1 Introduction

Exactly solvable models have a special place in statistical physics [1]. One of these models is the two-dimensional square ice model [2], which occupies a place of honor not only in physics, but also in mathematics [3]. For many years, this model was largely abstract in nature, because very approximately reflected the structure of real three-dimensional ice. However, more recently, square ice has ceased to be an abstraction, because was obtained in a laboratory experiment [4]. For ice monolayers, as well as other ice nanostructures, including water clusters and ice nanotubes, the concept of residual entropy takes on a new meaning. In the set of all proton configurations that differ in the arrangement of hydrogen atoms (protons) in H-bonds, the energy variation is very large. For a cluster in the shape of a pentagonal dodecahedron, the energy variation per molecule exceeds the heat of ice melting [5]. As a rule, only more stable proton configurations are realized in water clusters, nanotubes and nanolayers. However, a consistent theoretical analysis of the stability of ice nanostructures taking into account proton disorder requires an exhaustive enumeration and classification of all defect-free proton configurations by total binding energy [6,7,8]. Nanostructures of regular shape are of particular interest. The number of proton configurations increases exponentially with increasing size. In this situation, the residual entropy, as the logarithm of the total number of configurations, is a convenient quantity characterizing the set of all defect-free structures.

The problem of calculating the residual entropy for finite ice-like systems is certainly simpler than for infinite systems. For ice nanotubes and other quasi-one-dimensional systems, it is convenient to use the usual transfer matrix method [9, 10]. However, the size of the matrices is often quite large, which creates some inconvenience associated with the perception of these kind of results. The method of local conditional transfer matrices implemented in the MathCad software system is as clear as possible, since for its application it is enough to set small matrices: most often by size 2 × 2 [11, 12]. The method can be easily programmed outside of MathCad, which is sometimes necessary for very large matrices (more than 1024 × 1024). Using this method, detailed statistics of proton disorder were obtained for square and hexagonal ice nanotubes, as well as for a number of other nanostructures.

The entropy values ​​for tubes of different widths k and the fact that the entropy per one site converges to its limit value with a rate of 1/k2 by widening the strips made it possible to obtain a very good approximation to the well-known exact value for the infinite square ice model [11]. For a strip of a hexagonal monolayer with a width k = 10, the value S10 = 0.752749748 was obtained [12]. Recently, the exact residual entropy of ice hexagonal monolayer was calculated, where S = 0.752745 [13]. This value was obtained by multiplying by 3/2 the zero entropy of the antiferromagnetic kagome lattice, which is given by a double integral [14]. However, more careful multiplication of the exact value of the integral gives the value 0.752749747. The situation is strange: the approximate value S10 turned out to be “more accurate than the exact one [13]”. The purpose of the article is to show the fundamental difference between three- and four-coordinated systems in relation to the speed of convergence of approximate methods. The paper demonstrates various applications of the relatively simple computational method for the analysis of some complex problems.

2 The method of local conditional transfer matrices

The essence of the method of local conditional transfer matrices [11, 12] is that the transfer matrices between elements of a wide strip are themselves obtained as a result of multiplying very small matrices along perpendicularly oriented one-dimensional chains. The unusual nature of such multiplication lies in the fact that the matrix elements are not constant, but parametrically depend on the direction of perpendicular H-bonds. MathCad and other mathematical software is a convenient tool for implementing such algorithms.

Figure 1 shows a one-dimensional chain and a scheme for calculating the transfer matrices along this chain for different directions of external perpendicular H-bonds, shown by the dotted arrows. According to the Bernal-Fowler ice rules, at each lattice node two arrows are incoming and two are outgoing [15, 16].

