To overcome the limitations of solid-liquid phase change materials, such as liquid leakage, unstable shape, and serious corrosion at elevated temperatures, the design of a solid-solid phase change material using Ni-Ti-Zr as a metal element was utilized for its high thermal conductivity and stability to provide new thermal storage solutions for downhole tools. The influence of Zr content on the thermophysical properties and microstructure of the material was also investigated. The results showed that the designed solid-solid phase change material could reach 4129.93 × 106 J2K−1s−1m−4 quality factor, which was much higher than most of the phase change materials. This was attributed to the increase in Zr content, where Zr atoms took the place of Ti atoms, creating vacancy defect and leading to the distortion of the alloy lattice where the Ti2Ni phase gradually became the (Ti, Zr)2Ni phase, the rise in density and enthalpy of the phase transition, and the increase in the temperature of the phase transition. When the Zr content was 12%, the phase transition temperature of the alloy matched the downhole working temperature, demonstrating excellent thermal cycling stability, and is expected to be a heat storage material for downhole tools.
Citation: Wentao Qu, Qian Zhang, Guibian Li, Boyang Pan, Haiying Liu. Microstructure and thermophysical properties of NiTiZr phase change alloys for downhole tool heat storage[J]. AIMS Materials Science, 2025, 12(1): 85-100. doi: 10.3934/matersci.2025007
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To overcome the limitations of solid-liquid phase change materials, such as liquid leakage, unstable shape, and serious corrosion at elevated temperatures, the design of a solid-solid phase change material using Ni-Ti-Zr as a metal element was utilized for its high thermal conductivity and stability to provide new thermal storage solutions for downhole tools. The influence of Zr content on the thermophysical properties and microstructure of the material was also investigated. The results showed that the designed solid-solid phase change material could reach 4129.93 × 106 J2K−1s−1m−4 quality factor, which was much higher than most of the phase change materials. This was attributed to the increase in Zr content, where Zr atoms took the place of Ti atoms, creating vacancy defect and leading to the distortion of the alloy lattice where the Ti2Ni phase gradually became the (Ti, Zr)2Ni phase, the rise in density and enthalpy of the phase transition, and the increase in the temperature of the phase transition. When the Zr content was 12%, the phase transition temperature of the alloy matched the downhole working temperature, demonstrating excellent thermal cycling stability, and is expected to be a heat storage material for downhole tools.
Let V denote a finite dimensional representation of a finite group G over a field F in characteristic p such that p||G|. Then, V is called a modular representation. We choose a basis {x1,…,xn} for the dual space V∗. The action of G on V induces an action on V∗ and it extends to an action by algebra automorphisms on the symmetric algebra S(V∗), which is equivalent to the polynomial ring
F[V]=F[x1,…,xn]. |
The ring of invariants
F[V]G:={f∈F[V]|g⋅f=f∀g∈G} |
is an F-subalgebra of F[V]. F[V] as a graded ring has degree decomposition
F[V]=∞⨁d=0F[V]d |
and
dimFF[V]d<∞ |
for each d. The group action preserves degree, so F[V]d is a finite dimensional FG-module. A classical problem is to determine the structure of F[V]G, and the construction of generators of the invariant ring mainly relies on its Noether's bound, denoted by β(F[V]G), which is defined as follows:
β(F[V]G)=min{d|F[V]Gisgeneratedbyhomogeneousinvariantsofdegreeatmostd}. |
The bound reduces the task of finding invariant generators to pure linear algebra. If char(F)>|G|, the famous "Noether bound" says that
β(F[V]G)⩽|G|(Noether [1]). |
Fleischmann [2] and Fogarty [3] generalized this result to all non-modular characteristic (|G|∈F∗), with a much simplified proof by Benson. However, at first shown by Richman [4,5], in the modular case (|G| is divisible by char(F)), there is no bound that depends only on |G|. Some results have implied that
β(F[V]G)⩽dim(V)⋅(|G|−1). |
It was conjectured by Kemper [6]. It was proved in the case when the ring of invariants is Gorenstein by Campbell et al. [7], then in the Cohen-Macaulay case by Broer [8]. Symonds [9] has established that
β(F[V]G)⩽max{dim(V)⋅(|G|−1),|G|} |
for any representation V of any group G. This bound cannot be expected to be sharp in most cases. The Noether bound has been computed for every representation of Cp in Fleischmann et al. [10]. It is in fact 2p−3 for an indecomposable representation V with dim(V)>4. Also in Neusel and Sezer [11], an upper bound that applies to all indecomposable representations of Cp2 is obtained. This bound, as a polynomial in p, is of degree two. Sezer [12] provides a bound for the degrees of the generators of the invariant ring of the regular representation of Cpr.
