achieved. Analogously to the sample No. 2, the validity of the model at the highest temperatures is limited; thus, values of τ can be estimated only in the range 500 K ≤ T ≤ 700 K.

Finally, no features related to electrical conductivity are observed in the spectra of the sample No. 4 with the highest doping. By comparing the spectra at highest temperatures with those of the sample No. 3, and assuming the same values of γ, we can estimate that the carrier concentration decreased by a factor of four or more.

Discussion

In the undoped sample, the activation energy of 16 meV is very close to that determined from transport measurements in intentionally Si-doped samples [START_REF] Ng | The role of dislocation scattering in n-type GaN films[END_REF]. By contrast, the THz mobility is one order of magnitude higher, while the room-temperature concentration of carriers in THz experiments is about two orders of magnitude lower. This means that the interactions of carriers dominantly contributing to the THz conductivity, where local motion across about tens of nanometers is sensed, are different from those occurring in the long-range transport; we attribute this fact to short-scale spatial doping inhomogeneities. The energy level of 16 meV apparently corresponds to the residual donor-either O or Si-as reported earlier based on Hall-mobility measurements on electron-irradiated samples [START_REF] Look | Defect donor and acceptor in GaN[END_REF]. The activation energy of 60 meV in doped samples (Nos. 2 and 3) is clearly linked to the Fe doping; in agreement with previous works [START_REF] Look | Defect donor and acceptor in GaN[END_REF][START_REF] Look | On the nitrogen vacancy in GaN[END_REF] we attribute it to the N vacancy donor V N . The concentration of these donors appears to increase at low level of Fe doping; this might be because the doping leads to a decrease of the Fermi level whereby the vacancies would become energetically more stable. In this way, Fe atoms could stabilize the N vacancies, or even Fe-V N complexes might form. Since the Fig. 3 of Ref. [START_REF] Kumagai | Fe-doped semi-insulating GaN substrates prepared by hydride vapor-phase epitaxy using GaAs starting substrates[END_REF] displays only a partial agreement between experimental and simulated EXAFS spectra, a part of the Fe dopants probably occupy other crystal sites. Therefore, our hypotheses are not in contradiction with the observation of Fe Ga reported in Ref. [START_REF] Kumagai | Fe-doped semi-insulating GaN substrates prepared by hydride vapor-phase epitaxy using GaAs starting substrates[END_REF].

For the undoped sample (No. 1), τ (T ) data obtained from fits (Fig. 2) show a monotonic decrease, apparently linked to the electron-phonon scattering increasing on heating. For the low doped sample (No. 2), the same trend is observed at lower temperatures, however, around 600 K, τ seems to level off. Upon heating above 700 K, the model described by Eq. 1 departs from our experimental data, therefore τ cannot be reliably determined. This holds also for the higher doped sample (No. 3); values of τ similar to those in samples No. 1, 2 are observed at medium temperatures. For sample No. 3 at low temperatures, τ cannot be determined owing to the absence of the Drude contribution. The deviation of our model from the data at the highest temperatures suggests that the energy scheme may be more complex than that we take into account; however, the current knowledge of the energy levels in Fe:GaN and the accuracy of our measurements do not enable us to set up a more appropriate model.

We also note that, as σ ′ (ν) according to the Drude model rises steeply towards low frequencies, measurements at microwave frequencies might provide even more sensitivity for doped samples.

Conclusion

In conclusion, we have studied the impact of Fe doping on the THz spectra of GaN crystals in view of their future use above room temperature. While the doping leads to a suppression of conductivity at room temperature, heating the crystals has the opposite effect. From the temperature dependences of the spectra, we have determined defect ionization energies consistent with those observed earlier and temperature dependences of the free carrier concentrations. Our results are expected to contribute to both the fundamental understanding of carrier transport and the technological use of GaN crystals.

Figure 2 .

 2 Figure 2. Free carrier scattering times τ (a) and concentrations (b) estimated from spectra fitting as a function of T .

Acknowledgment

Support by the Czech Academy of Sciences (project AVOZ 10100520) as well as by MDA under the HQ0147-09-C-0005 contract for bulk GaN characterization is acknowledged. We would like to thank Dr. David Look for useful discussions.


