Defect layer modified TiO2 nanowire array photo-anode and preparation method thereof

Abstract

The invention discloses a defect layer modified TiO2 nanowire array photo-anode and a preparation method thereof, and belongs to the technical field of photoelectrochemistry. The Ti thin film is deposited on the surface of TiO2 NWAs growing on the substrate in a magnetron sputtering mode, then annealing treatment is conducted in the N2 atmosphere, Ti can capture O atoms in TiO2 crystal lattices in the annealing process, a TiO2 defect layer containing oxygen vacancies is formed on the surface, the lower Fermi energy level is helped to be obtained, the Fermi energy level tends to be balanced when the Ti thin film and a main body TiO2 nanowire form a homojunction, and therefore the TiO2 defect layer with the oxygen vacancies is formed. Therefore, a built-in electric field is generated, migration of photon-generated carriers is promoted, and the utilization rate of light is enhanced. The surface defect state concentration is regulated and controlled by further regulating and controlling conditions such as Ti film deposition time and annealing temperature, so that the prepared photo-anode has better PEC performance; the invention provides an effective means for regulating and controlling the photoelectrocatalytic performance of TiO2, and also provides a theoretical basis for related applications.

Description

Technical Field

[0001] The invention belongs to the technical field of photoelectrochemistry, and in particular relates to a defect layer modified TiO2 nanowire array photoanode and a preparation method thereof. Background Art

[0002] Photoelectrochemical water splitting system is a clean energy production technology that converts solar energy into hydrogen energy. It has the characteristics of abundant sources, environmental friendliness and low cost, and is regarded as one of the most promising ways to achieve "carbon neutrality". One of the key factors affecting the operating efficiency of the system is the selection of photoanode materials.

[0003] Titanium dioxide (TiO2) has been widely used in the fields of photocatalysts, chemical sensors and luminescent materials due to its strong photoelectrochemical oxidation (PEC) ability, non-toxicity and strong stability. For example, under the irradiation of sunlight, TiO2 crystals can absorb sunlight energy to generate solar power, photodecompose water to produce hydrogen, and photocatalytically degrade pollutants by using the photoelectric conversion effect. Based on the above characteristics, TiO2 has become a photoanode material that has been studied in depth.

[0004] However, due to the wide band gap of TiO2 (~3.2 eV), it can only utilize the shorter wavelength part of solar radiation, which limits its use of solar energy. In addition, the photogenerated electron-hole pairs in the TiO2 photoanode material will recombine quickly, which is not conducive to the occurrence of redox reactions and will lead to low photoelectric conversion efficiency.

[0005] Modifying the photoelectric properties of materials is the most common method to improve the photoelectric conversion efficiency of materials. Common methods for modifying oxide semiconductors mainly include nanostructuring, doping with foreign elements, introducing defects, constructing heterojunctions, and loading co-catalysts. Among them, defects are a common structural feature in materials. Introducing defects and rationally regulating defects can effectively improve the physicochemical properties and photoelectrocatalytic performance of TiO2. Vacancies are the most common point defects in metal oxide semiconductors. They are defects formed at a specific temperature in the lattice in order to minimize the Helmholtz free energy for establishing thermodynamic equilibrium. Unlike element doping, vacancies can effectively change the charge distribution and electronic energy levels near the defect site, affecting the electronic structure and surface properties of the material. In addition, vacancy defects can also change the optical absorption and emission properties of the crystal, affect the carrier concentration of the semiconductor, and then adjust the Fermi level of the semiconductor.

[0006] In semiconductor oxides, oxygen vacancies (Vo) have the lowest formation energy among various defects that act as donors and are the most common vacancy defects. They play a vital role in controlling and changing various physical and chemical properties, affecting every step of the photoelectrochemical water splitting reaction of metal oxide semiconductors, including light absorption, separation and transport of photogenerated carriers, and semiconductor-electrolyte interface reactions. It is generally believed that Vo promotes the PEC water splitting process by causing changes in the energy levels of TiO2 and promoting the separation of electron-hole pairs. In addition, Vo introduces localized states in the energy band, changes the optical absorption of the crystal, and can expand the action light range of nano-TiO2 to visible light, thereby improving the utilization rate of sunlight. Therefore, defect engineering is a feasible method to solve the low solar energy utilization rate and high recombination rate of photogenerated carriers of TiO2 photoanode.