In the previous paper Zhang et al. [13], we extended the periodicity property of the symmetric algebra F[V] to the case
G≅Cp×H |
if V is a direct sum of m indecomposable G-modules such that the norm polynomials of the simple H-modules are the power of the product of the basis elements of the dual. Now, H permutes the power of the basis elements of simple H-modules, for example, H is a monomial group. For the cyclic group Cp, the periodicity property and the proof of degree bounds mainly relies on the fact that the unique indecomposable projective FCp-module Vp is isomorphic to the regular representation of Cp. Then, every invariant in VCpp is in the image of the transfer map (see Hughes and Kemper [14]). Inspired by this, with the assumption above, we find that every invariant in projective G-modules is in the image of the transfer. In this paper, since the periodicity leads to degree bounds for the generators of the invariant ring, we show that F[V]G is generated by m norm polynomials together with homogeneous invariants of degree at most m|G|−dim(V) and transfer invariants, which yields the well-known degree bound dim(V)⋅(|G|−1). Moreover, we find that this bound gets less sharp as the dimensions of simple H-modules increase, which is presented in the end of the paper.
Let G=Cp×H be a finite group, whose form is a direct sum of the cyclic group of order p, Cp, and a p′-group H. Let F be an algebraically closed field in characteristic p. The complete set of indecomposable modules of Cp is well-known, which are exactly the Jordan blocks Vn of degree n, for 1⩽n⩽p, with 1's in the diagonal. Let Sim(H) be the complete set of non-isomorphic simple FH-modules. Since p∤|H|, there is no difference between irreducibility and indecomposibility of FH-modules. Then, by Huppert and Blackburn [15, Chapter Ⅶ, Theorem 9.15], the FG-modules
Vn⊗W(1⩽n⩽p,W∈Sim(H)) |
form a complete set of non-isomorphic indecomposable FG-modules.
Notice that the FH-module W is simple if and only if W∗ is simple, i.e., ∀0≠w∈W∗, w generates W∗ as an FH-module. For a fixed 0≠w∈W∗, let
H={h1=e,h2,…,h|H|} |
and
{w1=w,w2=h2w,…,wk=hkw} |
be a basis of W∗ with dim(W∗)=k. The norm polynomial
NH(w)=∏h∈Hh(w) |
is an H-invariant polynomial.
Lemma 2.1. ([13, Lemma 3.1]) Let W∗ be a simple FH-module with a vector space basis {w1,w2,…,wk} as above. Then, the norm polynomial NH(w1) is of the form (w1,w2,…,wk)lf, where l⩾1 and f is a polynomial in F[w1,w2,…,wk] such that wi∤f for 1⩽i⩽k.
Choose the triangular basis {w11, w12, …, w1k, …, wn1, wn2, …, wnk} for (Vn⊗W)∗, where
V∗n=<v1,…,vn>, |
such that
Cp=<σ>,σ(vj)=vj+vj−1,σ(v1)=v1,andwji=vj⊗wi. |
Then, we obtain
σ⋅(wji)=wji+ϵjwj−1,iwithϵj=1for2⩽j⩽n,ϵ1=0. |
Then,
w∈<wn1,wn2,…,wnk> |
is a distinguished variable in (Vn⊗W)∗, which means that w generates the indecomposable FG-module (Vn⊗W)∗. If
k=|H|l |
in Lemma 2.1, then
NH(wn1)=(wn1wn2⋯wnk)l |
and
Nn△=NCp(NH(wn1))=NCp((wn1wn2⋯wnk)l)∈F[Vn⊗W]G. |
Note that degwni(Nn)=lp for 1⩽i⩽k.