The unified transfer matrix can be written as follows:

$$\:\:\varvec{A}(i,j)=\left\{\begin{array}{c}\varvec{B}0,\:\:\:\:i=j\\\:\varvec{B}1,\:\:\:\:i<j\\\:\varvec{B}2,\:\:\:\:i>j\end{array}\right.$$
(1)

Here, the equality of matrices B0 and B0′ is taken into account. For a strip of width k = 3 cyclically closed into a tube, the resultant transfer matrix is ​​of the form.

$$\:{\varvec{M}}_{4i1+2i3+i5,\:\:4i2+2i4+i6}=Tr(\varvec{A}\left(i1,i2\right)\ \varvec{A}\left(i3,i4\right)\varvec{A}\left(i5,i6\right))$$
(2)

The elements of this transfer matrix for the entire strip M will be sequentially calculated if a list of cross-link directions (left and right) is specified at the very beginning.

$$\:i1=\text{0,1}\:\:\:\:\:i2=\text{0,1}\:\:\:\:\:i3=\text{0,1}\:\:\:\:\:i4=\text{0,1}\:\:\:\:\:i5=\text{0,1}\:\:\:\:\:i6=\text{0,1}$$
Fig. 1
figure 1

(a )Strait chain of square ice lattice. (b) The diagrams for the calculation of the local transfer matrices for each direction of outer H-bonds (dotted arrows)

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The trace of the n-th power of the transfer matrix, as is known, determines the number of configurations for an n-element tube cyclically closed into a torus.

$$\:{X}_{\text{n}}=Tr\left({\varvec{M}}^{\text{n}}\right)=\sum\:_{\text{i}=1}^{\text{N}}{\lambda\:}_{\text{i}}^{\text{n}}$$
(3)

where λi are the 2k (size the matrix) eigenvalues of the transfer-matrix. The asymptotic value of the residual entropy (in dimensionless form) for a strip of width k is determined by the maximum eigenvalue of the transfer matrix.

$$\:{S}_{\text{k}\:}\approx\:\text{l}\text{n}\left({\lambda\:}_{\text{m}\text{a}\text{x}}\right)/(k\:{n}_{\text{s}})$$
(4)

Here, ns is the number of nodes in one element of a one-dimensional chain. For a square lattice ns = 1. In this way, the asymptotic values ​​of the entropy S for stripes of square ice with widths up to k = 12 were calculated. Extrapolating these values ​​taking into account the convergence rate 1/k2, for constant W (S = ln W) a value of 1.53959 was obtained [11], which is very close to the exact value obtained by Lieb WL = (4/3)3/2 ≈ 1.53960 [2]. As is well-known, according to Pauling’s approximation, WP = 1.5, i.e. S = ln(3/2) = 0.405 [15].

For a monolayer of ice and other three-connected ice-like systems, an analogue of Pauling’s formula is the expression \(\:\text{l}\text{n}(3/\sqrt{2})\) = 0.7520 [17, 18]. The method for calculating the entropy of a zigzag tube of hexagonal ice is distinguished by the selection of two transverse elements (Fig. 2a) and two local conditional transfer matrices A1 and A2 for each transverse element. Note that for a tube in a chair conformation, you can use only one transverse element with the same matrices A1 and A2 [12]. But for equal sizes of the resultant transfer matrices, the convergence to the limit value according to the second variant is somewhat slower.

Fig. 2
figure 2

(a) Zigzag-like chains of hexagonal ice monolayer. (b) Diagrams for calculation of the transfer matrices B1 and B2 for each direction of outer H-bond (dotted arrows)

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$$\:\varvec{A}1\left(\text{i}\right)=\left\{\begin{array}{c}\varvec{B}1\:\:\:\text{i}=0\\\:\varvec{B}2\:\:\:\text{i}=1\end{array}\right.\:\varvec{A}2\left(\text{i}\right)=\left\{\begin{array}{c}\varvec{B}2\:\:\:\text{i}=0\\\:\varvec{B}1\:\:\:\text{i}=1\end{array}\right.$$
(5)