[0007] The methods for introducing oxygen vacancies in TiO2 photoanode mainly include heat treatment, doping and chemical reduction. Heat treatment refers to heating TiO2 in a reducing atmosphere (such as hydrogen) to induce the formation of oxygen vacancies. This method does not introduce impurities, but it has high energy consumption and certain risks. The reducing agent NaBH4 can generate Vo on the surface of TiO2 at room temperature, but it often generates surface Vo, which has poor repeatability and stability. The chemical reduction method is to directly introduce oxygen vacancies into the TiO2 nanostructure through wet chemical method or solid-state oxidation-reduction method. This method is simple in process, but foreign chemical substances may introduce impurities. Doping refers to the addition of a certain element during the synthesis process of TiO2 or the post-synthesis treatment, and the introduction of oxygen vacancies by replacing titanium elements or destroying Ti-O bonds. Common doping methods for introducing oxygen vacancies include metal doping, non-metal doping and co-doping. For example, reducing metal powders (Li, Mg, Zn, Al) can introduce Vo by grinding and capturing O atoms on the surface of TiO2 powder. Grinding doping is a direct and easy operation method, but it is difficult to introduce Vo into TiO2 nanowires generated on fluorine-doped tin oxide (FTO) substrates by this method, because nanowires on FTO substrates usually have higher structural stability, and compared with powdered TiO2, their surface atoms may be more firmly bound to the substrate, so it is difficult to capture O atoms through a simple grinding process; and for nanowires on FTO substrates, due to their arrangement and space limitations, reducing metal powders may not be able to fully contact the surface of nanowires, thereby limiting the reaction; in addition, in nanowires on FTO substrates, due to structural limitations and long diffusion paths between reactants, the reaction rate may be significantly reduced. In summary, the grinding method cannot effectively complete the introduction of Vo into nanowires on FTO substrates. As a highly conductive and transparent photoelectrocatalytic substrate material, FTO has a strong ability to bind to the photoanode or photocathode, can effectively transfer electrons, reduce the recombination of electrons and holes, and improve the photoelectric efficiency, and is very valuable for application. Therefore, improving the preparation process to successfully introduce Vo into the TiO2 nanowire photoanode based on FTO substrate to improve the photoelectric properties of the corresponding photoanode material still has very good industrial application prospects. This requires more in-depth research on how to successfully modify the Vo defect layer on the corresponding material. Summary of the invention

[0008] In view of the above-mentioned problems, the present invention aims to provide a defect layer modified TiO2 nanowire array photoanode and discloses its specific preparation method. A TiO2 defect layer can be formed on the surface of TiO2 NWAs by depositing a metal Ti film on the surface of TiO2 NWAs on a FTO substrate and annealing in a nitrogen atmosphere. The oxygen vacancy defect layer can increase the light utilization rate of the photoanode and fully improve the PEC performance of the photoanode.

[0009] In order to achieve the above object, the technical solution adopted by the present invention is as follows: A method for preparing a defect layer modified TiO2 nanowire array photoanode comprises the following steps: 1) Add the titanium source into the acid solution and mix to obtain a precursor solution; 2) Pre-treat the substrate; 3) Place the substrate with the conductive surface facing downward in the reactor lining, add the precursor solution, seal the reactor, and heat the reaction. After the reaction is completed, wash the sample and dry it to grow TiO2NWAs on the substrate surface; 4) Ti thin film was deposited on the surface of TiO2NWAs by magnetron sputtering, and the obtained sample was named TiO2@Ti; 5) Annealing the TiO2@Ti material to obtain the final product.

[0010] Furthermore, the substrate material is any one of fluorine-doped tin oxide (FTO), indium tin oxide (ITO), aluminum zinc oxide (AZO), and gallium zinc oxide (GZO).

[0011] Furthermore, in step 3), the heating reaction temperature is 130-200° C. and the heating reaction time is 10-24 h.

[0012] Furthermore, in step 3), after drying the sample, annealing treatment is performed in a muffle furnace to improve the crystallinity of the nanowires on the substrate surface, and the annealing temperature is 450-550° C. and the annealing time is 0.5-3 h.