Example 2.1. Consider the monomial group M generated by
π=(010⋱110),τ=(α10⋱0αk), |
where π is a k-rotation and αi are lth roots of unity in C. M acts on the module V=Ck and also on the symmetric algebra
C[V]=C[x1,…,xk] |
in the standard basis vectors xi of V∗. Then {xl1,…,xlk} is an M-orbit and (x1,…,xk)l∈C[V]M. For more details about degree bounds and Hilbert series of the invariant ring of monomial groups, it can be referred to Kemper [6] and Stanley [16].
Let F[Vn⊗W]♯ be the principal ideal of F[Vn⊗W] generated by Nn. The set {Nn} is a Gr¨obner basis for the ideal it generates. Then, we may divide any given f ∈ F[Vn⊗W] to obtain f = qNn + r for some q, r ∈ F[Vn⊗W] with degwni(r) < lp for at least one i.
We define
F[Vn⊗W]♭:={r∈F[Vn⊗W]|degwni(r)<lpforatleastonei}. |
Note that both F[Vn⊗W]♯ and F[Vn⊗W]♭ are vector spaces and that as vector spaces,
F[Vn⊗W]=F[Vn⊗W]♯⊕F[Vn⊗W]♭. |
Moreover, they are G-stable. Therefore, we have the FG-module decomposition
F[Vn⊗W]=F[Vn⊗W]♯⊕F[Vn⊗W]♭. |
Lemma 2.2. With the above notation, we have the FG-module direct sum decomposition
F[Vn⊗W]=Nn⋅F[Vn⊗W]⊕F[Vn⊗W]♭. |
Proof. It is clear from the definitions of F[Vn⊗W]♯ and F[Vn⊗W]♭.
Lemma 2.3. ([13, Lemma 3.2]) The FG-module F[Vn⊗W]♭d can be decomposed as a direct sum of indecomposable projective modules if the following conditions are satisfied:
1) d+kn ⩾ lkp+1;
2) NH(w1)=(w1w2⋯wk)l for some basis {w1,w2,…,wk} of W∗.
Theorem 2.1. ([13, Theorem 3.5]) Let G=Cp×H be a finite group described above. Let
V=(Vn1⊗W1)⊕(Vn2⊗W2)⊕…⊕(Vnm⊗Wm) |
be an FG-module such that the norm polynomial of the simple H-module Wi is the power of the product of the basis elements of the dual. Vni are indecomposable Cp-modules. Let d1,d2…,dm be non-negative integers and write di = qi(likip) + ri, where 0 ⩽ ri ⩽ likip−1 for i=1,2,…,m. Then,
F[V](d1,d2,…,dm)≅F[V](r1,r2,…,rm)⊕(⨁W∈Sim(H)sWVp⊗W) |
as FG-modules for some non-negative integers sW.
A set
{f1,f2,…,fn}⊂F[V]G |
of homogeneous polynomials is called a homogeneous system of parameters (in the sequel abbreviated by hsop) if the fi are algebraically independent over F and F[V]G as an F[f1,f2,…,fn]-module is finitely generated. This implies n = dim(V).
The next lemma provides a criterion about systems of parameters.
Lemma 2.4. ([17, Proposition 5.3.7]) f1,f2,…,fn∈F[z1,z2,…,zn] are an hsop if and only if for every field extension ¯F⊃F the variety
V(f1,f2,…,fn;¯F)={(x1,x2,…,xn)∈¯F|fi(x1,x2,…,xn)=0fori=1,2,…,n} |
consists of the point (0,0,…,0) alone.
With the previous lemma, Dade's algorithm provides an implicit method to obtain an hsop. We refer to Neusel and Smith [18, p. 99-100] for the description of the existence of Dade's bases when F is infinite.
Lemma 2.5. ([18, Proposition 4.3.1]) Suppose ρ: G↪GL(n,F) is a representation of a finite group G over a field F, and set V=Fn. Suppose that there is a basis z1,z2,…,zn for the dual representation V∗ that satisfies the condition
zi∉⋃g1,…,gi−1∈GSpanF{g1⋅z1,…,gi−1⋅zi−1},i=2,3,…,n. |
Then the top Chern classes
ctop([z1]),…,ctop([zn])∈F[V]G |
of the orbits [z1],…,[zn] of the basis elements are a system of parameters.