The scheme for calculating matrices B corresponding to a specific direction of the cross-linked H-bond is shown in Fig. 2b. For three-coordinated systems, local restrictions on the direction of arrows at each site are only that three incoming and three outgoing arrows are considered prohibited. The resultant matrix for doubled elements M = M1∙M2. For strips of width k = 8, 9 and 10 under toroidal boundary conditions, the asymptotic entropy values ​​are S8 = 0.752749775, S9 = 0.752749752, S10 = 0.752749748 (see introduction). Comparison of last value, obtained using MathCad without any extrapolation, with the exact value for the entropy of a hexagonal monolayer [13]

$$\:\frac{3}{2\:}\frac{1}{24\:{\pi\:}^{2}}{\int\:}_{0}^{2\pi\:}{\int\:}_{0}^{2\pi\:}\text{ln}\left[21-4\:\left(\text{cos}\left(x\right)+\text{cos}\left(y\right)+\text{cos}\left(x+y\right)\right)\right]dydx=0.752749747$$
(6)

indicates a remarkably high rate of convergence. It will be shown below that this is equally true for other three-coordinated systems.

In Fig. 3 different type of four- and three-coordinated lattices are shown. It is easy to see that the lattices shown in the center are decorated in relation to those shown on the left. The last lattice at the bottom is also obtained by decorating the upper so-called kagome lattice.

Fig. 3
figure 3

Two dimensional lattices. The following is the result of the analysis of three decorated lattices. Their names are determined by the type of polygons at each node: 3-12-12, 4-8-8 and 4-6-12. All of them, like the hexagonal monolayer, are three-coordinated

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3 Results and discussion

3.1 Lattice 3-12-12

The methods for calculating the entropy of this lattice and the entropy of the original hexagonal lattice are the same. It is enough just to recalculate the matrices B1 and B2. Fragments of the decorated lattice are shown in Fig. 4. The number of allowed configurations in triangles for different directions of H-bonds along the chain and different directions of the external bond (dotted line) is easy to determine. These values ​​are shown in Fig. 4b. They are the elements of matrices B1 and B2.

The results of calculating the entropy of this lattice using formula (4) are presented in Table 1 with fifteen significant digits. It can be seen that the limiting value is reached even faster than for the hexagonal ice monolayer. Certainly, there is no need to use any extrapolation methods here. A large number of significant digits will be needed to estimate the rate of convergence.

Fig. 4
figure 4

(a) Fragments of lattice 3-12-12. (b) Diagrams for calculation of the transfer matrices B1 and B2 for each direction of outer H-bond (dotted arrows)

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Table 1 Asymptotic values of entropy for infinite strips of width up to k = 10 with cylindrical boundary conditions
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3.2 Lattice 4-8-8

The entropy calculation for this decorated lattice is also slightly different from calculation of entropy for the original square lattice. Schemes for calculating the elements of matrices B for this case are shown in Fig. 5. The calculation is also not very difficult. Asymptotic entropy values for strips of different widths under periodic conditions in the perpendicular direction (nanotubes), calculated using formula (4), are presented in Table 1 with twelve significant digits. The speed of convergence to the limit value is also very high, although it is somewhat inferior to the 3-12-12 lattice.

Fig. 5
figure 5

Diagrams for calculation of the transfer matrices B0, B1 and B2 in lattice 4-8-8 (decorated square lattice)

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3.3 Speed of convergence

It is natural to assume that for three-coordinated lattices the speed of convergence of the entropy to the limit value is exponential if we consider a sequence of infinite stripes of increasing width. In this case, S = S0 + A exp(–α∙k), where S0 is the limit value, and k is the strip width, more precisely, the number of longitudinal chains of the tube. Taking logarithms, we obtain ln(S – S0) = lnA – α∙k. In the general case, S0 is unknown. But this is not important here. Instead of the exact value, you can substitute any of the last calculated values. For most of the obtained values, the difference S – S0 does not change much. Figure 6a shows the dependences of y ≡ ln(S – S0) on the width k for lattices 3-12-12 and 4-8-8, as well as for a hexagonal ice monolayer. The linear dependencies confirm the assumption of exponential convergence. For the considered lattices, the coefficient α is approximately equal to 4.07, 2.55 and 1.78, respectively. This speed is determined not only by the strip width, but also by the number of sites ns (see formula 4) in the element of the transverse chain, which determines the resultant transition matrix. For the lattices considered, ns is equal to 6, 4 and 2. Often the convergence rate is determined not for the specific entropy S itself, but for W, where S = ln(W). It is not difficult to verify that the convergence of S and W is actually the same, since W = exp(S0 + A∙exp(–α∙k)) ≈ W0 + A1∙exp(–α∙k).