[0013] Furthermore, in step 4), DC magnetron sputtering is used. During the deposition process, the substrate rotation speed is 6-10 r / min, the deposition power is 220-240 W, and the deposition time is 1-6 min.

[0014] Furthermore, in step 5), annealing is performed in a nitrogen atmosphere, the annealing temperature is 250-450° C., and the annealing time is 0.5-3 h.

[0015] The defect layer modified TiO2 nanowire array photoanode prepared by the above method includes a substrate, a TiO2NWAs layer and a TiO2 defect layer on the surface arranged in sequence, and the defect layer contains oxygen vacancies. The photoanode material can be efficiently used for photoelectrocatalytic decomposition of water to produce hydrogen and oxygen, and the photoelectrocatalytic process can be realized in neutral solution, acidic solution and alkaline solution.

[0016] The beneficial effects of the present invention are: 1. This application successfully deposited a metal Ti film on the surface of TiO2NWAs grown on a FTO glass substrate by magnetron sputtering and annealed the sample. Annealing of the Ti film in N2 can capture O atoms in the TiO2 lattice and form a TiO2 defect layer on the surface. The presence of the oxygen vacancy defect layer significantly improves the photoelectrocatalytic performance of TiO2. 2. The photoanode material prepared after constructing an oxygen vacancy defect layer on the surface of TiO2NWAs on the FTO glass substrate has a lower Fermi level than the TiO2 photoanode without a defect layer. When forming a homojunction with the main TiO2 nanowire, the Fermi level tends to balance, thereby generating a built-in electric field. Under the action of the electric field, the energy band of the defect layer moves to a more negative potential. Under the excitation of light, the photogenerated electrons migrate from the conduction band of the defect layer to the conduction band of the TiO2 nanowire, while the holes migrate in the opposite direction. This process promotes the migration of photogenerated carriers and greatly increases the utilization rate of light. 3. The thickness of the Ti film on the surface of TiO2NWAs can be regulated by controlling the time of magnetron deposition, thereby improving the PEC performance of the defect layer-modified TiO2NWAs photoanode; 4. The TiO2@Ti-3-N350 photoanode material obtained by adjusting the deposition time and the annealing conditions after Ti film deposition exhibited good photoelectrocatalytic activity. The defect layer modification increased the hydrogen generation rate of TiO2 by 2.9 times, and the constructed defect layer still maintained good structural and photoelectric stability during the catalytic water decomposition process; 5. This application modifies the TiO2NWAs photoanode through an oxygen vacancy defect layer. This process provides an effective means for regulating the photoelectrocatalytic performance of TiO2, and can be achieved in neutral solution, acidic solution and alkaline solution, which provides the possibility for large-scale application in the future and also provides a theoretical basis for related applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 The scanning electron microscope (SEM) images of the photoanode, wherein the a-panel is the SEM plane image of the TiO2 photoanode with TiO2NWAs grown on the FTO substrate, and the b-panel is the SEM plane image of the TiO2@Ti-3 photoanode obtained after a Ti film is deposited on the top of the TiO2 nanowires; Figure 2 Panel a in the middle is the Raman spectra of TiO2, TiO2@Ti-3, TiO2@Ti-3-N350 and TiO2@Ti-3-A350; Panel b is the main peak half-width of the Raman spectra of TiO2, TiO2@Ti-3, TiO2@Ti-3-N350 and TiO2@Ti-3-A350; Figure 3The photocurrent density data of different photoanode materials under 1.23 V vs. RHE bias; among them, the photocurrent density data of photoanode materials TiO2, TiO2@Ti-1-N350, TiO2@Ti-3-N350, TiO2@Ti-6-N350, and TiO2@Ti-9-N350 under 1.23 V vs. RHE bias; the photocurrent density data of photoanode material TiO2@Ti-1-A350 under 1.23 V vs. RHE bias; the photocurrent density data of photoanode material TiO2@Ti-3, TiO2@Ti-3-N250, TiO2@Ti-3-N350, TiO2@Ti-3-N450, and TiO2@Ti-3-N550 under 1.23 V vs. RHE bias are shown in Figure 1. Figure 4 Panel a in the middle shows the total number of electrons and gas evolution over time obtained from the photoelectrocatalytic test of TiO2@Ti-3-N350; Panel b is a comparison of the hydrogen generation rates of the TiO2 photoanode and the defect layer modified TiO2@Ti-3-N350 photoanode; Panel c is the test result of the water decomposition stability of the TiO2@Ti-3-N350 photoanode under illumination conditions; Panel d is the test result of the photocurrent stability of the TiO2@Ti-3-N350 photoanode under illumination conditions. DETAILED DESCRIPTION