An hsop exists in any invariant ring of a finite group, which is by no means uniquely determined. After an hsop has been chosen, it is important for subsequent computations to have an upper bound for the degrees of homogeneous generators of F[V]G as an F[f1,f2,…,fn]-module. In the non-modular case, F[V]G is a Cohen-Macaulay algebra. It means that for any chosen hsop {f1,f2,…,fn}, there exists a set {h1,h2,…,hs} ⊂ F[V]G of homogeneous polynomials such that
F[V]G≅s⊕j=1F[f1,f2,…,fn]⋅hj, |
where
s=dimF(Tot(F⊗F[f1,f2,…,fn]F[V]G)). |
Using Dade's construction to produce an hsop f1,f2,…,fn of F[V]G, the following lemmas imply that in the non-modular case, F[V]G as F[f1,f2,…,fn]-module is generated by homogeneous polynomials of degree less than or equal to dim(V)(|G|−1). This result is very helpful to consider the degree bound for G=Cp×H in the modular case (see Corollary 2).
Lemma 2.6. ([16, Proposition 3.8]) Suppose ρ: G↪GL(n,F) is a representation of a finite group G over a field F whose characteristic is prime to the order of G. Let
F[V]G≅⊕sj=1F[f1,f2,…,fn]⋅hj, |
where deg(fi)=di, deg(hj)=ej, 0=e1⩽e2⩽⋯es. Let μ be the least degree of a det−1 relative invariant of G. Then
es=n∑i=1(di−1)−μ. |
Lemma 2.7. ([17, Proposition 6.8.5]) With the notations of the preceding lemma, then
es⩽n∑i=1(di−1) |
with equality if and only if G⊆SL(n,F).
In this section, with the method of Hughes and Kemper [14], we proceed with the proof of degree bounds for the invariant ring.
Throughout this section, let G=Cp×H be a direct sum of a cyclic group of prime order and a p′-group, V any FG-module with a decomposition
V=(Vn1⊗W1)⊕(Vn2⊗W2)⊕⋯⊕(Vnm⊗Wm), |
where Wi's are simple H-modules satisfying that there is wi1∈W∗i such that
NH(wi1)=(wi1wi2⋯wiki)li |
with ki=dim(Wi) and
dim(Vni⊗Wi)>1. |
Then {wlii1,wlii2,…,wliiki} are H-orbits.
Consider the Chern classes of
Oi={wlii1,wlii2,…,wliiki}. |
Set
φOi(X)=ki∏t=1(X+wliit)=ki∑s=0cs(Oi)⋅Xki−s. |
Define classes cs(Oi)∈F[Wi]H, 1⩽s⩽ki, the orbit Chern classes of the orbit Oi.
Consider the twisted derivation
△:F[V]→F[V] |
defined as △=σ−Id, where Cp is generated by σ. For a basis x1…,xni of V∗ni such that
σ(x1)=x1andσ(xni)=xni+xni−1, |
and a basis y1,…,yki of W∗i, {xs⊗yt|1⩽s⩽ni,1⩽t⩽ki} is a basis of V∗ni⊗W∗i. On the other hand, since xni is a generator of indecomposable Cp-module V∗ni and yt,1⩽t⩽ki, are all generators of simple H-module W∗i, then
zit:=xni⊗yt,1⩽t⩽ki |
are all generators of the indecomposable G-module V∗ni⊗W∗i such that
zji1:=△j(zi1)=△j(xni)⊗y1=xni−j⊗y1,⋮zjiki:=△j(ziki)=△j(xni)⊗yki=xni−j⊗yki |
for 0⩽j⩽ni−1. Since for FG-modules dual is commutative with direct sum and tensor product, hence
{zjit=△j(zit)|0⩽j⩽ni−1,1⩽i⩽m,1⩽t⩽ki} |
can be considered as a vector space basis for V∗. Moreover,
Ni=NCp((zi1zi2⋯ziki)li) |
is the norm polynomial of (Vni⊗Wi)∗. Therefore for the H-orbits
Bi={zlii1,zlii2,…,zliiki}, |
we have
NCp(cs(Bi))∈F[V]G. |
Notice that
NCp(cki(Bi))=Ni |
is the top Chern class of the orbit Bi.