Zdenko Machala
email: machala@fmph.uniba.sk

Lenka Chládeková

Michal Pelach

Plasma agents in bio-decontamination by DC discharges in atmospheric air

Keywords: . 52.80.Hc Glow, corona -52.80.Mg Arcs, sparks, lightning, atmospheric electricity-52.70.Kz Optical (ultraviolet, visible, infrared) measurements -87.53.Ay Biophysical mechanisms of interaction

Bio-decontamination of water and surfaces contaminated by bacteria (Salmonella typhimurium) was investigated in two types of positive DC discharges in atmospheric pressure air, in needle-to-plane geometry: the streamer corona and its transition to a novel regime called transient spark with short high current pulses of limited energy. Both generate cold non-equilibrium plasma. Electro-spraying of the treated water through the needle electrode was applied for the first time and resulted in fast bio-decontamination. Experiments providing separation of various biocidal plasma agents, along with the emission spectra and coupled with oxidation stress measurements in the cell membranes helped better understanding of the mechanisms of microbial inactivation. The indirect exposure of contaminated surfaces to neutral active species was almost as efficient as the direct exposure to the plasma, whereas applying only UV radiation from the plasma had no biocidal effects. Radicals and reactive oxygen species were identified as dominant biocidal agents.

Introduction

Nonequilibrium plasmas, thanks to their reactive nature, find recently numerous biological and bio-medical applications, especially when operated at atmospheric pressure providing no need of costly vacuum equipment. Atmospheric pressure plasmas applied for sterilization and bio-decontamination are mostly generated by radio-frequency (RF) [1][2][3][4][5][6] discharges, and various plasma jets and afterglows [1][2][3][7][8][9][10][11], usually in rare gases (He or Ar) with/without admixtures of O 2 (or H 2 O). The plasma jets are typically blown into the ambient air where the rare gas plasma entrains air components. Air plasmas at atmospheric pressure have additional advantages of no need of special gases and an easy application in ambient environment. Biodecontamination by atmospheric air plasmas was tested in DC [7,12], dielectric barrier (DBD) [9,[13][14][15][16][17][18][19][20], RF [1], and pulsed discharges [8][9][10]15]. The plasmas can be also generated directly in water or on water-air boundary [7,16,17,21,22], which is of great interest for water decontamination. Atmospheric pressure plasmas have been tested on a large variety of prokaryotic microorganisms, such as bacteria, spores, viruses, and some eukaryotic yeasts, fungi and microalgae, resulting in partial disinfection (1-2 log reduction of microbial population) up to complete sterilization. Interaction of the plasmas with the tissues of higher organisms including humans and plasma applications for skin disinfection, wound healing, blood coagulation, dentistry, surgery, inducing apoptosis pathways for cancer treatment, etc. are targeted by a revolutionary novel discipline -plasma medicine [23][24][25][26][27]. In bio-decontamination by nonequilibrium plasma, it is very important to assess the role of various mechanisms involved. Despite there is no general consensus among the authors, the significant mechanisms depend strongly on the plasma composition (gas), temperature, treated microorganisms and the environment, in which they are (air, water, surfaces, etc.). In atmospheric pressure plasmas, the major role is typically attributed to radicals and reactive oxygen species (ROS, e.g. OH, O, O 3 ) [1, 3, 6-10, 12, 17-19, 22] and to charged particles, especially O 2 - [14,27] affecting the cell membranes. UV radiation plays a role in atmospheric pressure plasmas only if photons in UV C germicide region (220-280 nm) or in vacuum UV are produced [18,22,28]. UV C comes typically from NO γ system, which is formed in air at high plasma powers if both O and N radicals are generated. It is typical in RF and microwave discharges and plasma jets blown into the air, and high power DBDs. UV C and VUV radiation are also partially produced in discharges in Ar [3,4,28] or in water [22]. In cold (temperature near ambient) atmospheric air discharges (corona, DBDs, pulsed discharges), NO γ and other sources of UV C or VUV radiation are usually not generated, so radicals and reactive oxygen species are identified as the dominant bio-inactivation agents [6-9, 11, 16, 17, 29]. Our former bio-decontamination investigations with DC discharges in atmospheric air with water were in agreement with this statement and demonstrated the dominant role of radicals and ROS [30]. NO γ radiation was only detected in our DC atmospheric pressure glow discharge [31], which was found efficient for biodecontamination but consumed large power [30]. In this paper, the biocidal effects of two plasma sources in atmospheric air with water are investigatedpositive DC streamer corona (SC) and a novel regime named transient spark (TS). Despite DC applied voltage, these discharges have a pulsed character with nanosecond repetitive pulses. The pulsed plasmas have been demonstrated as an effective source of active species and provide a variety of uses in pollution control, plasma assisted combustion, and bio-decontamination. They are especially convenient for penetration into topographically non-uniform surfaces and cavities, which is applicable e.g. in teeth root canal disinfection [9,10,15,32]. The effects of SC and TS on selected bacteria in water solutions and on surfaces were tested. Biodecontamination of water is important from the viewpoint of waste water cleaning or drinking water disinfection and is involved in all bio-medical applications and food technology, since cells and most foods contain water. Plasma decontamination of various surfaces is important in medicine, e.g. for the sterilization of endoscopes, implants and other heat sensitive materials. The surfaces can be also potentially used as carriers of bio-terrorism agents (e.g. anthrax contaminated letters). The plasmas generated by the studied DC discharges, especially TS, induce chemical effects that are crucial in bio-decontamination and environmental applications, such as VOC abatement that have been successfully tested in the past [33]. We focus on the identification of the dominant plasma agents in bio-inactivation by coupling their electrical characteristics, emission spectra, and biocidal effects in various regimes. Comparing direct with indirect plasma effects enables separation of various biocidal plasma agents (electric field, charged particles, neutral active species, UV radiation). In addition, measurements of the oxidative stress induced in microbial cells applied for the first time in plasma bio-decontamination enable further indicate their respective roles.