[0018] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0019] Unless otherwise specified, the reagents and raw materials used in the following examples are commercially available. Example

[0020] This embodiment discloses a defect layer modified TiO2 nanowire array photoanode. First, a Ti film of a certain thickness is deposited on the surface of TiO2 NWAs grown on a FTO substrate by magnetron sputtering and annealed. Ti atoms are used to capture O atoms in the TiO2 lattice to form a TiO2 defect layer on the surface of the nanowire. The specific preparation process is as follows: 1) Mix 4.5 mL of hydrochloric acid and 4.5 mL of deionized water, add 0.15 mL of tetrabutyl titanate, and stir until clear to obtain a precursor solution.

[0021] 2) Pretreatment of FTO glass: FTO glass was cut into small pieces of 1 cm×3 cm×1.6 mm, first ultrasonically treated in acetone for 30 min to remove most of the organic pollutants on the FTO surface, then ultrasonically treated in ethanol for 30 min to remove the remaining organic matter and the residual acetone from the previous cleaning step, and finally ultrasonically cleaned in deionized water for 30 min to remove the final residue, and dried at 70°C.

[0022] 3) The precursor solution prepared in step 1) was transferred into a polytetrafluoroethylene liner, and the conductive surface of the FTO glass was placed downward into the polytetrafluoroethylene liner, with the edge forming a 45° angle with the liner wall. The hydrothermal reactor was sealed, and the hydrothermal reactor was placed in an oven and heated at 150°C for 15 h. After the reaction was completed, the reactor was cooled naturally, and the liner was removed. The FTO glass was removed from the liner, washed with anhydrous ethanol and deionized water, dried in a 70°C oven, and annealed in a muffle furnace at 500°C for 1 h to improve the crystallinity. TiO2 nanowire arrays (TiO2NWAs) were successfully grown on the surface of the FTO substrate.

[0023] The SEM plan view of the TiO2 nanowire photoanode (denoted as TiO2) based on the FTO substrate obtained in this step is shown in Figure 1 As shown in the middle panel a, its microscopic morphology is a nanorod structure, and the TiO2 nanowires grow vertically and closely on the FTO substrate with a height of about 3.5 μm.

[0024] The Ti film was deposited on the surface of TiO2NWAs by DC magnetron sputtering. The deposition chamber was evacuated to 5.0×10 -3 Pa, high-purity Ar (99.99%) was introduced into the chamber at a flow rate of 45 sccm to make the chamber pressure reach 3.0×10 -1 Pa; before magnetron deposition, the target material was pre-sputtered with Ar plasma for 10 min. During the deposition process, the substrate rotation speed was 8 r / min, the deposition power was 220 W, and the deposition time was 1 min, 3 min, 6 min, and 9 min, respectively, to deposit Ti films of different thicknesses on the surface of TiO2NWAs. The samples obtained based on different deposition times were correspondingly recorded as TiO2@Ti-1, TiO2@Ti-3, TiO2@Ti-6, and TiO2@Ti-9.

[0025] Observe the SEM plan view of TiO2@Ti-3 photoanode ( Figure 1(B) It is found that after the Ti film is deposited on the top of the TiO2 nanowire, the Ti film wraps the tip of the TiO2, making the top of the nanowire smoother. And by observing the microscopic morphology of the TiO2@Ti-6 and TiO2@Ti-9 photoanodes, it is found that with the increase of deposition time, the top of the nanowire is completely wrapped, and the thickness of the shell layer increases significantly.

[0026] 6) The four different samples obtained in step 5) were annealed at 350 °C for 1 h in a nitrogen atmosphere, and the obtained photoanode materials were respectively recorded as TiO2@Ti-1-N350, TiO2@Ti-3-N350, TiO2@Ti-6-N350, and TiO2@Ti-9-N350.