Fig. 6
figure 6

Approximation of calculated data by exponential (a) and power (b) equations: S = S0 + A∙exp(–α∙k) and S = S0 + A/kα. Here, α is equal to the coefficient at x in the linear trend equation. The equations of the averaged linear trend are given on the right. (Color figure online)

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As noted, the square ice entropy converges to its limit value with a rate of 1/k2 by widening the strips. For comparison, we can calculate the power-law dependences of the rate convergence of entropy to the limit value for different lattices. Figure 6b shows the dependence of ln(S – S0) on ln(k). In this case, the slope coefficient α determines the power-low function 1/kα. The slope coefficients in Fig. 6b (20.23, 13.88 and 10.38) correspond to the average values ​​of α for 4 ≤ k ≤ 9. Such large values of α most clearly show the qualitative difference between four- and three-coordinated systems in the rate of convergence to the limit value of entropy.

The reason for the high rate of entropy convergence for three-coordinate systems is due to the weaker structural correlation and new Bernal-Fowler ice rules. In this case, unlike four-coordinated systems, the number of incoming and outgoing hydrogen bonds in each site is not strictly fixed, but has some, very important non-uniqueness.

3.4 Cluster methods

Pauling’s formula for entropy of four-coordinated ice S = ln(3/2) and its analogue for three-coordinated systems S = \(\:\text{l}\text{n}(3/\sqrt{2})\) = 0.75203870 were obtained by analyzing correlations at one lattice site. For lattices 3-12-12 and 4-8-8, it is not difficult to take into account the correlations for a separate decorated ring: a triangle or a square. Here, it is important to take into account that the number of allowed configurations of an n-gon, given the direction of external hydrogen bonds, is determined by the formula Xn = 3n + 1. This formula was also obtained by the transfer matrix method [9, 19]. In a cluster approximation, for the two considered lattices with the number of nodes N, the entropy per one lattice site (molecule) is as follows:

$$\:S=N\:\text{l}\text{n}\left[{2}^{3/2}{\left(\frac{{3}^{n}+1}{{2}^{2n}}\right)}^{1/n}\right]$$
(7)

It is taken into account here that for three-coordinated lattices the number of hydrogen bonds is one and a half times greater than the number of site (molecules) N of the entire lattice, the number of internal and external bonds of a separate n-gon is 2n, and the total number of clusters is N/n. It is easy to see that the difference in the values ​​of S for various lattices is entirely due to the presence of a unit in the expression for Xn. For triangular cycles the effect of this addition is maximum. For lattices 3-12-12 and 4-8-8, formula (7) gives 0.76416125 and 0.75510622, respectively. The italic digits here are those that coincide with the digits obtained for the most accurate values ​​in Table 1. Obviously, this approximation is more accurate compared to the value \(\:\text{l}\text{n}(3/\sqrt{2})\) = 0.75203870.

Using the cluster approximation, it is not difficult to obtain even more accurate entropy values. However, clusters that combine only two triangles, as well as two or three squares, are not suitable for this purpose. Such clusters do not increase accuracy because they do not take into account additional cyclic correlations. Similarly, using an arbitrary tree does not increase the accuracy of the Pauling estimates based on the analysis of an individual site. As a cluster of the 3-12-12 lattice, it is natural to choose a decorated hexagonal ring (Fig. 2) combining six triangles. The number of configurations of this cluster is Tr(M6) = 7,529,600, where the matrix M is equal to the sum of the matrices in Fig. 4 for fragments that differ only in the direction of external bond: \(\:\varvec{M}=\left(\begin{array}{cc}8&\:6\\\:6&\:8\end{array}\right)\). Therefore, S = ln[23/2(752,9600/230)1/18] = 0.76416172, which is more closer to the limit value.