Experiments

Experimental set-ups

The experimental setup for fundamental investigations of the DC discharges in point-to-plane geometry, with a high voltage (HV) hollow needle electrode enabling water flowing through the discharge zone and a plane or mesh electrode is depicted in figure 1. The inter-electrode spacing was varied from 5-10 mm. A positive DC high voltage was applied through the ballast resistor R (20 M for SC and ~5 M for TS). The discharge voltage was measured by a high voltage probe Tektronix P6015A. The discharge current was measured: on a 50 (SC) or 1 (TS) resistor and by a Rogowski current monitor PEARSON 2877. The current and voltage signals were processed by a digitizing 200 MHz oscilloscope Tektronix TDS 2024. The discharges were photo-and video-documented with digital cameras Olympus E410 or Nikon Coolpix S10. The emission spectroscopy optical system comprised a dual fibre-optic compact spectrometer Ocean Optics SD2000 with CCD detector for fast scanning in the UV and VIS-NIR regions (200-500 and 500-1050 nm, resolution 0.6-1.2 nm), filters, fused silica lenses, irises, and fibre optics. The discharge setup was placed in the Faraday cage together with the optical components mounted on translation stages, which enabled lateral and vertical scanning capabilities. The bio-decontamination effects of the DC discharges were tested on flowing water and on direct/indirect treatment of the contaminated solid medium. The discharge setup shown in figure 1 enabled the contaminated water to flow directly through the high voltage hollow needle electrode, and so through the corona active region in its proximity. The effect of electrostatic spraying (electro-hydro-dynamic atomization, EHDA) occurred when the high voltage was applied on the needle electrode [34]. We also used point-to-plane electrode geometry for the comparison of direct and indirect plasma effects on contaminated solid agar surfaces. A needle electrode was placed about 1 cm above the agar surface in the centre of the Petri dish and the discharge was applied for 1 or 2 min. In direct treatment, the agar was grounded with a wire. Indirect plasma effects on the contaminated agar were tested by: 1) placing the grounded mesh electrode ~2 mm above the agar, this shielded the electric field and trapped the charged particles, letting but neutral particles and partial UV light to reach the agar surface, this concept is similar to [14]; 2) placing the 3 mm thick quartz window onto the agar surface and a grounded ring electrode on its top, this let only the light emitted from the discharge to reach the agar, including UV. We also tested MgF 2 window transmitting vacuum UV, this concept is similar to [3]. Figure 2 schematically depicts these arrangements. The agar in the direct treatment (Figure 2a) represents a certain electrical resistance between the discharge and the ground, its typical value was 1-2 kΩ depending on the water content. In indirect set-ups (Figure 2b,c), a small resistor r was inserted between the mesh (or ring) and the ground to simulate the agar's resistance. Its exact value was set empirically from case to case to make the discharge pulses of about the same amplitude and shape as in the direct treatment on agar. This ensured the same discharge properties in all three set-ups.