Consider the transfer
TrG:F[V]→F[V]G |
defined as
f↦∑σ∈Gσ(f). |
Note that the transfer is an F[V]G-module homomorphism. The homomorphism is surjective in the non-modular case, while the image of transfer is a proper ideal of F[V]G in the modular case.
Lemma 2.8. Let V be a projective FG-module, then VG⊂ImTrG.
Proof. Since V is projective, then it has a direct sum decomposition
V=⨁W∈Sim(H)sW(Vp⊗W) |
as FG-modules for some non-negative integers sW. It is easy to see that
(Vp⊗W)G=(Vp⊗W)Cp×H=VCpp⊗WH. |
Since every invariant in Vp is a multiple of the sum over a basis which is permuted by Cp and WH=ImTrH, the claim is proved.
The following result is mainly a consequence of Lemmas 2.2, 2.3 and 3.1.
Theorem 3.1. In the above setting, F[V]G is generated as a module over the subalgebra F[N1,N2,…,Nm] by homogeneous invariants of degree less than or equal to m|G|−dim(V) and TrG(F[V]).
Proof. Let M⊆F[V]G be the F[N1,N2,…,Nm]-module generated by all homogeneous invariants of degree less than or equal to m|G|−dim(V) and TrG(F[V]). We will prove F[V]Gd⊆M by induction on d. So we can assume d>m|G|−dim(V). By Lemma 2.2, there is a direct sum decomposition of F[V] as FG-module
F[V]≅m⨂i=1F[Vni⊗Wi]≅m⨂i=1(Ni⋅F[Vni⊗Wi]⊕F[Vni⊗Wi]♭)≅⨁I⊆{1,2,…,m}(⊗i∈INi⋅F[Vni⊗Wi]⊗⊗i∈¯IF[Vni⊗Wi]♭) |
with ¯I={1,2,…,m}∖I. Therefore, any f∈F[V]Gd can be written as
f=∑I⊆{1,2,…,m}fI |
and
fI∈⨁|I|⋅|G|+∑i∈Iqi+∑i∈¯Iri=d(⊗i∈INi⋅F[Vni⊗Wi]qi⊗⊗i∈¯IF[Vni⊗Wi]♭ri)G, |
where qi,ri are non-negative integers. For I≠∅, fI lies in M by the induction hypothesis. For I=∅, there is an i such that ri>|G|−kini for each summand ⊗i∈¯IF[Vni⊗Wi]♭ri, since otherwise one would have
d⩽m∑i=1(|G|−kini)=m|G|−dim(V), |
which contradicts the hypothesis. By Lemma 2.3, F[Vni⊗Wi]♭ri is projective if
ri>likip−kini=|G|−kini, |
and so by Alperin [19, Lemma 7.4] the summand ⊗i∈¯IF[Vni⊗Wi]♭ri is projective. Then every invariant in it is in the image of transfer by Lemma 3.1.
Corollary 3.1. Let G=Cp×H be a finite group, where H is a cyclic group such that p∤|H|,
V=(Vn1⊗W1)⊕(Vn2⊗W2)⊕⋯⊕(Vnm⊗Wm), |
a module over FG such that Wi's are permutation modules over FH. Then dim(Wi)=|H| and F[V]G is generated by N1,N2,…,Nm together with homogeneous invariants of degree less than or equal to ∑mi=1|H|(p−ni) and transfer invariants. Moreover, if
V=(Vp⊗W1)⊕(Vp⊗W2)⊕⋯⊕(Vp⊗Wm), |
such that Wi's are permutation modules over FH, then F[V]G is generated by N1,N2,…,Nm together with transfer invariants.