The entropy of lattice 4-8-8 is calculated similarly in the approximation of a cluster in the form of a decorated square ring (Fig. 3). The scheme for calculating the elements of matrix B is shown in Fig. 7. Since changing the direction of all bonds (arrows) does not violate the Bernal-Fowler ice rules, the diagonal matrix elements are equal to each other, as are the other two elements. Considering that the total number of configurations of a square ring is known and equal to 34 + 1 = 82, it is sufficient to determine only one matrix element. In the second diagram (B12), (B12), the direction of the lower bond is selected, and then the direction of another bond is clearly determined (gray arrows). Near the top left site, 6 configurations of adjacent bonds are possible. Of which, for three configurations, the direction of the right external bond is determined uniquely, and for the other three configurations, two directions of the external right bond are possible. Therefore, with a fixed direction of the gray arrows, the number of configurations is 9 and the same number of configurations is possible with the opposite direction of the lower gray arrow. Therefore, this matrix element is equal to 18. The sum of the first two matrix elements (B11 and B12)  is 82/2 = 41, i.e. the diagonal elements of the transfer matrix are 23. The number of configurations of the decorated square ring is Tr(M4) = 2,826,386, so S = ln[23/2(2,826,386/228)1/16] = 0.7551200437, which also brings us closer to the limit value. In passing, we note that the sum of all numbers in Fig. 5 is also equal to 82.

Fig. 7
figure 7

Diagrams for calculation of the transfer matrix for square decorated rings

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3.5 Lattice 4-6-12

This lattice is a decorated version of kagome lattice. This can be easily verified by replacing each four-coordinated node with a square. For the kagome lattice itself, you can also apply the method of local conditional transfer matrices if you divide it into stripes, including straight lines and triangles on one side of the straight line (Fig. 3, dotted line). In this case, the conditional transfer matrices also have the size of 2 × 2, but the number of matrices B will be equal to 16, according to the number of options for the direction of external bonds 24. The exact value of entropy for the kagome lattice is well known [20], so we will limit ourselves to calculating the entropy of the decorated 4-6-12 lattice using the cluster approximation.

As for all three-coordinated lattices, the simplest is the Pauling-type estimate \(\:\text{l}\text{n}(3/\sqrt{2})\) = 0.75203870. The next approximation is based on a cluster in the form of a separate square and therefore exactly coincides with the same approximation for the 4-8-8 lattice: S = 0.75510622. The 4-6-12 lattice has two rings decorated with squares: a triangular ring and a hexagonal one. Approximate entropy values for these clusters are calculated through the matrix M, the elements of which are shown in Fig. 7. The numbers of configurations in these rings are equal to Tr(M3) = 69,046 and Tr(M6) = 4,750,119,866. It is easy to obtain two approximate values of the residual entropy of this lattice: S3 = ln[23/2(69,046/221)1/12] = 0.7552572239 and S6 = ln[23/2(4,750,119,866/242)1/24] = 0.7551063586. The remarkable thing is that the first approximation brings us much closer to the limit value. Indeed, it differs more from the approximation based on a single square. The reason is that the triangular decorated ring takes into account additional cyclic correlations in hexagonal rings, while the second approximation only adds correlations in 12-gonal rings.

4 Conclusions

Calculating the entropy of square ice is rightfully considered extremely difficult. The methods developed to solve this and other similar problems are also applicable to the analysis of three-coordinated ice-like systems. In this article we use a much simpler computational method that allows us to obtain approximate entropy values ​​for infinite systems. For the hexagonal ice monolayer and for other three-coordinated systems, the convergence rate of the approximate method and the accuracy of the obtained estimates are amazingly high. For these systems, the convergence is of exponential type, in contrast to square ice, for which the convergence is much slower (1/k2). This is due to less rigid topological restrictions on the direction of bonds at each lattice site, which results in a significantly weaker the system’s total correlations. The high speed of convergence makes it possible to obtain almost exact entropy values ​​for three-coordinated systems.

In conclusion, we note that the method of local conditional transition matrices was developed to accurately calculate the number of defect-free configurations in ice nanotubes and other finite systems with periodic boundary conditions. Exact statistics of proton disorder is a primary task in the consistent analysis of the properties of ice nanostructures.