Treated microorganisms, microbial handling and cultivation procedure

Bio-decontamination effects of investigated DC discharges were tested on selected Gram-negative bacteria Salmonella typhimurium (Salmonella enterica, serovar Typhimurium, strains TA 98 and 100) in water with initial populations from 10 3 to 10 7 colony forming units per mL (CFU/mL), or directly spread on the solid nutrient medium (agar, Roth Ltd.) on a Petri dish, about 10 6 per dish. S. typhimurium is a pathogen causing salmonellas diseases; its inactivation is important from the viewpoint of drinking water and food disinfection.

The microbial cultivation was carried out in a sterile environment in the following steps: an overnight bacterial culture was first prepared in a shaker with sterile liquid nutrient. A hot sterile nutrient mediumagar -was poured into sterile Petri dishes, on which the bacteria were grown, and solidified. Cultivated bacteria in the liquid nutrient were compared with McFarland turbidity scale to assess their initial population per mL. They were then diluted in water to obtain desired concentrations. The plasma experiments were performed with both discharges, at various parameters and treatment times and repeated 5-8 times. Usually, 50 µL of both plasma treated and reference (control) samples were spread on Petri dishes with agar. 3-4 Petri dishes from each sample were taken for statistical evaluation. These were incubated during 12-24 h in a thermostat at 37 °C. The grown CFUs on the treated and reference samples were counted and evaluated.

Measurements of the oxidative stress

Interaction of ROS with the bacterial cell membranes results in the peroxidation of membrane lipids. The final product of lipoperoxidation is malondialdehyde (MDA), quantifiable by spectrophotometry after the reaction with thiobarbituric acid (TBA) at 90-100 C [35]. This method of thiobarbituric acid reactive substances (TBARS) was applied for the first time to measure the oxidative stress induced in bacteria in distilled water exposed to SC and TS. We assigned the TBARS concentrations from the absorbance of MDA at 532 by using Lambert-Beer's law with the absorption coefficient 1.57×10 5 mol -1 L cm -1 [35].

Results and discussion

Applied DC discharges

Two types of DC discharges of both polarities operating in atmospheric air with water were investigated: a well-known streamer corona (SC), and a novel transient spark (TS). These discharges generate nonequilibrium plasmas inducing various chemical and biological effects important in bio-decontamination. Their electrical parameters and emission spectra were documented in detail in our previous works [31,36].

The typical voltage and current waveforms of SC and TS discharges in 10 mm gap with electro-spray of water are shown in figure 3. When a high voltage of a few kV is applied to the point electrode, SC appears, typical with small current pulses of streamers (~10 mA) with a repetitive frequency of 10-30 kHz, during which the discharge voltage remains fairly constant and generates cold plasma (~300 K). As the voltage is further increased (to ~12 kV in 10 mm gap), the streamers establish a conductive channel that gradually heats, thus enhancing the reduced electric field E/N, which eventually leads to a spark breakdown with excessive current pulse [37]. In our case, the spark pulse current is limited by (1) the ballast resistor R that drops the voltage as the current increases, and (2) the capacity C between the electrodes that is small (order of 10 pF). C is a sum of the internal capacity of the discharge gap (C int ~1 pF) and the capacities of the high voltage cable (C cable ≈5-20 pF depending on the length) and the voltage probe (C probe = 3 pF). Thus, when the sparks forms, it is only transient since the discharged energy is small (0.1-1 mJ). After the pulse, C is recharged by a growing potential on the stressed electrode and triggers a new pulse. This transient spark becomes then a repetitive streamer-to spark transition discharge, with each spark pulse (~1 A) preceded by one or a sequence of streamer pulses. The repetitive frequency of pulses is 0.5-10 kHz, and increases with the applied voltage. Thanks to the very short pulse duration (~10-100 ns) given by the small C and a limiting R, the plasma cannot reach equilibrium conditions and remains at relatively low gas temperature, depending on frequency, i.e. dissipated power (~500-1500 K).

Optical emission spectroscopy (OES)

OES in UV-VIS region is a powerful technique of plasma diagnostics, because it gives valuable information on excited atomic and molecular states, enables to determine the rotational and vibrational temperatures and thus the level of nonequilibrium in the plasma, and gives insight in ongoing plasma chemistry [31,38]. We employed OES, together with the comparison of the biocidal effects of the DC discharges to better understand the dominant mechanisms involved in bio-decontamination. Both SC and TS generate cold, nonequilibrium plasmas (300-550 K) in the discharge channel. OES characteristics of the applied discharges described in detail in [31] showed that electrons with the highest energies were present in TS. These electrons initiate dissociations, ionizations and excitations of various species. Atomic O, N and H radicals, and the N 2 + ions have only been detected in TS, and there were a lot of OH radicals. Part of O radicals reacts with air O 2 and form ozone O 3 . There was no UV C radiation detected from SC and TS.