[0027] 7) PEC performance was measured using a typical three-electrode setup on a CHI660E electrochemical workstation (Shanghai Chenhua). Photoanode material, Ag / AgCl (saturated potassium chloride) and Pt wire were used as working electrode, reference electrode and counter electrode respectively. The light source was a 300W Xe lamp (CHF-XM300) to simulate full-spectrum illumination. The photoelectrochemical performance of the sample was analyzed. The test light intensity was stable at 100 mW·cm -2 , the test area is 1.5 cm 2 The PEC measurement was performed using 0.5 M Na2SO4 solution (pH=7) as the electrolyte.

[0028] The obtained photocurrent density data can be found in Figure 3 In the middle panel a, we can see that under full spectrum illumination (100 mW cm -2 ) and 0.5 M Na2SO4 (pH = 7) electrolyte, the photocurrent density of the TiO2 photoanode was 1.09 mA·cm −2 The photocurrent density of TiO2@Ti-3-N350 photoanode reaches 3.26 mA·cm −2 . This indicates that the PEC water splitting performance and stability of TiO2NWAs photoanode under neutral reaction conditions are significantly improved by modifying the TiO2NWAs photoanode with oxygen vacancy defects. It has also been verified that it has good PEC water splitting performance and stability in acidic and alkaline solutions.

[0029] Photoelectrocatalytic tests using TiO2 and TiO2@Ti-3-N350 photoanodes showed that TiO2@Ti-3-N350 photoanode exhibited good photoelectrocatalytic activity, with oxygen and hydrogen generation rates of 18.3 μmol·cm -2 ·h -1 and 37.6 μmol·cm -2 ·h -1 , which is consistent with the stoichiometric ratio of 1:2 for water splitting ( Figure 4 The initial hydrogen generation rate of the TiO2 photoanode is 9.6 μmol·cm -2 ·h -1 , defect layer modification increased the hydrogen generation rate of TiO2 by 2.9 times ( Figure 4 The stability of water decomposition and photocurrent of TiO2@Ti-3-N350 photoanode under illumination conditions was tested ( Figure 4 (Figures c and d in the middle), it can be seen that the hydrogen generation and photocurrent density of the photoelectrode remain stable over time, indicating that the constructed defect layer still maintains good structural and photoelectric stability during the catalytic water decomposition process. Example

[0030] The difference between this embodiment and embodiment 1 is that after TiO2@Ti-3 is prepared according to the same method as in embodiment 1, it is annealed at 250°C, 350°C, 450°C and 550°C for 1 h in a nitrogen atmosphere, and the obtained photoanode materials are respectively recorded as TiO2@Ti-3-N250, TiO2@Ti-3-N350, TiO2@Ti-3-N450 and TiO2@Ti-3-N550. The PEC performance of the material is tested according to the same test method as step 7) in embodiment 1, and the photocurrent density data is shown in Figure 3 Panel c shows the full spectrum illumination (100 mW cm -2 ) and 0.5 M Na2SO4 (pH=7) electrolyte conditions, the photocurrent densities of TiO2@Ti-3-N250, TiO2@Ti-3-N350, TiO2@Ti-3-N450, and TiO2@Ti-3-N550 at 1.23 V vs. RHE bias are 2.46, 3.26, 2.93, and 1.46 mA·cm, respectively. −2 .

[0031] Comparative Example 1 The difference between this comparative example and Example 1 is that after TiO2@Ti-3 is prepared in the same method as Example 1, the sample is annealed in air instead of nitrogen: the sample is annealed at 350°C in an air atmosphere for 1 h, and the obtained photoanode material is recorded as TiO2@Ti-3-A350.

[0032] The PEC determination method is the same as in Example 1. The photocurrent density of the photoanode material TiO2@Ti-3-A350 obtained in this comparative example is as follows: Figure 3 As shown in panel b, under full spectrum illumination (100 mW cm -2) and 0.5 M Na2SO4 (pH = 7) electrolyte conditions, the photocurrent density of the TiO2@Ti-3-A350 photoanode was 1.77 mA·cm -2 .

[0033] Figure 2 The small figure in the middle a shows the Raman spectra of TiO2, TiO2@Ti-3, TiO2@Ti-3-N350 and TiO2@Ti-3-A350. It can be seen from the figure that compared with TiO2, TiO2@Ti-3 E 1g The red shift of the characteristic peak of the vibration mode indicates that Ti deposition destroys the symmetry of the O-Ti-O bond. After further annealing, the red shift of the peak disappears, indicating that the Ti layer is oxidized. 1g The half-width of the characteristic peak often reflects the disorder of the sample surface.