Corollary 3.2. Let G=Cp×H be a finite group,
V=(Vn1⊗W1)⊕(Vn2⊗W2)⊕⋯⊕(Vnm⊗Wm) |
be a finite-dimensional vector space in the setting of the beginning of this section. Assume that F is an infinite field. We have
β(F[V]G)⩽dim(V)⋅(|G|−1). |
Proof. Let zi1,zi2,…,ziki be distinguished variables such that
Ni=NCp((zi1zi2⋯ziki)li) |
are the norm polynomials of Vni⊗Wi for 1⩽i⩽m. Let
Bi={zlii1,zlii2,…,zliiki},1⩽i⩽m |
be H-orbits. We use Dade's algorithm to extend the set
{NCp(cs(Bi))|1⩽s⩽ki,1⩽i⩽m}⊂F[V]G |
to an hsop for F[V]G in the following way. We extend the sequence
v1=z11,v2=z12,…vk1=z1k1,…vk1+k2+⋯+km−1+1=zm1,vk1+k2+⋯+km−1+2=zm2,…vk1+k2+⋯+km=zmkm |
in V∗ by vectors vk1+⋯+km+1,vk1+⋯+km+2,…,vdim(V) such that
vi∉⋃g1,…,gi−1∈GSpanF{g1⋅v1,…,gi−1⋅vi−1} |
for all
i∈{k1+…+km+1,k1+…+km+2,…,dim(V)}, |
and set
fj=∏g∈Gg(vj) |
for
k1+⋯+km<j⩽dim(V). |
Applying Lemmas 2.4 and 2.5, we see that NCp(cs(Bi)), 1⩽s⩽ki,1⩽i⩽m, together with fk1+k2+⋯+km+1,…,fdim(V) form an hsop of F[V]G since their common zero in ¯Fdim(V) is the origin. Let A denote the polynomial algebra generated by the elements of this hsop. Since F[V] as an F[V]G-module is finitely generated, then we have that F[V] is a finitely generated A-module. View F[V] as the invariant ring of the trivial group. By Lemma 2.7 the upper degree bound of module generators satisfies that
β(F[V]/A)⩽∑s,ideg(NCp(cs(Bi)))+∑jdeg(fj)−dim(V). |
Since the transfer preserves the degree of polynomials and it is clear that
m|G|=m∑i=1deg(NCp(cki(Bi)))⩽∑s,ideg(NCp(cs(Bi)))+∑jdeg(fj), |
we conclude from Theorem 3.1 that the upper degree bound of module generators of F[V]G as an A-module satisfies that
β(F[V]G/A)⩽∑s,ideg(NCp(cs(Bi)))+∑jdeg(fj)−dim(V)=m∑i=1pli(1+2+⋯+ki)+|G|(dim(V)−k1−k2−⋯−km)−dim(V)=m∑i=1pliki(ki+1)2−|G|m∑i=1ki+dim(V)⋅(|G|−1)=|G|m∑i=11−ki2+dim(V)⋅(|G|−1)⩽dim(V)⋅(|G|−1). |
Note that we have assumed that |G|⩾p and dim(Vni⊗Wi)>1. Since
deg(NCp(cs(Bi))=psli⩽|G|anddeg(fj)=|G|, |
we obtain
β(A)⩽|G|⩽dim(V)⋅(|G|−1), |
which completes the proof.
Remark 3.1. The proof of the previous corollary implies that Symonds' bound is far too large. Since
β(F[V]G/A)⩽|G|m∑i=11−ki2+dim(V)⋅(|G|−1), |
it seems that this bound is less sharp as the dimensions of simple H-modules increase. Moreover, the construction of fj for k1+…+km<j⩽dim(V) is more than necessary to obtain invariants for F[V]G. It is enough to take the product of the G-orbit of vj.
In this paper, we consider degree bounds of the invariant rings of finite groups Cp×H with the projectivity property of symmetric algebras, and we show that this bound is sharper than Symonds' bound as the dimensions of simple H-modules increase.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Thanks to Dr. Haixian Chen for reminding us that the projectivity property of the symmetric powers leads to the degree bounds of the ring of invariants. Thanks to the referees for their helpful remarks. The authors are supported by the National Natural Science Foundation of China (Grant No. 12171194).
The authors declare no conflicts of interest in this paper.
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