Flowing water treatment through the stressed electrode

The contaminated water flew directly through the high voltage hollow needle electrode, and so through the plasma active zone in its proximity, which substantially improved the volume efficiency compared to our previous set-ups for water treatment [30]. This idea was first introduced for polluted gas treatment through the high voltage needle electrodes [39], we first time applied it for water treatment. The effect of electrostatic spraying (electro-hydro-dynamic atomization, EHDA) occurred when the high voltage was applied on the needle electrode [34]. The electrostatic spraying of contaminated water through the corona active zone contributed to the improved efficiency of bio-decontamination. Figure 4 shows the photographs of the EHDA of water when a high voltage is applied, EHDA with SC, a transition SC-TS perturbing the EHDA and then TS discharge with water sprayed. The temperature of the treated water did not change in SC and was increased by maximum 10 K in TS. The lethal heat effect of the discharges to bacteria cannot be considered at all. The electro-spray water treatment resulted in very short D-values (0.1-0.4 s in both SC and TS), and low energy costs (0.7-20 J/mL per log reduction) in SC. Further improvements of the electro-spray and EHDAdischarge interaction will expectantly enhance the inactivation at lower energy costs.

Oxidative stress induced in bacteria

Figure 5 shows the TBARS concentration gain ∆c(TBARS) correlated with the bio-decontamination efficiency of SC and TS applied to the electro-sprayed water. The same bacterial samples were irradiated by biocidal UV C radiation (Hg lamp, 254 nm, 1 min) for comparison.

UV C radiation induced almost no ∆c(TBARS) despite its efficiency was very high. Obviously, UV dominant biocidal mechanism is not peroxidation of cell membranes, albeit radiation induction of ROS in cell nuclei is possible [27]. On the contrary, SC and TS plasma treatments significantly enhanced the TBARS concentration. This indicates that oxidations of cell membranes by ROS are important in microbial inactivation. More ROS is linked with the higher efficiency.

Direct vs. indirect plasma treatment

A comparison of direct and indirect plasma effects on contaminated agar surfaces was aimed at separation of various biocidal plasma agents. We compared direct SC and TS plasma treatment with 2 types of indirect exposure obtained by 1) filtering the charged particles and electric field from neutral radicals and excited species; and 2) applying only UV radiation from the plasma. The corresponding schematics of the electrode arrangements were shown in figure 2. Figure 6 shows the photographs of the Petri dishes with contaminated agar after direct and both indirect treatments together with the untreated control sample, and a sample treated with 50 µL drop of liquid ethanol (96%) for comparison. The effects of plasma (and ethanol) on contaminated agar are clearly visible as dark voids, whereas control sample is homogeneously covered by cultivated bacteria (bright). Dark voids with bright spots represent incomplete decontamination (e.g. with ethanol), the spots are CFUs grown from single bacteria. With respect to the total number of bacteria spread on one Petri dish (10 6 ), a few tens of survived bacteria that cause these bright CFUs in the voids are quite negligible. Both direct plasma and indirect exposures to neutral reactive species (and partial UV) caused apparent biodecontamination (voids). SC resulted in a larger treated area on the Petri dish, likely due to the electric wind that drives the active species from the point electrode towards and along the agar surface in one preferential direction. TS treatment is more localized in the centre of the dish but more intense. The heat effect of TS on contaminated agar is possible in the very small area under the discharge. TS usually resulted in a tiny hole (1-2 mm diameter) of dried agar. Nevertheless, the area of decontamination (void) was always much larger than this tiny hole in its centre. The void kept the ambient temperature after treatment, so we can neglect the heat effect on bacteria. In both indirect treatments, the TS heat was lead out through the mesh (ring) electrode which was not in the direct contact with the agar surface: the heat effect can be excluded. Interestingly, there was very little difference between the direct and indirect plasma treatments with both discharges, which indicates that neutral reactive species are crucial even in the direct exposure. Exposure to the UV light only, transmitted by quartz or MgF 2 windows demonstrated no visible decontamination; the dishes were almost identical with the control. This correlates with the emission spectra lacking any UV C or VUV radiation in SC and TS. A similarity of the effects of direct plasma and indirect treatment by reactive neutrals agrees with the emission spectra, oxidation stress measurements and our previous findings [30]. Apparently, radicals and active species