[0034] like Figure 2 As shown in the middle panel b, Ti deposition makes the surface of TiO2 disordered. After annealing in N2, the half-peak width of the characteristic peak decreases, but it is still greater than the initial TiO2, indicating that the shell surface formed is a defect layer containing Vo. The half-peak width of the sample annealed in air is consistent with that of TiO2, indicating that its shell has a good lattice structure. This shows that in the sample annealed in air, Ti will be oxidized by oxygen in the air, which has the effect of passivating the surface.

[0035] Comparative Example 2 The difference between this comparative example and Example 1 is that in this comparative example, TiO2@Ti-3 is prepared by referring to the method in Example 1 without annealing, and the PEC performance of the TiO2@Ti-3 photoanode is directly tested by the same test method as Example 1.

[0036] The PEC determination method is the same as in Example 1. The photocurrent density of the photoanode material TiO2@Ti-3 obtained in this comparative example is as follows: Figure 3 As shown in panel c, under full spectrum illumination (100 mW cm -2 ) and 0.5 M Na2SO4 (pH = 7) electrolyte conditions, the photocurrent density of the TiO2@Ti-3 photoanode was 1.31 mA·cm -2 .

[0037] The above shows and describes the basic principles, main features and advantages of the present invention. However, the above is only a specific embodiment of the present invention, and the technical features of the present invention are not limited thereto. Any other implementation methods derived by any technician in the field without departing from the technical solution of the present invention should be included in the scope of the present invention.

Claims

1. A method for preparing a defect layer-modified TiO2 nanowire array photoanode, characterized in that: The steps include: 1) Add the titanium source into the acid solution and mix to obtain a precursor solution; 2) Pre-treat the substrate; 3) Place the substrate with the conductive surface facing downward in the reactor lining, add the precursor solution, seal the reactor, and heat the reaction. After the reaction is completed, wash the sample and dry it to grow TiO2NWAs on the substrate surface; 4) Ti thin film was deposited on the surface of TiO2 NWAs by magnetron sputtering, and the obtained sample was named TiO2@Ti; 5) Annealing the TiO2@Ti material to obtain the final product.

2. The method for preparing the defect layer modified TiO2 nanowire array photoanode according to claim 1, characterized in that: The substrate material is any one of fluorine-doped tin oxide, indium tin oxide, aluminum zinc oxide, and gallium zinc oxide.

3. The method for preparing the defect layer modified TiO2 nanowire array photoanode according to claim 1, characterized in that: In step 3), the heating reaction temperature is 130-200° C. and the heating reaction time is 10-24 h.

4. The method for preparing the defect layer modified TiO2 nanowire array photoanode according to claim 1, characterized in that: In step 3), after drying the sample, annealing treatment is performed in a muffle furnace to improve the crystallinity of the nanowires on the substrate surface. The annealing temperature is 450-550° C. and the annealing time is 0.5-3 h.

5. The method for preparing the defect layer modified TiO2 nanowire array photoanode according to claim 1, characterized in that: In step 4), DC magnetron sputtering is used. During the deposition process, the substrate rotation speed is 6-10 r / min, the deposition power is 220-240 W, and the deposition time is 1-6 min.

6. The method for preparing the defect layer modified TiO2 nanowire array photoanode according to claim 1, characterized in that: In step 5), annealing is performed in a nitrogen atmosphere at a temperature of 250-450° C. and an annealing time of 0.5-3 h.

7. A defect layer modified TiO2 nanowire array photoanode, characterized in that: The TiO2 NWAs layer is prepared based on the preparation method described in any one of claims 1 to 6, and comprises a substrate, a TiO2 NWAs layer and a TiO2 defect layer on the surface which are arranged in sequence, and the defect layer contains oxygen vacancies.

8. Use of the defect layer modified TiO2 nanowire array photoanode as claimed in claim 7 in the photoelectrocatalytic decomposition of water to produce hydrogen and oxygen.

9. The use according to claim 8, characterized in that The photoelectrocatalytic process is applied in neutral, acidic or alkaline solutions.

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