Conclusions

Bio-decontamination of water and surfaces contaminated by bacteria (S. typhimurium) was tested in two types of positive DC discharges in atmospheric pressure air in point-to-plane geometry with one electrode submerged in water or in a new arrangement with water sprayed through the plasma zone. The bacteria were handled and their population evaluated by standard microbiology thermostatic growth cultivation procedures. The streamer corona with small current pulses (~10 mA) and 10-30 kHz repetitive frequency generates cold (300-350 K) nonequilibrium plasma. With increasing applied voltage, the streamers transit to the novel regime transient spark with short (<100 ns) current pulses (~1-10 A) of 0.5-10 kHz repetitive frequency and very limited energy. Thanks to the very short spark pulse duration given by the small internal capacity of the discharge system and the limiting series resistor, the TS plasma remains relatively cold (~500 K). The emission spectra of the discharges and the measured temperatures indicate that both SC and TS generate non-equilibrium plasmas with various excited species (N 2 * ), and molecular (OH) and atomic (O, N, H) radicals.

Both SC and TS were found very efficient when the treated water was sprayed directly through the high voltage needle electrode and thus through the active discharge zone. Considerable efficiencies at short treatment times resulted in very short D-values and relatively low energy costs. EHDA effect occurring with corona discharge in this regime applied for the first time enhanced the efficiency of the process. The comparisons of direct and two types of indirect exposure of contaminated agar plates to the plasma of SC and TS enabled the separation of the various biocidal agents. It was demonstrated that the direct exposure and indirect exposure to separated active neutral species generated in the plasma had almost the same effect on bacteria. On the other hand, separated plasma radiation, including UV, had no significant effect. These investigations, together with the emission spectra of the discharges, indicated the major role of radicals and reactive oxygen species (O, OH, O 3 ). Their role in the plasma treatment was confirmed by the absorption spectroscopic detection of the products of microbial cell membrane oxidation stress in TBARS (thiobarbituric acid reactive substances) method. In summary, we demonstrated that cold atmospheric air DC discharges can be efficiently used for biodecontamination of water and surfaces. The dominant biocidal plasma agents are radicals and ROS, which agrees with findings published by many other authors.

Figure 1 .

 1 Figure 1. Experimental setup for DC discharges, with a high voltage hollow needle electrode enabling water flowing through the discharge zone and a plane or mesh electrode.

Figure 2 .

 2 Figure 2. Schematics of electrode arrangements for (a) direct and (b, c) indirect plasma treatment of contaminated agar plates. (b) Mesh electrode ~2 mm above agar surface trapped the charged particles and shielded the electric field. (c) Quartz (or MgF 2 ) window only transmitted light from the discharge (including UV).

Figure 3 .

 3 Typical voltage and current waveforms of (a) SC and (b) TS discharges with electro-spray of water, 10 mm gap.

Figure 4 .

 4 Photographs of the EHDA of water in 8 mm gap, water flow rate 0.5 mL/min: (a) droplet without a high voltage, (b) EHDA spray with a high voltage applied, 5.5 kV, (c) EHDA together with SC, 6.5 kV, (d) transition SC-TS, 7.8 kV, (e) spray with TS, 9 kV.

EfficiencyFigure 5 .

 5 Figure 5. TBARS concentration gains and decontamination efficiencies of S. typhimurium in water treated by SC and TS with electro-spray, compared with 1 min exposure to UV C (shown with standard error of the mean, n -number of repeated experiments).

  (O, N, H, OH, O 3 , O 2 -), especially ROS, represent the dominant biocidal mechanism in atmospheric air SC and TS discharges.

Figure 6 .

 6 Figure 6. Photographs of Petri dishes with contaminated agar; left side: untreated control and treated by ethanol, right side: treated by SC (upper row) and TS (lower row) direct, neutral species only, and UV light only.

Acknowledgements

Effort sponsored by Slovak grant agency VEGA 1/0293/08. We thank I. Jedlovský, B. Pongrác, L. Šikurová, P. Pol ic, M. Janda and K. Hensel for their assistance and fruitful discussions.